Subgroups — C₆₇×#60309 #30739560

Order 17152.

2142 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2990	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2989	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2974	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2973	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2970	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2965	C₂	1	2	yes	maximal, complemented
9	D₄×M₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₂×#435×C₆₇	C₂	1	4	yes	maximal, complemented
11	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	complemented
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
15	#520×C₆₇	C₄	1	4	yes	—
16	#520×C₆₇	C₄	1	2	yes	—
17	#518×C₆₇	C₄	1	4	yes	—
18	#518×C₆₇	C₄	1	4	yes	—
19	#518×C₆₇	C₄	1	1	yes	characteristic
20	C₄×#41×C₆₇	C₂²	1	2	yes	—
21	C₄×#41×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
25	#446×C₆₇	C₂²	1	2	yes	complemented
26	#446×C₆₇	C₂²	1	2	yes	complemented
27	#446×C₆₇	C₂²	1	4	yes	complemented
28	#436×C₆₇	C₂²	1	2	yes	complemented
29	#436×C₆₇	C₂²	1	4	yes	complemented
30	#436×C₆₇	C₂²	1	2	yes	complemented
31	#435×C₆₇	C₂²	1	4	yes	complemented
32	#435×C₆₇	C₂²	1	8	yes	complemented
33	#435×C₆₇	C₂²	1	8	yes	complemented
34	#435×C₆₇	C₂²	1	8	yes	complemented
35	#435×C₆₇	C₂²	1	8	yes	complemented
36	#435×C₆₇	C₂²	1	4	yes	complemented
37	#435×C₆₇	C₂²	1	8	yes	complemented
38	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
39	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
40	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
41	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
42	#432×C₆₇	C₂²	1	8	yes	complemented
43	#432×C₆₇	C₂²	1	4	yes	complemented
44	#431×C₆₇	C₂²	1	2	yes	complemented
45	#431×C₆₇	C₂²	1	4	yes	complemented
46	#423×C₆₇	C₂²	1	4	yes	complemented
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
49	#407×C₆₇	C₂²	1	8	yes	complemented
50	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
51	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
52	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
53	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
54	C₂×#97×C₆₇	C₂²	1	4	yes	complemented
55	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
56	H(5,𝔽₂)×C₆₇	—	2	8	no	—
57	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
60	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
61	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
64	C₂×M₁₆×C₆₇	—	4	8	no	—
65	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
68	C₂²×C₈×C₆₇	—	4	4	no	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
76	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
79	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
82	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
83	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
84	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
85	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
86	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
87	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
88	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
89	C₄×D₄×C₆₇	C₂³	1	4	yes	—
90	C₄×D₄×C₆₇	C₂³	1	4	yes	—
91	C₄×D₄×C₆₇	C₂³	1	8	yes	—
92	C₄×D₄×C₆₇	C₂³	1	2	yes	—
93	#117×C₆₇	C₂×C₄	1	4	yes	—
94	#117×C₆₇	C₂×C₄	1	4	yes	—
95	#117×C₆₇	C₂×C₄	1	8	yes	—
96	#117×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
107	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
108	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
110	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
111	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
112	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
114	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
115	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
116	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
117	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
118	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
119	#97×C₆₇	C₂³	1	4	yes	—
120	#97×C₆₇	C₂³	1	8	yes	—
121	#97×C₆₇	C₂³	1	4	yes	—
122	C₄×C₈×C₆₇	C₂³	1	2	yes	—
123	C₄×C₈×C₆₇	C₂³	1	2	yes	—
124	C₄×C₈×C₆₇	C₂³	1	4	yes	—
125	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
126	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
127	#41×C₆₇	—	2	16	no	—
128	#41×C₆₇	—	2	8	no	—
129	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₆₇	—	2	16	no	—
131	C₂×D₄×C₆₇	—	2	16	no	—
132	C₂×D₄×C₆₇	M₁₆	1	8	yes	complemented
133	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₆₇	—	4	2	no	—
138	C₂²×C₄×C₆₇	—	4	4	no	—
139	C₂²×C₄×C₆₇	—	4	2	no	—
140	C₂²×C₄×C₆₇	—	4	4	no	—
141	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
147	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
154	M₁₆×C₆₇	—	2	8	no	—
155	M₁₆×C₆₇	—	4	8	no	—
156	M₁₆×C₆₇	—	4	4	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	8	no	—
159	C₂×C₈×C₆₇	—	4	4	no	—
160	C₂×C₈×C₆₇	—	4	2	no	—
161	C₂×C₈×C₆₇	—	4	4	no	—
162	C₂×C₈×C₆₇	—	4	1	no	—
163	C₂×C₈×C₆₇	—	4	4	no	—
164	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
166	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
167	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
168	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
169	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
172	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
173	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
174	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
175	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
176	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
177	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
178	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
179	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
180	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
182	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
183	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
184	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
185	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
186	C₄²×C₆₇	C₂⁴	1	2	yes	—
187	C₄²×C₆₇	C₂⁴	1	4	yes	—
188	C₂³×C₆₇	—	2	8	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	—	4	2	no	—
191	C₂³×C₆₇	—	4	4	no	—
192	C₂³×C₆₇	—	4	4	no	—
193	C₂³×C₆₇	C₂×M₁₆	1	8	yes	complemented
194	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
195	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
196	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
197	Q₈×C₆₇	—	2	4	no	—
198	D₄×C₆₇	—	2	16	no	—
199	D₄×C₆₇	—	2	8	no	—
200	D₄×C₆₇	—	2	4	no	—
201	D₄×C₆₇	C₂×M₁₆	1	16	yes	complemented
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	8	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	4	no	—
216	C₂×C₄×C₆₇	—	4	4	no	—
217	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	complemented
218	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
220	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
221	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
224	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
226	C₈×C₆₇	—	2	1	no	—
227	C₈×C₆₇	—	2	4	no	—
228	C₈×C₆₇	—	2	4	no	—
229	C₈×C₆₇	—	2	4	no	—
230	C₈×C₆₇	—	2	2	no	—
231	C₈×C₆₇	H(5,𝔽₂)	1	2	yes	complemented
232	C₂²×C₆₇	—	2	1	no	—
233	C₂²×C₆₇	—	2	4	no	—
234	C₂²×C₆₇	—	2	1	no	—
235	C₂²×C₆₇	—	2	4	no	—
236	C₂²×C₆₇	—	2	4	no	—
237	C₂²×C₆₇	—	2	4	no	—
238	C₂²×C₆₇	—	2	4	no	—
239	C₂²×C₆₇	—	4	8	no	—
240	C₂²×C₆₇	—	8	8	no	—
241	C₂²×C₆₇	C₂²×M₁₆	1	8	yes	—
242	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
243	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	4	no	—
246	C₄×C₆₇	—	2	4	no	—
247	C₄×C₆₇	—	2	4	no	—
248	C₄×C₆₇	—	2	4	no	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	C₂²×M₁₆	1	4	yes	—
251	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
252	C₄×C₆₇	#585	1	1	yes	characteristic
253	#60309	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂×C₆₇	—	2	2	no	—
255	C₂×C₆₇	—	2	8	no	—
256	C₂×C₆₇	—	4	4	no	—
257	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
259	C₂×C₆₇	#3573	1	1	yes	characteristic
260	C₄×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic
261	#2990	C₂×C₆₇	1	4	yes	complemented
262	#2989	C₂×C₆₇	1	2	yes	complemented
263	#2974	C₂×C₆₇	1	4	yes	complemented
264	#2973	C₂×C₆₇	1	4	yes	complemented
265	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#2970	C₂×C₆₇	1	4	yes	complemented
267	#2965	C₂×C₆₇	1	2	yes	complemented
268	D₄×M₁₆	C₂×C₆₇	1	4	yes	complemented
269	C₂×#435	C₂×C₆₇	1	4	yes	complemented
270	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₆₇	#60309	1	1	yes	characteristic, direct product factor, Sylow
272	C₂×H(5,𝔽₂)	C₄×C₆₇	1	1	yes	characteristic
273	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	complemented
274	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
275	#520	C₄×C₆₇	1	4	yes	—
276	#520	C₄×C₆₇	1	2	yes	—
277	#518	C₄×C₆₇	1	4	yes	—
278	#518	C₄×C₆₇	1	4	yes	—
279	#518	C₄×C₆₇	1	1	yes	characteristic
280	C₄×#41	C₂²×C₆₇	1	2	yes	—
281	C₄×#41	C₂²×C₆₇	1	4	yes	—
282	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
284	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
285	#446	C₂²×C₆₇	1	2	yes	complemented
286	#446	C₂²×C₆₇	1	2	yes	complemented
287	#446	C₂²×C₆₇	1	4	yes	complemented
288	#436	C₂²×C₆₇	1	2	yes	complemented
289	#436	C₂²×C₆₇	1	4	yes	complemented
290	#436	C₂²×C₆₇	1	2	yes	complemented
291	#435	C₂²×C₆₇	1	4	yes	complemented
292	#435	C₂²×C₆₇	1	8	yes	complemented
293	#435	C₂²×C₆₇	1	8	yes	complemented
294	#435	C₂²×C₆₇	1	8	yes	complemented
295	#435	C₂²×C₆₇	1	8	yes	complemented
296	#435	C₂²×C₆₇	1	4	yes	complemented
297	#435	C₂²×C₆₇	1	8	yes	complemented
298	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
299	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
300	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
301	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
302	#432	C₂²×C₆₇	1	8	yes	complemented
303	#432	C₂²×C₆₇	1	4	yes	complemented
304	#431	C₂²×C₆₇	1	2	yes	complemented
305	#431	C₂²×C₆₇	1	4	yes	complemented
306	#423	C₂²×C₆₇	1	4	yes	complemented
307	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
308	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
309	#407	C₂²×C₆₇	1	8	yes	complemented
310	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
311	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
312	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
313	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
314	C₂×#97	C₂²×C₆₇	1	4	yes	complemented
315	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
316	H(5,𝔽₂)	—	2	8	no	—
317	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
318	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
319	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
320	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
321	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
324	C₂×M₁₆	—	4	8	no	—
325	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
326	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
327	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
328	C₂²×C₈	—	4	4	no	—
329	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
331	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
332	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
333	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
334	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
336	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
337	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
338	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
339	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
340	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
341	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
342	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
343	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
344	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
345	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
346	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
347	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
348	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
349	C₄×D₄	C₂³×C₆₇	1	4	yes	—
350	C₄×D₄	C₂³×C₆₇	1	4	yes	—
351	C₄×D₄	C₂³×C₆₇	1	8	yes	—
352	C₄×D₄	C₂³×C₆₇	1	2	yes	—
353	#117	C₂×C₄×C₆₇	1	4	yes	—
354	#117	C₂×C₄×C₆₇	1	4	yes	—
355	#117	C₂×C₄×C₆₇	1	8	yes	—
356	#117	C₂×C₄×C₆₇	1	2	yes	—
357	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
358	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
359	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
362	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
363	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
364	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
366	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
367	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
368	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
369	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
370	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
371	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
372	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
373	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
374	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
375	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
376	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
377	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
378	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
379	#97	C₂³×C₆₇	1	4	yes	—
380	#97	C₂³×C₆₇	1	8	yes	—
381	#97	C₂³×C₆₇	1	4	yes	—
382	C₄×C₈	C₂³×C₆₇	1	2	yes	—
383	C₄×C₈	C₂³×C₆₇	1	2	yes	—
384	C₄×C₈	C₂³×C₆₇	1	4	yes	—
385	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
386	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
387	#41	—	2	16	no	—
388	#41	—	2	8	no	—
389	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
390	C₂×D₄	—	2	16	no	—
391	C₂×D₄	—	2	16	no	—
392	C₂×D₄	M₁₆×C₆₇	1	8	yes	complemented
393	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
394	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
395	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
396	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
402	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
406	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
407	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
408	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
410	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
414	M₁₆	—	2	8	no	—
415	M₁₆	—	4	8	no	—
416	M₁₆	—	4	4	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	8	no	—
419	C₂×C₈	—	4	4	no	—
420	C₂×C₈	—	4	2	no	—
421	C₂×C₈	—	4	4	no	—
422	C₂×C₈	—	4	4	no	—
423	C₂×C₈	—	4	1	no	—
424	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
425	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
426	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
427	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
428	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
429	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
430	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
431	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
432	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
434	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
435	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
436	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
437	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
438	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
439	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
440	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
441	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
442	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
443	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
444	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
445	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
446	C₄²	C₂⁴×C₆₇	1	2	yes	—
447	C₄²	C₂⁴×C₆₇	1	4	yes	—
448	C₂³	—	2	8	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	4	4	no	—
453	C₂³	C₂×M₁₆×C₆₇	1	8	yes	complemented
454	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
455	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
456	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
457	Q₈	—	2	4	no	—
458	D₄	—	2	16	no	—
459	D₄	—	2	8	no	—
460	D₄	—	2	4	no	—
461	D₄	C₂×M₁₆×C₆₇	1	16	yes	complemented
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	4	8	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	complemented
478	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
480	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
481	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
482	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
483	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
484	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
485	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
486	C₈	—	2	1	no	—
487	C₈	—	2	4	no	—
488	C₈	—	2	4	no	—
489	C₈	—	2	4	no	—
490	C₈	—	2	2	no	—
491	C₈	H(5,𝔽₂)×C₆₇	1	2	yes	complemented
492	C₂²	—	2	1	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	4	8	no	—
500	C₂²	—	8	8	no	—
501	C₂²	C₂²×M₁₆×C₆₇	1	8	yes	—
502	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
503	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
504	C₄	—	2	4	no	—
505	C₄	—	2	4	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	1	no	—
508	C₄	—	2	4	no	—
509	C₄	—	2	4	no	—
510	C₄	C₂²×M₁₆×C₆₇	1	4	yes	—
511	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
512	C₄	#585×C₆₇	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	2	8	no	—
515	C₂	—	4	4	no	—
516	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
517	C₂	C₂³×M₁₆×C₆₇	1	1	yes	characteristic
518	C₂	C₆₇×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₂×H(5,𝔽₂)×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2990	C₂	semidirect
3. C₆₇×#2989	C₂	semidirect
4. C₆₇×#2974	C₂	semidirect
5. C₆₇×#2973	C₂	semidirect
6. C₆₇×#2972	C₂	semidirect
7. C₆₇×#2970	C₂	semidirect
8. C₆₇×#2965	C₂	semidirect
9. D₄×M₁₆×C₆₇	C₂	semidirect
10. C₂×#435×C₆₇	C₂	semidirect
11. C₂×C₈×D₄×C₆₇	C₂	semidirect
13. C₂²×M₁₆×C₆₇	C₂²	semidirect
14. C₂³×C₈×C₆₇	C₂²	semidirect
25. #446×C₆₇	C₂²	semidirect
26. #446×C₆₇	C₂²	semidirect
27. #446×C₆₇	C₂²	semidirect
28. #436×C₆₇	C₂²	semidirect
29. #436×C₆₇	C₂²	semidirect
30. #436×C₆₇	C₂²	semidirect
31. #435×C₆₇	C₂²	semidirect
32. #435×C₆₇	C₂²	semidirect
33. #435×C₆₇	C₂²	semidirect
34. #435×C₆₇	C₂²	semidirect
35. #435×C₆₇	C₂²	semidirect
36. #435×C₆₇	C₂²	semidirect
37. #435×C₆₇	C₂²	semidirect
38. C₈×D₄×C₆₇	C₂²	semidirect
39. C₈×D₄×C₆₇	C₂²	semidirect
40. C₈×D₄×C₆₇	C₂²	semidirect
41. C₈×D₄×C₆₇	C₂²	semidirect
42. #432×C₆₇	C₂²	semidirect
43. #432×C₆₇	C₂²	semidirect
44. #431×C₆₇	C₂²	semidirect
45. #431×C₆₇	C₂²	semidirect
46. #423×C₆₇	C₂²	semidirect
47. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
48. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
49. #407×C₆₇	C₂²	semidirect
50. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
51. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
52. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
53. C₄×M₁₆×C₆₇	C₂²	semidirect
54. C₂×#97×C₆₇	C₂²	semidirect
55. C₂×C₄×C₈×C₆₇	C₂²	semidirect
132. C₂×D₄×C₆₇	M₁₆	semidirect
193. C₂³×C₆₇	C₂×M₁₆	semidirect
201. D₄×C₆₇	C₂×M₁₆	semidirect
217. C₂×C₄×C₆₇	C₂×M₁₆	semidirect
231. C₈×C₆₇	H(5,𝔽₂)	semidirect
253. #60309	C₆₇	direct (N × H)
261. #2990	C₂×C₆₇	semidirect
262. #2989	C₂×C₆₇	semidirect
263. #2974	C₂×C₆₇	semidirect
264. #2973	C₂×C₆₇	semidirect
265. #2972	C₂×C₆₇	semidirect
266. #2970	C₂×C₆₇	semidirect
267. #2965	C₂×C₆₇	semidirect
268. D₄×M₁₆	C₂×C₆₇	semidirect
269. C₂×#435	C₂×C₆₇	semidirect
270. C₂×C₈×D₄	C₂×C₆₇	semidirect
271. C₆₇	#60309	direct (N × H)
273. C₂²×M₁₆	C₂²×C₆₇	semidirect
274. C₂³×C₈	C₂²×C₆₇	semidirect
285. #446	C₂²×C₆₇	semidirect
286. #446	C₂²×C₆₇	semidirect
287. #446	C₂²×C₆₇	semidirect
288. #436	C₂²×C₆₇	semidirect
289. #436	C₂²×C₆₇	semidirect
290. #436	C₂²×C₆₇	semidirect
291. #435	C₂²×C₆₇	semidirect
292. #435	C₂²×C₆₇	semidirect
293. #435	C₂²×C₆₇	semidirect
294. #435	C₂²×C₆₇	semidirect
295. #435	C₂²×C₆₇	semidirect
296. #435	C₂²×C₆₇	semidirect
297. #435	C₂²×C₆₇	semidirect
298. C₈×D₄	C₂²×C₆₇	semidirect
299. C₈×D₄	C₂²×C₆₇	semidirect
300. C₈×D₄	C₂²×C₆₇	semidirect
301. C₈×D₄	C₂²×C₆₇	semidirect
302. #432	C₂²×C₆₇	semidirect
303. #432	C₂²×C₆₇	semidirect
304. #431	C₂²×C₆₇	semidirect
305. #431	C₂²×C₆₇	semidirect
306. #423	C₂²×C₆₇	semidirect
307. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
308. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
309. #407	C₂²×C₆₇	semidirect
310. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
311. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
312. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
313. C₄×M₁₆	C₂²×C₆₇	semidirect
314. C₂×#97	C₂²×C₆₇	semidirect
315. C₂×C₄×C₈	C₂²×C₆₇	semidirect
392. C₂×D₄	M₁₆×C₆₇	semidirect
453. C₂³	C₂×M₁₆×C₆₇	semidirect
461. D₄	C₂×M₁₆×C₆₇	semidirect
477. C₂×C₄	C₂×M₁₆×C₆₇	semidirect
491. C₈	H(5,𝔽₂)×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×H(5,𝔽₂)×C₆₇	2
C₆₇×#2990	2
C₆₇×#2989	2
C₆₇×#2974	2
C₆₇×#2973	2
C₆₇×#2972	2
C₆₇×#2970	2
C₆₇×#2965	2
D₄×M₁₆×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
#60309	67

Subgroups — C67×#60309 #30739560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60309 #30739560