Subgroups — C₂×C₆₇×#2990 #30739508

Order 17152.

2210 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2990	C₂	direct (N × H)
3. C₂×#446×C₆₇	C₂	semidirect
4. C₂×#435×C₆₇	C₂	semidirect
5. C₂×C₈×D₄×C₆₇	C₂	semidirect
6. C₂×#432×C₆₇	C₂	semidirect
7. C₂×#431×C₆₇	C₂	semidirect
8. C₂×#423×C₆₇	C₂	semidirect
9. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
25. #446×C₆₇	C₂²	semidirect
26. #435×C₆₇	C₂²	semidirect
27. C₈×D₄×C₆₇	C₂²	semidirect
28. #432×C₆₇	C₂²	semidirect
29. #431×C₆₇	C₂²	semidirect
30. #423×C₆₇	C₂²	semidirect
36. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
103. C₂×D₄×C₆₇	M₁₆	semidirect
167. D₄×C₆₇	C₂×M₁₆	semidirect
224. C₂×#2990	C₆₇	direct (N × H)
227. C₂×C₆₇	#2990	direct (N × H)
232. #2990	C₂×C₆₇	direct (N × H)
233. C₂×#446	C₂×C₆₇	semidirect
234. C₂×#435	C₂×C₆₇	semidirect
235. C₂×C₈×D₄	C₂×C₆₇	semidirect
236. C₂×#432	C₂×C₆₇	semidirect
237. C₂×#431	C₂×C₆₇	semidirect
238. C₂×#423	C₂×C₆₇	semidirect
239. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
240. C₆₇	C₂×#2990	direct (N × H)
256. #446	C₂²×C₆₇	semidirect
257. #435	C₂²×C₆₇	semidirect
258. C₈×D₄	C₂²×C₆₇	semidirect
259. #432	C₂²×C₆₇	semidirect
260. #431	C₂²×C₆₇	semidirect
261. #423	C₂²×C₆₇	semidirect
267. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
334. C₂×D₄	M₁₆×C₆₇	semidirect
398. D₄	C₂×M₁₆×C₆₇	semidirect
457. C₂	C₆₇×#2990	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₆₇×#2990	2
C₂×#446×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₂×#423×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂×#2990	67

Subgroups — C2×C67×#2990 #30739508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2990 #30739508