Subgroups — C₆₇×#22652 #30701903

Order 17152.

964 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435×C₆₇	C₂	1	2	yes	maximal
2	C₂×#424×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2293	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1940	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1851	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1850	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1798	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	2	yes	—
14	#435×C₆₇	—	2	8	no	—
15	#424×C₆₇	—	2	8	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	#390×C₆₇	C₄	1	1	yes	characteristic
28	#390×C₆₇	C₄	1	1	yes	characteristic
29	#389×C₆₇	C₄	1	1	yes	characteristic
30	#389×C₆₇	C₄	1	1	yes	characteristic
31	#388×C₆₇	C₄	1	2	yes	—
32	#388×C₆₇	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₄³×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	—	2	4	no	—
43	C₂×M₁₆×C₆₇	—	4	2	no	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	—	4	2	no	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄×D₄×C₆₇	D₄	1	4	yes	—
52	C₄×D₄×C₆₇	Q₈	1	4	yes	—
53	#117×C₆₇	—	2	4	no	—
54	#117×C₆₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₆₇	—	2	4	no	—
74	C₄⋊C₈×C₆₇	—	2	4	no	—
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	C₄⋊C₈×C₆₇	—	2	4	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	—	4	1	no	—
79	C₄⋊C₈×C₆₇	—	4	1	no	—
80	C₄⋊C₈×C₆₇	D₄	1	4	yes	complemented
81	C₂²⋊C₈×C₆₇	—	2	4	no	—
82	C₂²⋊C₈×C₆₇	—	2	4	no	—
83	#97×C₆₇	—	2	4	no	—
84	#95×C₆₇	C₂×C₄	1	2	yes	—
85	#95×C₆₇	C₂×C₄	1	2	yes	—
86	#95×C₆₇	C₂×C₄	1	2	yes	—
87	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₆₇	—	4	8	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	4	8	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₄⋊C₄×C₆₇	—	4	1	no	—
125	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
126	C₂²⋊C₄×C₆₇	—	2	4	no	—
127	C₂²⋊C₄×C₆₇	—	2	4	no	—
128	C₂²⋊C₄×C₆₇	—	2	4	no	—
129	C₂²⋊C₄×C₆₇	—	4	2	no	—
130	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	1	no	—
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	—	4	1	no	—
137	C₄²×C₆₇	—	4	1	no	—
138	C₄²×C₆₇	—	4	1	no	—
139	C₄²×C₆₇	C₂×D₄	1	2	yes	—
140	C₄²×C₆₇	C₂×D₄	1	2	yes	—
141	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
142	C₄²×C₆₇	#41	1	2	yes	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
150	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
151	D₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
180	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₆₇	#131	1	2	yes	—
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
189	C₈×C₆₇	—	2	4	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₈×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	2	4	no	—
194	C₂²×C₆₇	—	2	4	no	—
195	C₂²×C₆₇	—	2	4	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₆₇	#520	1	1	yes	characteristic
203	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
204	C₂²×C₆₇	#568	1	1	yes	characteristic
205	C₂²×C₆₇	#568	1	1	yes	characteristic
206	C₂²×C₆₇	#568	1	1	yes	characteristic
207	C₄×C₆₇	—	2	4	no	—
208	C₄×C₆₇	—	2	2	no	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	2	2	no	—
213	C₄×C₆₇	—	4	2	no	—
214	C₄×C₆₇	—	4	2	no	—
215	C₄×C₆₇	—	4	1	no	—
216	C₄×C₆₇	D₄²	1	1	yes	characteristic
217	C₄×C₆₇	#547	1	1	yes	characteristic
218	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
219	C₄×C₆₇	#554	1	1	yes	characteristic
220	#22652	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₆₇	—	4	4	no	—
222	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₆₇	#2905	1	1	yes	characteristic
224	C₂×C₆₇	#2906	1	1	yes	characteristic
225	C₂×C₆₇	#2932	1	1	yes	characteristic
226	C₂×C₆₇	#2935	1	1	yes	characteristic
227	C₂×C₆₇	#2992	1	1	yes	characteristic
228	C₂×C₆₇	#2995	1	1	yes	characteristic
229	C₂×#435	C₂×C₆₇	1	2	yes	—
230	C₂×#424	C₂×C₆₇	1	2	yes	complemented
231	#2293	C₂×C₆₇	1	1	yes	characteristic
232	#1940	C₂×C₆₇	1	2	yes	complemented
233	#1851	C₂×C₆₇	1	2	yes	complemented
234	#1850	C₂×C₆₇	1	2	yes	complemented
235	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#1798	C₂×C₆₇	1	2	yes	complemented
237	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₆₇	#22652	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
240	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
241	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#117	C₂²×C₆₇	1	2	yes	—
243	#435	—	2	8	no	—
244	#424	—	2	8	no	—
245	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
248	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
249	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
250	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
255	C₂×#97	C₂²×C₆₇	1	2	yes	—
256	#390	C₄×C₆₇	1	1	yes	characteristic
257	#390	C₄×C₆₇	1	1	yes	characteristic
258	#389	C₄×C₆₇	1	1	yes	characteristic
259	#389	C₄×C₆₇	1	1	yes	characteristic
260	#388	C₄×C₆₇	1	2	yes	—
261	#388	C₄×C₆₇	1	2	yes	—
262	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
263	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
264	C₄³	C₂²×C₆₇	1	1	yes	characteristic
265	#336	C₂²×C₆₇	1	2	yes	—
266	#336	C₂²×C₆₇	1	2	yes	—
267	#336	C₂²×C₆₇	1	2	yes	—
268	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
269	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	—	2	4	no	—
272	C₂×M₁₆	—	4	2	no	—
273	C₂²×C₈	—	2	4	no	—
274	C₂²×C₈	—	2	4	no	—
275	C₂²×C₈	—	4	2	no	—
276	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
277	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
278	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
280	C₄×D₄	D₄×C₆₇	1	4	yes	—
281	C₄×D₄	Q₈×C₆₇	1	4	yes	—
282	#117	—	2	4	no	—
283	#117	—	2	4	no	—
284	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
290	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
291	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
292	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
293	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
294	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
299	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
302	C₄⋊C₈	—	2	4	no	—
303	C₄⋊C₈	—	2	4	no	—
304	C₄⋊C₈	—	2	4	no	—
305	C₄⋊C₈	—	2	4	no	—
306	C₄⋊C₈	—	2	2	no	—
307	C₄⋊C₈	—	4	1	no	—
308	C₄⋊C₈	—	4	1	no	—
309	C₄⋊C₈	D₄×C₆₇	1	4	yes	complemented
310	C₂²⋊C₈	—	2	4	no	—
311	C₂²⋊C₈	—	2	4	no	—
312	#97	—	2	4	no	—
313	#95	C₂×C₄×C₆₇	1	2	yes	—
314	#95	C₂×C₄×C₆₇	1	2	yes	—
315	#95	C₂×C₄×C₆₇	1	2	yes	—
316	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
327	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
334	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
336	M₁₆	—	4	8	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	8	no	—
349	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
355	C₂²⋊C₄	—	2	4	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	—	4	1	no	—
368	C₄²	C₂×D₄×C₆₇	1	2	yes	—
369	C₄²	C₂×D₄×C₆₇	1	2	yes	—
370	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
371	C₄²	#41×C₆₇	1	2	yes	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	2	no	—
375	C₂³	—	2	2	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	2	no	—
378	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
379	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
380	D₄	—	4	4	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
407	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
408	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
409	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
410	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
411	C₂×C₄	#131×C₆₇	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
416	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
417	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	2	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	4	no	—
424	C₂²	—	2	4	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
430	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
431	C₂²	#520×C₆₇	1	1	yes	characteristic
432	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
433	C₂²	#568×C₆₇	1	1	yes	characteristic
434	C₂²	#568×C₆₇	1	1	yes	characteristic
435	C₂²	#568×C₆₇	1	1	yes	characteristic
436	C₄	—	2	4	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	1	no	—
445	C₄	D₄²×C₆₇	1	1	yes	characteristic
446	C₄	#547×C₆₇	1	1	yes	characteristic
447	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
448	C₄	#554×C₆₇	1	1	yes	characteristic
449	C₂	—	4	4	no	—
450	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
451	C₂	C₆₇×#2905	1	1	yes	characteristic
452	C₂	C₆₇×#2906	1	1	yes	characteristic
453	C₂	C₆₇×#2932	1	1	yes	characteristic
454	C₂	C₆₇×#2935	1	1	yes	characteristic
455	C₂	C₆₇×#2992	1	1	yes	characteristic
456	C₂	C₆₇×#2995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#424×C₆₇	C₂	semidirect
4. C₆₇×#1940	C₂	semidirect
5. C₆₇×#1851	C₂	semidirect
6. C₆₇×#1850	C₂	semidirect
7. C₆₇×#1842	C₂	semidirect
8. C₆₇×#1798	C₂	semidirect
9. C₆₇×#1767	C₂	semidirect
80. C₄⋊C₈×C₆₇	D₄	semidirect
220. #22652	C₆₇	direct (N × H)
230. C₂×#424	C₂×C₆₇	semidirect
232. #1940	C₂×C₆₇	semidirect
233. #1851	C₂×C₆₇	semidirect
234. #1850	C₂×C₆₇	semidirect
235. #1842	C₂×C₆₇	semidirect
236. #1798	C₂×C₆₇	semidirect
237. #1767	C₂×C₆₇	semidirect
238. C₆₇	#22652	direct (N × H)
309. C₄⋊C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#435×C₆₇	2
C₂×#424×C₆₇	2
C₆₇×#2293	2
C₆₇×#1940	2
C₆₇×#1851	2
C₆₇×#1850	2
C₆₇×#1842	2
C₆₇×#1798	2
C₆₇×#1767	2
#22652	67

Subgroups — C67×#22652 #30701903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22652 #30701903