Subgroups — C₆₇×#22439 #30701690

Order 17152.

1466 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#440×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2341	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2339	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#135×C₆₇	—	2	8	no	—
15	C₂×#135×C₆₇	C₄	1	8	yes	complemented
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	#440×C₆₇	—	2	8	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	#391×C₆₇	C₂²	1	2	yes	—
32	#390×C₆₇	—	2	2	no	—
33	#390×C₆₇	C₂²	1	2	yes	—
34	#386×C₆₇	—	2	2	no	—
35	#381×C₆₇	—	2	1	no	—
36	#380×C₆₇	—	2	1	no	—
37	#380×C₆₇	C₂²	1	1	yes	characteristic
38	#380×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#95×C₆₇	—	2	1	no	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	C₂×#41×C₆₇	—	2	2	no	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	4	2	no	—
49	C₂×#41×C₆₇	—	4	2	no	—
50	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×#41×C₆₇	C₂×C₄	1	4	yes	complemented
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	#135×C₆₇	—	2	32	no	—
63	C₂×Q₁₆×C₆₇	—	2	4	no	—
64	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
65	C₂×SD₁₆×C₆₇	—	2	4	no	—
66	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
68	C₂×SD₁₆×C₆₇	D₄	1	8	yes	—
69	C₂×D₈×C₆₇	—	2	4	no	—
70	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
71	C₂²×C₈×C₆₇	—	2	4	no	—
72	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₆₇	—	2	4	no	—
77	C₄×Q₈×C₆₇	—	4	2	no	—
78	C₄×D₄×C₆₇	—	2	4	no	—
79	C₄×D₄×C₆₇	—	4	2	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
89	C₂×C₄²×C₆₇	—	2	2	no	—
90	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
91	#107×C₆₇	—	2	4	no	—
92	#103×C₆₇	—	2	4	no	—
93	#103×C₆₇	—	4	2	no	—
94	#102×C₆₇	—	2	4	no	—
95	#102×C₆₇	—	4	2	no	—
96	C₂²⋊C₈×C₆₇	—	2	2	no	—
97	#97×C₆₇	—	2	4	no	—
98	#95×C₆₇	—	2	1	no	—
99	#95×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	2	no	—
102	#95×C₆₇	D₄	1	1	yes	characteristic
103	#95×C₆₇	D₄	1	1	yes	characteristic
104	C₂⁴×C₆₇	—	2	1	no	—
105	C₂⁴×C₆₇	—	2	1	no	—
106	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
107	#41×C₆₇	—	2	8	no	—
108	#41×C₆₇	—	2	8	no	—
109	#41×C₆₇	—	4	4	no	—
110	#41×C₆₇	—	4	4	no	—
111	#41×C₆₇	—	4	8	no	—
112	#41×C₆₇	—	4	8	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
121	C₂×D₄×C₆₇	—	2	4	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	4	4	no	—
129	C₂×D₄×C₆₇	—	4	4	no	—
130	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	4	4	no	—
142	C₂²×C₄×C₆₇	—	4	4	no	—
143	C₂²×C₄×C₆₇	—	4	4	no	—
144	C₂²×C₄×C₆₇	—	4	4	no	—
145	C₂²×C₄×C₆₇	—	8	2	no	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	#41	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	Q₁₆×C₆₇	—	2	8	no	—
153	SD₁₆×C₆₇	—	2	8	no	—
154	SD₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
155	D₈×C₆₇	—	2	8	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	8	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
163	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	4	4	no	—
169	C₂²⋊C₄×C₆₇	—	4	2	no	—
170	C₂²⋊C₄×C₆₇	—	4	4	no	—
171	C₂²⋊C₄×C₆₇	—	4	4	no	—
172	C₂²⋊C₄×C₆₇	—	8	2	no	—
173	C₂²⋊C₄×C₆₇	—	8	1	no	—
174	C₂²⋊C₄×C₆₇	—	8	1	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₄²×C₆₇	—	4	4	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	1	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	1	no	—
187	C₂³×C₆₇	—	4	1	no	—
188	C₂³×C₆₇	—	4	2	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	—	4	1	no	—
191	C₂³×C₆₇	—	4	1	no	—
192	C₂³×C₆₇	C₄×D₄	1	2	yes	—
193	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₆₇	—	2	2	no	—
195	Q₈×C₆₇	—	2	4	no	—
196	Q₈×C₆₇	—	2	2	no	—
197	Q₈×C₆₇	—	2	2	no	—
198	Q₈×C₆₇	—	4	2	no	—
199	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
200	D₄×C₆₇	—	2	2	no	—
201	D₄×C₆₇	—	2	2	no	—
202	D₄×C₆₇	—	2	2	no	—
203	D₄×C₆₇	—	2	4	no	—
204	D₄×C₆₇	—	4	8	no	—
205	D₄×C₆₇	—	4	8	no	—
206	D₄×C₆₇	—	4	2	no	—
207	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	4	8	no	—
222	C₂×C₄×C₆₇	—	4	8	no	—
223	C₂×C₄×C₆₇	—	4	2	no	—
224	C₂×C₄×C₆₇	—	4	2	no	—
225	C₂×C₄×C₆₇	—	4	2	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	8	8	no	—
231	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
232	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
236	C₈×C₆₇	—	2	2	no	—
237	C₈×C₆₇	—	4	2	no	—
238	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
239	C₂²×C₆₇	—	2	2	no	—
240	C₂²×C₆₇	—	2	4	no	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	2	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	4	4	no	—
246	C₂²×C₆₇	—	4	2	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	8	2	no	—
253	C₂²×C₆₇	—	8	2	no	—
254	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₆₇	D₄²	1	2	yes	—
256	C₂²×C₆₇	#575	1	2	yes	—
257	C₂²×C₆₇	#576	1	1	yes	characteristic
258	C₂²×C₆₇	#578	1	1	yes	characteristic
259	C₄×C₆₇	—	2	2	no	—
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	4	2	no	—
262	C₄×C₆₇	—	4	2	no	—
263	C₄×C₆₇	—	4	2	no	—
264	C₄×C₆₇	—	8	2	no	—
265	C₄×C₆₇	—	8	2	no	—
266	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
267	C₄×C₆₇	#518	1	1	yes	characteristic
268	C₄×C₆₇	#546	1	2	yes	—
269	#22439	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₆₇	—	4	2	no	—
271	C₂×C₆₇	—	4	4	no	—
272	C₂×C₆₇	—	8	2	no	—
273	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₆₇	#2943	1	1	yes	characteristic
275	C₂×C₆₇	#2951	1	1	yes	characteristic
276	C₂×C₆₇	#3268	1	2	yes	—
277	C₂×C₆₇	#3269	1	2	yes	—
278	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
279	C₂×#440	C₂×C₆₇	1	2	yes	complemented
280	#2341	C₂×C₆₇	1	1	yes	characteristic, complemented
281	#2339	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1973	C₂×C₆₇	1	2	yes	complemented
283	#1879	C₂×C₆₇	1	2	yes	complemented
284	#1878	C₂×C₆₇	1	2	yes	complemented
285	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#1788	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1788	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#1752	C₂×C₆₇	1	1	yes	characteristic, complemented
289	C₆₇	#22439	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
291	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×#135	—	2	8	no	—
293	C₂×#135	C₄×C₆₇	1	8	yes	complemented
294	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
296	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
297	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
300	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
301	#440	—	2	8	no	—
302	C₂×#107	C₂²×C₆₇	1	2	yes	—
303	C₂×#103	C₂²×C₆₇	1	2	yes	—
304	C₂×#103	C₂²×C₆₇	1	2	yes	—
305	C₂×#102	C₂²×C₆₇	1	2	yes	—
306	C₂×#102	C₂²×C₆₇	1	2	yes	—
307	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×#97	C₂²×C₆₇	1	2	yes	—
309	#391	C₂²×C₆₇	1	2	yes	—
310	#390	—	2	2	no	—
311	#390	C₂²×C₆₇	1	2	yes	—
312	#386	—	2	2	no	—
313	#381	—	2	1	no	—
314	#380	—	2	1	no	—
315	#380	C₂²×C₆₇	1	1	yes	characteristic
316	#380	C₂²×C₆₇	1	1	yes	characteristic
317	C₄×C₂²⋊C₄	—	2	1	no	—
318	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#95	—	2	1	no	—
320	#340	C₂²×C₆₇	1	2	yes	—
321	#336	C₂²×C₆₇	1	2	yes	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	2	4	no	—
325	C₂×#41	—	2	4	no	—
326	C₂×#41	—	4	2	no	—
327	C₂×#41	—	4	2	no	—
328	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
329	C₂×#41	C₂×C₄×C₆₇	1	4	yes	complemented
330	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	1	no	—
334	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
336	C₂³×C₄	—	2	1	no	—
337	C₂³×C₄	—	2	1	no	—
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
340	#135	—	2	32	no	—
341	C₂×Q₁₆	—	2	4	no	—
342	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
343	C₂×SD₁₆	—	2	4	no	—
344	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
345	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
346	C₂×SD₁₆	D₄×C₆₇	1	8	yes	—
347	C₂×D₈	—	2	4	no	—
348	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
349	C₂²×C₈	—	2	4	no	—
350	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
351	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
352	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
354	C₄×Q₈	—	2	4	no	—
355	C₄×Q₈	—	4	2	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	4	2	no	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
367	C₂×C₄²	—	2	2	no	—
368	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
369	#107	—	2	4	no	—
370	#103	—	2	4	no	—
371	#103	—	4	2	no	—
372	#102	—	2	4	no	—
373	#102	—	4	2	no	—
374	C₂²⋊C₈	—	2	2	no	—
375	#97	—	2	4	no	—
376	#95	—	2	1	no	—
377	#95	—	2	2	no	—
378	#95	—	2	1	no	—
379	#95	—	2	2	no	—
380	#95	D₄×C₆₇	1	1	yes	characteristic
381	#95	D₄×C₆₇	1	1	yes	characteristic
382	C₂⁴	—	2	1	no	—
383	C₂⁴	—	2	1	no	—
384	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
385	#41	—	2	8	no	—
386	#41	—	2	8	no	—
387	#41	—	4	4	no	—
388	#41	—	4	4	no	—
389	#41	—	4	8	no	—
390	#41	—	4	8	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
397	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
409	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
410	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	8	2	no	—
424	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	#41×C₆₇	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
430	Q₁₆	—	2	8	no	—
431	SD₁₆	—	2	8	no	—
432	SD₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
433	D₈	—	2	8	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	8	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
440	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
441	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	4	4	no	—
447	C₂²⋊C₄	—	4	2	no	—
448	C₂²⋊C₄	—	4	4	no	—
449	C₂²⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	—	8	2	no	—
451	C₂²⋊C₄	—	8	1	no	—
452	C₂²⋊C₄	—	8	1	no	—
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	4	4	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	1	no	—
465	C₂³	—	4	1	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	2	no	—
468	C₂³	—	4	1	no	—
469	C₂³	—	4	1	no	—
470	C₂³	C₄×D₄×C₆₇	1	2	yes	—
471	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
472	Q₈	—	2	2	no	—
473	Q₈	—	2	4	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	2	2	no	—
476	Q₈	—	4	2	no	—
477	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
478	D₄	—	2	2	no	—
479	D₄	—	2	2	no	—
480	D₄	—	2	2	no	—
481	D₄	—	2	4	no	—
482	D₄	—	4	8	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	2	no	—
485	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	8	8	no	—
509	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
510	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
512	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
513	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
514	C₈	—	2	2	no	—
515	C₈	—	4	2	no	—
516	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	2	no	—
532	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
533	C₂²	D₄²×C₆₇	1	2	yes	—
534	C₂²	#575×C₆₇	1	2	yes	—
535	C₂²	#576×C₆₇	1	1	yes	characteristic
536	C₂²	#578×C₆₇	1	1	yes	characteristic
537	C₄	—	2	2	no	—
538	C₄	—	2	4	no	—
539	C₄	—	4	2	no	—
540	C₄	—	4	2	no	—
541	C₄	—	4	2	no	—
542	C₄	—	8	2	no	—
543	C₄	—	8	2	no	—
544	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
545	C₄	#518×C₆₇	1	1	yes	characteristic
546	C₄	#546×C₆₇	1	2	yes	—
547	C₂	—	4	2	no	—
548	C₂	—	4	4	no	—
549	C₂	—	8	2	no	—
550	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
551	C₂	C₆₇×#2943	1	1	yes	characteristic
552	C₂	C₆₇×#2951	1	1	yes	characteristic
553	C₂	C₆₇×#3268	1	2	yes	—
554	C₂	C₆₇×#3269	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₂²⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#440×C₆₇	C₂	semidirect
3. C₆₇×#2341	C₂	semidirect
4. C₆₇×#2339	C₂	semidirect
5. C₆₇×#1973	C₂	semidirect
6. C₆₇×#1879	C₂	semidirect
7. C₆₇×#1878	C₂	semidirect
8. C₆₇×#1808	C₂	semidirect
9. C₆₇×#1788	C₂	semidirect
10. C₆₇×#1788	C₂	semidirect
11. C₆₇×#1752	C₂	semidirect
15. C₂×#135×C₆₇	C₄	semidirect
51. C₂×#41×C₆₇	C₂×C₄	semidirect
64. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
67. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
72. C₂²×C₈×C₆₇	C₂×C₄	semidirect
154. SD₁₆×C₆₇	C₂²⋊C₄	semidirect
269. #22439	C₆₇	direct (N × H)
279. C₂×#440	C₂×C₆₇	semidirect
280. #2341	C₂×C₆₇	semidirect
281. #2339	C₂×C₆₇	semidirect
282. #1973	C₂×C₆₇	semidirect
283. #1879	C₂×C₆₇	semidirect
284. #1878	C₂×C₆₇	semidirect
285. #1808	C₂×C₆₇	semidirect
286. #1788	C₂×C₆₇	semidirect
287. #1788	C₂×C₆₇	semidirect
288. #1752	C₂×C₆₇	semidirect
289. C₆₇	#22439	direct (N × H)
293. C₂×#135	C₄×C₆₇	semidirect
329. C₂×#41	C₂×C₄×C₆₇	semidirect
342. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
345. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
350. C₂²×C₈	C₂×C₄×C₆₇	semidirect
432. SD₁₆	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂×#440×C₆₇	2
C₆₇×#2341	2
C₆₇×#2339	2
C₆₇×#1973	2
C₆₇×#1879	2
C₆₇×#1878	2
C₆₇×#1808	2
C₆₇×#1788	2
C₆₇×#1752	2
#22439	67

Subgroups — C67×#22439 #30701690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22439 #30701690