Subgroups — #30824946

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568840	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863452	C₂	1	1	yes	maximal, characteristic
3	#431×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558993	C₂	1	2	yes	maximal, complemented
5	#22558991	C₂	1	2	yes	maximal, complemented
6	#22558988	C₂	1	2	yes	maximal, complemented
7	#22558986	C₂	1	2	yes	maximal, complemented
8	#22558962	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558959	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
13	#16863452	—	2	1	no	—
14	#16863452	—	2	1	no	—
15	#16863452	C₂²	1	2	yes	—
16	#16863452	C₂²	1	2	yes	—
17	#16863451	C₄	1	2	yes	—
18	#16863451	C₄	1	2	yes	—
19	#16863449	C₂²	1	1	yes	characteristic
20	#16863448	C₂²	1	1	yes	characteristic
21	#16863448	C₂²	1	1	yes	characteristic
22	#16863447	—	2	1	no	—
23	#16863447	C₂²	1	2	yes	—
24	#16863447	C₂²	1	2	yes	—
25	#16863446	—	2	1	no	—
26	#16863419	C₄	1	1	yes	characteristic
27	#16863419	C₄	1	1	yes	characteristic
28	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
31	#16863359	C₄	1	1	yes	characteristic
32	#16863359	C₄	1	1	yes	characteristic
33	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
34	C₂²⋊C₈×D₆₇	—	2	1	no	—
35	C₄×#11761612	C₂²	1	1	yes	characteristic
36	C₂×#14523953	—	2	1	no	—
37	C₄×C₈×D₆₇	—	2	1	no	—
38	#16863149	C₂²	1	2	yes	complemented
39	#16863148	C₂²	1	2	yes	complemented
40	#16863148	C₂²	1	2	yes	complemented
41	#16863147	C₂²	1	2	yes	complemented
42	#16863146	C₂²	1	2	yes	complemented
43	#16863146	C₂²	1	2	yes	complemented
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	#335×C₆₇	C₂²	1	2	yes	complemented
48	#334×C₆₇	C₂²	1	2	yes	complemented
49	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×D₆₇	Q₈	1	1	yes	characteristic
52	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂×C₈×D₆₇	—	2	1	no	—
56	C₂×C₈×D₆₇	—	2	1	no	—
57	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
60	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₆₇	D₄	1	1	yes	characteristic
63	C₄²×D₆₇	D₄	1	1	yes	characteristic
64	C₄²×D₆₇	Q₈	1	1	yes	characteristic
65	C₄²×D₆₇	Q₈	1	1	yes	characteristic
66	C₄²×D₆₇	C₂³	1	1	yes	characteristic
67	C₄²×D₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
72	Dic₆₇⋊C₈	—	2	1	no	—
73	Dic₆₇⋊C₈	—	2	1	no	—
74	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
75	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
76	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
77	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
78	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
79	(C₂×D₆₇)⋊C₈	—	2	1	no	—
80	(C₂×D₆₇)⋊C₈	—	2	1	no	—
81	(C₂×D₆₇)⋊C₈	—	2	2	no	—
82	(C₂×D₆₇)⋊C₈	—	2	2	no	—
83	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
84	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
85	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
86	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
87	#14523954	C₂×C₄	1	2	yes	—
88	#14523953	—	2	2	no	—
89	#14523953	—	2	2	no	—
90	C₄×C₆₇⋊C₈	—	2	1	no	—
91	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
92	C₈×Dic₆₇	—	2	1	no	—
93	C₈×Dic₆₇	—	2	1	no	—
94	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	1	no	—
99	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
101	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²⋊C₈×C₆₇	—	2	1	no	—
103	#97×C₆₇	C₂×C₄	1	2	yes	—
104	C₄×C₈×C₆₇	—	2	1	no	—
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×D₆₇	—	2	2	no	—
111	C₂×C₄×D₆₇	—	2	2	no	—
112	C₂×C₄×D₆₇	—	4	1	no	—
113	C₂×C₄×D₆₇	—	4	1	no	—
114	C₂×C₄×D₆₇	—	4	2	no	—
115	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
126	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
127	#11761612	—	2	2	no	—
128	#11761612	—	2	2	no	—
129	C₂×C₆₇⋊C₈	—	2	1	no	—
130	C₂×C₆₇⋊C₈	—	2	1	no	—
131	C₂×C₆₇⋊C₈	—	2	2	no	—
132	C₂×C₆₇⋊C₈	—	4	2	no	—
133	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
135	C₈×D₆₇	—	4	2	no	—
136	C₈×D₆₇	—	4	2	no	—
137	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
138	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
139	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
140	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
141	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
142	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
143	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
146	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
147	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
148	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
152	M₁₆×C₆₇	—	2	2	no	—
153	M₁₆×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	4	2	no	—
158	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
159	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
160	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
163	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
164	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×D₆₇	—	2	2	no	—
166	C₂²×D₆₇	—	2	2	no	—
167	C₂²×D₆₇	—	4	1	no	—
168	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	#131	1	2	yes	—
177	C₂×C₄×C₆₇	#131	1	2	yes	—
178	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
181	C₂×Dic₆₇	—	2	2	no	—
182	C₂×Dic₆₇	—	4	1	no	—
183	C₂×Dic₆₇	—	4	1	no	—
184	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄×D₆₇	—	2	2	no	—
191	C₄×D₆₇	—	2	2	no	—
192	C₄×D₆₇	—	2	1	no	—
193	C₄×D₆₇	—	2	1	no	—
194	C₄×D₆₇	—	2	1	no	—
195	C₄×D₆₇	—	2	1	no	—
196	C₄×D₆₇	—	2	1	no	—
197	C₄×D₆₇	—	2	1	no	—
198	C₄×D₆₇	—	2	1	no	—
199	C₄×D₆₇	—	2	1	no	—
200	C₄×D₆₇	—	2	2	no	—
201	C₄×D₆₇	—	2	2	no	—
202	C₄×D₆₇	—	4	1	no	—
203	C₄×D₆₇	—	4	1	no	—
204	C₄×D₆₇	—	4	1	no	—
205	C₄×D₆₇	—	4	1	no	—
206	C₈×C₆₇	—	2	2	no	—
207	C₈×C₆₇	—	2	2	no	—
208	C₈×C₆₇	—	4	2	no	—
209	C₈×C₆₇	—	4	1	no	—
210	C₈×C₆₇	—	4	1	no	—
211	C₆₇⋊C₈	—	2	2	no	—
212	C₆₇⋊C₈	—	2	2	no	—
213	C₆₇⋊C₈	—	4	1	no	—
214	C₆₇⋊C₈	—	4	1	no	—
215	C₆₇⋊C₈	—	4	2	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	#424	1	2	yes	—
219	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₂×D₆₇	—	2	1	no	—
221	C₂×D₆₇	—	2	1	no	—
222	C₂×D₆₇	—	2	1	no	—
223	C₂×D₆₇	—	2	1	no	—
224	C₂×D₆₇	—	2	1	no	—
225	C₂×D₆₇	—	4	4	no	—
226	C₂×D₆₇	#430	1	2	yes	—
227	C₄×C₆₇	—	2	1	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	#424	1	2	yes	—
231	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
232	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
233	C₄×C₆₇	#575	1	1	yes	characteristic
234	C₄×C₆₇	#575	1	1	yes	characteristic
235	Dic₆₇	—	2	1	no	—
236	Dic₆₇	—	2	1	no	—
237	Dic₆₇	—	2	1	no	—
238	Dic₆₇	—	2	1	no	—
239	Dic₆₇	—	2	1	no	—
240	Dic₆₇	—	2	1	no	—
241	#17810	—	67	1	no	maximal, Sylow
242	C₂×C₆₇	—	2	2	no	—
243	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
244	C₂×C₆₇	#2911	1	1	yes	characteristic
245	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
246	D₆₇	—	2	2	no	—
247	D₆₇	—	4	1	no	—
248	#2920	—	67	1	no	—
249	C₂×#432	—	67	1	no	—
250	C₂×#431	—	67	1	no	—
251	#1573	—	67	2	no	—
252	#1573	—	67	2	no	—
253	#1572	—	67	2	no	—
254	#1572	—	67	2	no	—
255	#1565	—	67	1	no	—
256	#1565	—	67	1	no	—
257	#1565	—	67	1	no	—
258	#1565	D₆₇	1	1	yes	characteristic, complemented
259	C₆₇	#17810	1	1	yes	characteristic, complemented, Sylow
260	C₂²×C₄²	—	67	1	no	—
261	#432	—	67	2	no	—
262	#432	—	67	2	no	—
263	#432	—	67	2	no	—
264	#432	—	67	2	no	—
265	#432	—	134	1	no	—
266	#432	—	134	1	no	—
267	#431	—	67	2	no	—
268	#431	—	67	2	no	—
269	#431	—	67	1	no	—
270	#431	—	67	1	no	—
271	#431	—	67	1	no	—
272	#431	—	134	1	no	—
273	#431	—	134	1	no	—
274	#431	C₂×D₆₇	1	1	yes	characteristic
275	#423	—	67	1	no	—
276	#423	—	67	1	no	—
277	C₂×C₄⋊C₈	—	67	1	no	—
278	C₂×C₄⋊C₈	—	67	1	no	—
279	C₂×C₄⋊C₈	—	67	1	no	—
280	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
281	#407	—	67	1	no	—
282	#407	—	67	1	no	—
283	C₂×C₂²⋊C₈	—	134	1	no	—
284	C₂×C₂²⋊C₈	—	134	1	no	—
285	C₄×M₁₆	—	67	1	no	—
286	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
287	C₂×#97	—	134	1	no	—
288	C₂×C₄×C₈	—	134	1	no	—
289	#335	—	67	2	no	—
290	#335	—	67	2	no	—
291	#335	—	67	2	no	—
292	#335	C₂×D₆₇	1	2	yes	complemented
293	#334	—	67	2	no	—
294	#334	—	67	2	no	—
295	#334	—	67	2	no	—
296	#334	C₂×D₆₇	1	2	yes	complemented
297	C₂³×C₄	—	67	1	no	—
298	C₂³×C₄	—	67	1	no	—
299	C₂³×C₄	—	67	1	no	—
300	C₂×M₁₆	—	67	1	no	—
301	C₂×M₁₆	—	67	1	no	—
302	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
303	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	134	1	no	—
306	C₂²×C₈	—	134	1	no	—
307	C₂²×C₈	—	134	1	no	—
308	C₂×C₄²	—	67	1	no	—
309	C₂×C₄²	—	67	1	no	—
310	C₂×C₄²	—	67	1	no	—
311	C₂×C₄²	—	67	1	no	—
312	C₂×C₄²	—	67	1	no	—
313	C₂×C₄²	—	67	1	no	—
314	C₂×C₄²	—	67	1	no	—
315	C₂×C₄²	—	67	1	no	—
316	C₂×C₄²	—	67	1	no	—
317	C₂×C₄²	—	67	1	no	—
318	C₂×C₄²	—	67	1	no	—
319	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
320	C₄⋊C₈	—	2	1	no	—
321	C₄⋊C₈	—	67	2	no	—
322	C₄⋊C₈	—	67	2	no	—
323	C₄⋊C₈	—	67	2	no	—
324	C₄⋊C₈	—	67	2	no	—
325	C₄⋊C₈	—	67	1	no	—
326	C₄⋊C₈	—	67	1	no	—
327	C₄⋊C₈	—	67	2	no	—
328	C₄⋊C₈	—	134	1	no	—
329	C₄⋊C₈	—	134	1	no	—
330	C₄⋊C₈	—	134	1	no	—
331	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
332	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
333	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
334	C₂²⋊C₈	—	2	1	no	—
335	C₂²⋊C₈	—	67	2	no	—
336	C₂²⋊C₈	—	67	2	no	—
337	C₂²⋊C₈	—	67	2	no	—
338	C₂²⋊C₈	—	67	2	no	—
339	C₂²⋊C₈	—	134	1	no	—
340	C₂²⋊C₈	—	134	1	no	—
341	C₂²⋊C₈	—	134	2	no	—
342	C₂²⋊C₈	—	134	2	no	—
343	C₂²⋊C₈	—	134	1	no	—
344	#97	—	67	2	no	—
345	#97	—	134	2	no	—
346	#97	—	134	2	no	—
347	#97	C₄×D₆₇	1	2	yes	—
348	C₄×C₈	—	2	1	no	—
349	C₄×C₈	—	67	2	no	—
350	C₄×C₈	—	134	1	no	—
351	C₄×C₈	—	134	1	no	—
352	C₄×C₈	—	134	1	no	—
353	C₄×C₈	C₂²×D₆₇	1	2	yes	—
354	C₂⁴	—	67	1	no	—
355	C₂²×C₄	—	268	1	no	—
356	C₂²×C₄	—	268	1	no	—
357	C₂²×C₄	—	268	2	no	—
358	C₂²×C₄	—	67	1	no	—
359	C₂²×C₄	—	67	1	no	—
360	C₂²×C₄	—	67	1	no	—
361	C₂²×C₄	—	67	1	no	—
362	C₂²×C₄	—	67	1	no	—
363	C₂²×C₄	—	67	1	no	—
364	C₂²×C₄	—	67	1	no	—
365	C₂²×C₄	—	67	1	no	—
366	C₂²×C₄	—	67	1	no	—
367	C₂²×C₄	—	67	1	no	—
368	C₂²×C₄	—	67	1	no	—
369	C₂²×C₄	—	67	1	no	—
370	C₂²×C₄	—	67	1	no	—
371	C₂²×C₄	—	67	1	no	—
372	C₂²×C₄	—	67	1	no	—
373	C₂²×C₄	—	134	2	no	—
374	C₂²×C₄	—	134	2	no	—
375	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
376	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
377	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
378	M₁₆	—	2	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	134	2	no	—
381	M₁₆	—	134	2	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	4	2	no	—
386	C₂×C₈	—	268	2	no	—
387	C₂×C₈	—	268	2	no	—
388	C₂×C₈	—	268	2	no	—
389	C₂×C₈	—	67	2	no	—
390	C₂×C₈	—	67	2	no	—
391	C₂×C₈	—	134	1	no	—
392	C₂×C₈	—	134	1	no	—
393	C₂×C₈	—	134	2	no	—
394	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
395	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
396	C₄²	—	67	1	no	—
397	C₄²	—	67	1	no	—
398	C₄²	—	67	1	no	—
399	C₄²	—	67	1	no	—
400	C₄²	—	67	1	no	—
401	C₄²	—	67	1	no	—
402	C₄²	—	67	1	no	—
403	C₄²	—	67	1	no	—
404	C₄²	—	67	1	no	—
405	C₄²	—	67	1	no	—
406	C₄²	—	67	1	no	—
407	C₄²	—	67	1	no	—
408	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
409	C₄²	D₄×D₆₇	1	1	yes	characteristic
410	C₄²	Q₈×D₆₇	1	1	yes	characteristic
411	C₄²	C₂³×D₆₇	1	1	yes	characteristic
412	C₂³	—	268	1	no	—
413	C₂³	—	67	1	no	—
414	C₂³	—	67	1	no	—
415	C₂³	—	134	2	no	—
416	C₂³	—	134	2	no	—
417	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	268	1	no	—
422	C₂×C₄	—	268	1	no	—
423	C₂×C₄	—	268	1	no	—
424	C₂×C₄	—	268	1	no	—
425	C₂×C₄	—	268	1	no	—
426	C₂×C₄	—	268	1	no	—
427	C₂×C₄	—	67	1	no	—
428	C₂×C₄	—	67	1	no	—
429	C₂×C₄	—	67	1	no	—
430	C₂×C₄	—	67	1	no	—
431	C₂×C₄	—	67	1	no	—
432	C₂×C₄	—	67	1	no	—
433	C₂×C₄	—	134	2	no	—
434	C₂×C₄	—	134	2	no	—
435	C₂×C₄	—	134	2	no	—
436	C₂×C₄	—	134	1	no	—
437	C₂×C₄	—	134	1	no	—
438	C₂×C₄	—	134	1	no	—
439	C₂×C₄	—	134	1	no	—
440	C₂×C₄	—	134	2	no	—
441	C₂×C₄	—	134	2	no	—
442	C₂×C₄	—	134	1	no	—
443	C₂×C₄	—	134	1	no	—
444	C₂×C₄	—	134	1	no	—
445	C₂×C₄	—	134	1	no	—
446	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
448	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
449	C₂×C₄	#14524049	1	2	yes	—
450	C₂×C₄	#14524051	1	2	yes	—
451	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	4	2	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	268	1	no	—
460	C₈	—	268	1	no	—
461	C₈	—	268	2	no	—
462	C₈	—	134	2	no	—
463	C₈	—	134	2	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	268	4	no	—
467	C₂²	—	67	2	no	—
468	C₂²	—	134	1	no	—
469	C₂²	—	134	1	no	—
470	C₂²	—	134	1	no	—
471	C₂²	—	134	1	no	—
472	C₂²	—	134	1	no	—
473	C₂²	#16863421	1	2	yes	—
474	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	134	1	no	—
479	C₄	—	134	1	no	—
480	C₄	—	134	1	no	—
481	C₄	—	134	1	no	—
482	C₄	—	134	1	no	—
483	C₄	—	134	1	no	—
484	C₄	#16863421	1	2	yes	—
485	C₄	C₂×#14524049	1	1	yes	characteristic
486	C₄	C₂×#14524051	1	1	yes	characteristic
487	C₄	#16864192	1	1	yes	characteristic
488	C₄	#16864195	1	1	yes	characteristic
489	C₂	—	2	2	no	—
490	C₂	—	268	1	no	—
491	C₂	—	134	2	no	—
492	C₂	C₂×#16863421	1	1	yes	characteristic
493	C₂	#22568796	1	1	yes	characteristic
494	C₂	#430×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1565	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1565	D₆₇
hypocenter	C₆₇	#17810
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #22558993	C₂	semidirect
5. #22558991	C₂	semidirect
6. #22558988	C₂	semidirect
7. #22558986	C₂	semidirect
8. #22558962	C₂	semidirect
9. #22558961	C₂	semidirect
10. #22558959	C₂	semidirect
11. C₆₇×#1565	C₂	semidirect
38. #16863149	C₂²	semidirect
39. #16863148	C₂²	semidirect
40. #16863148	C₂²	semidirect
41. #16863147	C₂²	semidirect
42. #16863146	C₂²	semidirect
43. #16863146	C₂²	semidirect
47. #335×C₆₇	C₂²	semidirect
48. #334×C₆₇	C₂²	semidirect
258. #1565	D₆₇	semidirect
259. C₆₇	#17810	semidirect
292. #335	C₂×D₆₇	semidirect
296. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568840	2
C₂×#16863452	2
#431×D₆₇	2
#22558993	2
#22558991	2
#22558988	2
#22558986	2
#22558962	2
#22558961	2
#22558959	2
C₆₇×#1565	2
#17810	67