Subgroups — #30824868

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569323	C₂	1	2	yes	maximal
2	C₄×#14524049	C₂	1	1	yes	maximal, characteristic
3	#22558974	C₂	1	2	yes	maximal, complemented
4	#22558973	C₂	1	2	yes	maximal, complemented
5	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558969	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558963	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558958	C₂	1	2	yes	maximal, complemented
9	#22558954	C₂	1	1	yes	maximal, characteristic, complemented
10	#334×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	C₄	1	1	yes	characteristic
13	C₂×#14524049	C₄	1	1	yes	characteristic
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	#16863525	—	2	2	no	—
16	#16863472	—	2	2	no	—
17	#16863466	—	2	2	no	—
18	#16863465	—	2	2	no	—
19	#16863453	C₂²	1	2	yes	—
20	#16863452	C₂²	1	2	yes	—
21	#16863448	C₂²	1	2	yes	—
22	#16863418	C₂²	1	2	yes	—
23	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
24	#16863351	C₄	1	1	yes	characteristic
25	#16863351	C₄	1	1	yes	characteristic
26	#16863350	C₄	1	1	yes	characteristic
27	#16863350	C₄	1	1	yes	characteristic
28	#16863350	C₄	1	1	yes	characteristic
29	#16863350	C₄	1	1	yes	characteristic
30	C₄×#11761612	C₂²	1	1	yes	characteristic
31	C₄×#11761611	C₂²	1	1	yes	characteristic
32	C₄×#11761611	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
35	#16863149	C₂²	1	2	yes	—
36	#16863148	C₂²	1	2	yes	—
37	#16863148	C₂²	1	2	yes	—
38	#16863147	C₂²	1	1	yes	characteristic
39	#16863147	C₂²	1	1	yes	characteristic
40	#16863146	C₂²	1	2	yes	—
41	#16863146	C₂²	1	2	yes	—
42	#423×C₆₇	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#334×C₆₇	C₂²	1	1	yes	characteristic
45	#334×C₆₇	C₂²	1	1	yes	characteristic
46	#334×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
48	#14524049	—	2	4	no	—
49	#14524049	—	2	4	no	—
50	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
51	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
52	C₂×#11761611	—	2	2	no	—
53	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
56	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₆₇	—	2	2	no	—
58	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
63	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
64	C₄×D₂₆₈	D₄	1	1	yes	characteristic
65	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
66	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
67	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₆₇	Q₈	1	1	yes	characteristic
69	C₄×C₂²⋊D₆₇	Q₈	1	1	yes	characteristic
70	#14524002	C₂×C₄	1	1	yes	characteristic
71	#14524000	C₂×C₄	1	1	yes	characteristic
72	#14524000	C₂×C₄	1	1	yes	characteristic
73	C₄²×D₆₇	C₂³	1	1	yes	characteristic
74	C₄²×D₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
76	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
80	(C₂×D₆₇)⋊C₈	—	2	2	no	—
81	(C₂×D₆₇)⋊C₈	—	2	2	no	—
82	#14523954	—	2	2	no	—
83	#14523954	C₂×C₄	1	1	yes	characteristic
84	#14523954	C₂×C₄	1	1	yes	characteristic
85	#14523953	—	2	2	no	—
86	#14523953	C₂×C₄	1	1	yes	characteristic
87	#14523953	C₂×C₄	1	1	yes	characteristic
88	#14523953	C₂×C₄	1	1	yes	characteristic
89	#14523953	C₂×C₄	1	1	yes	characteristic
90	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
91	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₈×Dic₆₇	—	2	2	no	—
93	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
96	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×M₁₆×C₆₇	—	2	2	no	—
98	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	—	2	2	no	—
102	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
104	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	#11761628	—	2	2	no	—
109	#11761628	—	2	2	no	—
110	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
111	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
120	#11761612	—	2	2	no	—
121	#11761612	—	2	2	no	—
122	#11761611	—	2	2	no	—
123	#11761611	—	2	2	no	—
124	#11761611	—	2	2	no	—
125	#11761611	—	8	2	no	—
126	#11761611	M₁₆	1	4	yes	complemented
127	C₂×C₆₇⋊C₈	—	2	2	no	—
128	C₂×C₆₇⋊C₈	—	2	2	no	—
129	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
133	C₈×D₆₇	—	2	2	no	—
134	C₈×D₆₇	—	2	2	no	—
135	C₈×D₆₇	—	2	2	no	—
136	C₈×D₆₇	—	8	2	no	—
137	C₈×D₆₇	M₁₆	1	4	yes	complemented
138	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
139	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
140	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
141	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
142	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
143	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
144	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
145	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
146	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
147	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
148	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
149	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
150	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
154	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
155	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
159	M₁₆×C₆₇	—	8	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
171	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
185	C₂²⋊D₆₇	—	2	2	no	—
186	C₂²⋊D₆₇	—	2	2	no	—
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	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
193	D₂₆₈	—	2	2	no	—
194	C₄×D₆₇	—	2	1	no	—
195	C₄×D₆₇	—	2	1	no	—
196	C₄×D₆₇	—	2	2	no	—
197	C₄×D₆₇	—	2	2	no	—
198	C₄×D₆₇	—	4	1	no	—
199	C₄×D₆₇	—	4	1	no	—
200	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
201	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
202	Dic₁₃₄	—	2	2	no	—
203	C₈×C₆₇	—	2	1	no	—
204	C₈×C₆₇	—	2	1	no	—
205	C₈×C₆₇	—	4	2	no	—
206	C₈×C₆₇	—	4	2	no	—
207	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
208	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
209	C₆₇⋊C₈	—	2	1	no	—
210	C₆₇⋊C₈	—	2	1	no	—
211	C₆₇⋊C₈	—	4	2	no	—
212	C₆₇⋊C₈	—	4	2	no	—
213	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
214	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
219	C₂×D₆₇	—	2	1	no	—
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	#423	1	2	yes	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	2	1	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
230	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
231	C₄×C₆₇	#577	1	2	yes	—
232	Dic₆₇	—	2	1	no	—
233	Dic₆₇	—	2	1	no	—
234	Dic₆₇	—	2	1	no	—
235	Dic₆₇	—	2	1	no	—
236	Dic₆₇	—	2	1	no	—
237	Dic₆₇	#423	1	2	yes	—
238	#17798	—	67	1	no	maximal, Sylow
239	C₂×C₆₇	—	4	1	no	—
240	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
241	C₂×C₆₇	#2912	1	1	yes	characteristic
242	C₂×C₆₇	#2916	1	1	yes	characteristic
243	D₆₇	—	2	2	no	—
244	D₆₇	—	4	1	no	—
245	#2989	—	67	2	no	—
246	C₄×#131	—	67	1	no	—
247	#1569	—	67	2	no	—
248	#1569	—	67	2	no	—
249	#1568	—	67	1	no	—
250	#1568	—	67	1	no	—
251	#1566	—	67	1	no	—
252	#1566	D₆₇	1	1	yes	characteristic, complemented
253	#1565	—	67	2	no	—
254	#1564	—	67	1	no	—
255	C₂×#334	—	67	1	no	—
256	C₆₇	#17798	1	1	yes	characteristic, complemented, Sylow
257	C₂×#131	—	67	1	no	—
258	C₂×#131	—	67	1	no	—
259	C₄×#41	—	67	1	no	—
260	#446	—	134	2	no	—
261	#436	—	134	2	no	—
262	#435	—	134	2	no	—
263	#435	—	134	2	no	—
264	#432	—	67	2	no	—
265	#432	—	67	2	no	—
266	#431	—	67	2	no	—
267	#423	—	67	2	no	—
268	#423	C₂×D₆₇	1	2	yes	—
269	C₂×C₄⋊C₈	—	67	2	no	—
270	#405	—	67	1	no	—
271	#405	—	67	1	no	—
272	#405	—	67	1	no	—
273	#405	—	67	1	no	—
274	#405	—	67	1	no	—
275	#405	—	67	1	no	—
276	C₄×M₁₆	—	67	1	no	—
277	C₄×M₁₆	—	67	1	no	—
278	C₄×M₁₆	—	67	1	no	—
279	C₂×C₄×C₈	—	67	1	no	—
280	C₂×C₄×C₈	—	67	1	no	—
281	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
282	#335	—	67	2	no	—
283	#335	—	67	2	no	—
284	#335	—	67	2	no	—
285	#334	—	67	2	no	—
286	#334	—	67	2	no	—
287	#334	—	67	1	no	—
288	#334	—	67	1	no	—
289	#334	C₂×D₆₇	1	1	yes	characteristic
290	#334	C₂×D₆₇	1	1	yes	characteristic
291	#334	C₂×D₆₇	1	2	yes	—
292	C₂×#41	—	67	1	no	—
293	#131	—	134	4	no	—
294	#131	—	134	4	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂×M₁₆	—	67	1	no	—
297	C₂×M₁₆	—	67	1	no	—
298	C₂×M₁₆	—	67	1	no	—
299	C₂×M₁₆	—	67	1	no	—
300	C₂×M₁₆	—	67	1	no	—
301	C₂×M₁₆	—	67	1	no	—
302	C₂×M₁₆	—	134	2	no	—
303	C₂²×C₈	—	67	1	no	—
304	C₂²×C₈	—	67	1	no	—
305	C₂²×C₈	—	67	1	no	—
306	C₂²×C₈	—	67	1	no	—
307	C₂²×C₈	—	134	2	no	—
308	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
309	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
310	C₄×Q₈	—	67	1	no	—
311	C₄×Q₈	—	67	1	no	—
312	C₄×D₄	—	67	1	no	—
313	C₄×D₄	—	67	1	no	—
314	C₄×D₄	—	67	1	no	—
315	C₄×D₄	—	67	1	no	—
316	C₄×D₄	—	67	1	no	—
317	C₄×D₄	—	67	1	no	—
318	#117	—	67	1	no	—
319	#117	—	67	1	no	—
320	#117	—	67	1	no	—
321	C₂×C₄²	—	67	1	no	—
322	C₂×C₄²	—	67	1	no	—
323	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
324	C₄⋊C₈	—	2	2	no	—
325	C₄⋊C₈	—	67	2	no	—
326	C₄⋊C₈	—	67	2	no	—
327	C₄⋊C₈	—	134	2	no	—
328	C₄⋊C₈	—	134	2	no	—
329	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
330	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
331	C₂²⋊C₈	—	134	2	no	—
332	C₂²⋊C₈	—	134	2	no	—
333	C₂²⋊C₈	—	134	2	no	—
334	#97	—	67	1	no	—
335	#97	—	67	1	no	—
336	#97	—	67	1	no	—
337	#97	—	67	1	no	—
338	#97	—	67	1	no	—
339	#97	—	67	1	no	—
340	#97	—	134	2	no	—
341	#97	—	134	2	no	—
342	C₄×C₈	—	67	1	no	—
343	C₄×C₈	—	67	1	no	—
344	C₄×C₈	—	67	1	no	—
345	C₄×C₈	—	67	1	no	—
346	C₄×C₈	—	67	1	no	—
347	C₄×C₈	—	67	1	no	—
348	C₄×C₈	—	134	2	no	—
349	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
350	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
351	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
352	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
353	#41	—	134	2	no	—
354	#41	—	134	2	no	—
355	C₂×Q₈	—	67	1	no	—
356	C₂×D₄	—	67	1	no	—
357	C₂×D₄	—	67	1	no	—
358	C₂×D₄	—	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	C₂×C₄×D₆₇	1	1	yes	characteristic
366	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
368	M₁₆	—	8	2	no	—
369	M₁₆	—	536	2	no	—
370	M₁₆	—	67	4	no	—
371	M₁₆	—	134	2	no	—
372	M₁₆	—	134	2	no	—
373	M₁₆	—	134	2	no	—
374	M₁₆	—	134	2	no	—
375	M₁₆	—	134	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	536	2	no	—
381	C₂×C₈	—	67	1	no	—
382	C₂×C₈	—	67	1	no	—
383	C₂×C₈	—	67	4	no	—
384	C₂×C₈	—	67	1	no	—
385	C₂×C₈	—	67	1	no	—
386	C₂×C₈	—	134	2	no	—
387	C₂×C₈	—	134	2	no	—
388	C₂×C₈	—	134	2	no	—
389	C₂×C₈	—	134	2	no	—
390	C₂×C₈	—	134	2	no	—
391	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
392	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
393	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
394	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
395	C₄⋊C₄	—	67	1	no	—
396	C₄⋊C₄	—	67	1	no	—
397	C₄⋊C₄	—	67	1	no	—
398	C₄⋊C₄	—	67	1	no	—
399	C₄⋊C₄	—	67	1	no	—
400	C₄⋊C₄	—	67	1	no	—
401	C₂²⋊C₄	—	67	1	no	—
402	C₂²⋊C₄	—	67	1	no	—
403	C₂²⋊C₄	—	67	1	no	—
404	C₂²⋊C₄	—	67	1	no	—
405	C₂²⋊C₄	—	67	1	no	—
406	C₂²⋊C₄	—	67	1	no	—
407	C₄²	—	67	1	no	—
408	C₄²	—	67	1	no	—
409	C₄²	—	67	1	no	—
410	C₄²	—	67	1	no	—
411	C₄²	—	67	1	no	—
412	C₄²	—	67	1	no	—
413	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
414	C₄²	D₄×D₆₇	1	1	yes	characteristic
415	C₄²	Q₈×D₆₇	1	1	yes	characteristic
416	C₄²	C₂³×D₆₇	1	1	yes	characteristic
417	C₂³	—	67	1	no	—
418	C₂³	—	67	1	no	—
419	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
420	Q₈	—	134	2	no	—
421	D₄	—	134	2	no	—
422	D₄	—	134	2	no	—
423	D₄	—	134	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	268	1	no	—
429	C₂×C₄	—	268	1	no	—
430	C₂×C₄	—	67	1	no	—
431	C₂×C₄	—	67	2	no	—
432	C₂×C₄	—	67	2	no	—
433	C₂×C₄	—	67	1	no	—
434	C₂×C₄	—	67	1	no	—
435	C₂×C₄	—	67	1	no	—
436	C₂×C₄	—	67	1	no	—
437	C₂×C₄	—	67	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	C₄⋊C₄×D₆₇	1	1	yes	characteristic
443	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
444	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
445	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
448	C₈	—	2	1	no	—
449	C₈	—	2	1	no	—
450	C₈	—	4	2	no	—
451	C₈	—	4	2	no	—
452	C₈	—	268	2	no	—
453	C₈	—	268	2	no	—
454	C₈	—	67	2	no	—
455	C₈	—	67	2	no	—
456	C₈	—	134	1	no	—
457	C₈	—	134	1	no	—
458	C₈	M₁₆×D₆₇	1	2	yes	—
459	C₈	M₁₆×D₆₇	1	2	yes	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	67	2	no	—
464	C₂²	—	134	1	no	—
465	C₂²	—	134	1	no	—
466	C₂²	—	134	1	no	—
467	C₂²	—	134	1	no	—
468	C₂²	—	134	1	no	—
469	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	67	2	no	—
475	C₄	—	134	1	no	—
476	C₄	—	134	1	no	—
477	C₄	—	134	1	no	—
478	C₄	—	134	1	no	—
479	C₄	—	134	1	no	—
480	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
481	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
482	C₄	#16864202	1	2	yes	—
483	C₂	—	4	1	no	—
484	C₂	—	268	1	no	—
485	C₂	—	134	2	no	—
486	C₂	#423×D₆₇	1	1	yes	characteristic
487	C₂	#22568800	1	1	yes	characteristic
488	C₂	#22568817	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×#1566	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#17798
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #22558974	C₂	semidirect
4. #22558973	C₂	semidirect
5. #22558972	C₂	semidirect
6. #22558969	C₂	semidirect
7. #22558963	C₂	semidirect
8. #22558958	C₂	semidirect
9. #22558954	C₂	semidirect
10. #334×D₆₇	C₂	semidirect
11. C₆₇×#1566	C₂	semidirect
126. #11761611	M₁₆	semidirect
137. C₈×D₆₇	M₁₆	semidirect
252. #1566	D₆₇	semidirect
256. C₆₇	#17798	semidirect

Maximal subgroups

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

subgroup	index
#22569323	2
C₄×#14524049	2
#22558974	2
#22558973	2
#22558972	2
#22558969	2
#22558963	2
#22558958	2
#22558954	2
#334×D₆₇	2
C₆₇×#1566	2
#17798	67