Subgroups — #30839109

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571147	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863744	C₂	1	1	yes	maximal, characteristic
3	#22569144	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559751	C₂	1	4	yes	maximal, complemented
5	#22559627	C₂	1	4	yes	maximal, complemented
6	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558890	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558876	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#3230	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524040	C₂²	1	1	yes	characteristic
11	#16863770	—	2	1	no	—
12	#16863744	—	2	1	no	—
13	#16863744	—	2	1	no	—
14	#16863744	—	2	1	no	—
15	#16863744	C₂²	1	2	yes	—
16	#16863744	C₂²	1	2	yes	—
17	#16863596	C₂²	1	4	yes	—
18	#16863546	—	2	2	no	—
19	#16863527	—	2	1	no	—
20	#16863527	C₂²	1	2	yes	complemented
21	Q₈×C₆₇⋊C₈	—	2	1	no	—
22	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
23	#16863449	C₂²	1	1	yes	characteristic
24	#16863423	—	2	1	no	—
25	#16863420	C₂²	1	1	yes	characteristic
26	C₂×#14523963	—	2	1	no	—
27	C₂×#14523963	—	2	1	no	—
28	#16863352	—	2	1	no	—
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	#16863148	C₂²	1	2	yes	complemented
31	#16863146	C₂²	1	2	yes	complemented
32	#16863145	C₂²	1	4	yes	complemented
33	#16863139	C₂²	1	4	yes	—
34	#533×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
36	#500×C₆₇	—	2	1	no	—
37	#487×C₆₇	—	2	1	no	—
38	#484×C₆₇	—	2	2	no	—
39	#477×C₆₇	C₂²	1	4	yes	complemented
40	#431×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×D₂₆₈	—	2	1	no	—
42	C₂²×D₂₆₈	—	2	1	no	—
43	C₂²×D₂₆₈	—	2	1	no	—
44	C₂×#11761615	—	2	4	no	—
45	C₂×#11761612	—	2	1	no	—
46	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
48	#14524040	—	2	1	no	—
49	#14524040	—	2	1	no	—
50	#14524040	—	2	1	no	—
51	#14524040	C₂³	1	2	yes	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
55	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
56	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
57	#14523963	—	2	2	no	—
58	#14523963	—	2	2	no	—
59	#14523963	—	2	2	no	—
60	#14523963	—	2	2	no	—
61	#14523963	—	2	4	no	—
62	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
63	#14523954	—	2	2	no	—
64	C₄×C₆₇⋊C₈	—	2	2	no	—
65	C₄×C₆₇⋊C₈	—	2	1	no	—
66	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₇	—	2	1	no	—
71	C₄⋊Q₈×C₆₇	—	2	1	no	—
72	C₄⋊Q₈×C₆₇	—	2	1	no	—
73	C₄⋊Q₈×C₆₇	—	2	1	no	—
74	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
77	#125×C₆₇	—	2	1	no	—
78	C₂²⋊Q₈×C₆₇	—	2	1	no	—
79	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
80	C₄×Q₈×C₆₇	—	2	1	no	—
81	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
82	#117×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	#107×C₆₇	—	2	4	no	—
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#103×C₆₇	—	2	4	no	—
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₄×C₈×C₆₇	—	2	1	no	—
91	C₂³×D₆₇	—	4	1	no	—
92	C₂×D₂₆₈	—	2	2	no	—
93	C₂×D₂₆₈	—	2	2	no	—
94	C₂×D₂₆₈	—	2	2	no	—
95	C₂×D₂₆₈	—	4	2	no	—
96	C₂×D₂₆₈	—	4	2	no	—
97	C₂×D₂₆₈	—	4	1	no	—
98	C₂×D₂₆₈	—	4	1	no	—
99	C₂×D₂₆₈	—	4	1	no	—
100	C₂×D₂₆₈	—	8	2	no	—
101	C₂×D₂₆₈	—	8	2	no	—
102	#11761615	—	8	4	no	—
103	#11761612	—	4	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	4	1	no	—
115	C₂×Q₈×C₆₇	—	8	2	no	—
116	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	—	4	1	no	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	4	1	no	—
130	C₄⋊C₄×C₆₇	—	4	1	no	—
131	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
132	C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₆₇	#41	1	1	yes	characteristic
137	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂³×C₆₇	#121	1	1	yes	characteristic
139	C₂²×D₆₇	—	4	2	no	—
140	C₂²×D₆₇	—	8	2	no	—
141	C₂²×D₆₇	—	8	2	no	—
142	C₂²×D₆₇	—	8	2	no	—
143	Q₈×C₆₇	—	2	4	no	—
144	Q₈×C₆₇	—	4	2	no	—
145	Q₈×C₆₇	—	8	2	no	—
146	Q₈×C₆₇	—	8	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	8	2	no	—
153	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#135	1	2	yes	—
157	C₂×C₄×C₆₇	#137	1	2	yes	—
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
161	D₂₆₈	—	4	2	no	—
162	D₂₆₈	—	4	2	no	—
163	D₂₆₈	—	4	4	no	—
164	D₂₆₈	—	8	2	no	—
165	D₂₆₈	—	8	2	no	—
166	D₂₆₈	—	8	2	no	—
167	D₂₆₈	—	8	2	no	—
168	C₈×C₆₇	—	4	2	no	—
169	C₆₇⋊C₈	—	2	1	no	—
170	C₆₇⋊C₈	—	2	1	no	—
171	C₆₇⋊C₈	—	2	1	no	—
172	C₆₇⋊C₈	—	2	1	no	—
173	C₆₇⋊C₈	—	4	2	no	—
174	C₆₇⋊C₈	—	4	2	no	—
175	C₂²×C₆₇	—	2	1	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	#464	1	2	yes	—
178	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
179	C₂×D₆₇	—	8	2	no	—
180	C₂×D₆₇	—	8	2	no	—
181	C₂×D₆₇	—	8	2	no	—
182	C₂×D₆₇	—	16	4	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	—	8	2	no	—
188	C₄×C₆₇	#464	1	2	yes	—
189	C₄×C₆₇	#472	1	2	yes	—
190	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
191	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
192	C₄×C₆₇	#577	1	2	yes	—
193	#19212	—	67	1	no	maximal, Sylow
194	C₂×C₆₇	—	2	2	no	—
195	C₂×C₆₇	C₂×#464	1	1	yes	characteristic
196	C₂×C₆₇	#3057	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
198	D₆₇	—	16	2	no	—
199	#3230	D₆₇	1	1	yes	characteristic, complemented
200	#3208	—	67	1	no	—
201	C₂×#486	—	67	1	no	—
202	#2960	—	67	1	no	—
203	#1698	—	67	4	no	—
204	#1678	—	67	4	no	—
205	#1564	—	67	1	no	—
206	#1548	—	67	1	no	—
207	#1545	—	67	1	no	—
208	C₆₇	#19212	1	1	yes	characteristic, complemented, Sylow
209	#533	C₂×D₆₇	1	1	yes	characteristic
210	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
211	C₂×#127	—	67	1	no	—
212	#500	—	2	1	no	—
213	#492	—	134	1	no	—
214	#487	—	2	1	no	—
215	#486	—	67	2	no	—
216	#486	—	67	2	no	—
217	#486	—	134	1	no	—
218	#486	—	134	1	no	—
219	#486	—	134	1	no	—
220	#484	—	2	2	no	—
221	#477	C₂×D₆₇	1	4	yes	complemented
222	#460	—	67	4	no	—
223	#450	—	134	2	no	—
224	#446	—	67	2	no	—
225	#446	—	134	1	no	—
226	C₈×Q₈	—	67	2	no	—
227	C₈×Q₈	—	134	1	no	—
228	#431	—	67	1	no	—
229	#431	C₂×D₆₇	1	1	yes	characteristic
230	#424	—	134	1	no	—
231	#423	—	67	1	no	—
232	C₂×#102	—	134	1	no	—
233	C₂×#102	—	134	1	no	—
234	#405	—	134	1	no	—
235	C₂×C₄×C₈	—	67	1	no	—
236	#335	—	67	2	no	—
237	#334	—	67	2	no	—
238	#333	—	67	4	no	—
239	#331	—	67	4	no	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×D₄	—	134	1	no	—
242	C₂²×D₄	—	134	1	no	—
243	C₂²×D₄	—	134	1	no	—
244	C₂×SD₁₆	—	134	4	no	—
245	C₂×M₁₆	—	134	1	no	—
246	C₂²×C₈	—	67	1	no	—
247	C₂²×C₈	—	67	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
252	C₄⋊Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
253	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
254	#127	—	67	2	no	—
255	#127	—	134	1	no	—
256	#127	—	134	1	no	—
257	#127	—	134	1	no	—
258	#125	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	C₂²⋊D₆₇	1	2	yes	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
263	#117	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
267	#107	—	2	4	no	—
268	C₄⋊C₈	—	67	2	no	—
269	C₄⋊C₈	—	67	2	no	—
270	C₄⋊C₈	—	134	2	no	—
271	C₄⋊C₈	—	134	2	no	—
272	C₄⋊C₈	—	134	1	no	—
273	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
274	#103	—	2	4	no	—
275	#102	—	134	2	no	—
276	#102	—	134	2	no	—
277	#102	—	134	2	no	—
278	#102	—	134	2	no	—
279	#102	—	134	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	134	1	no	—
282	#97	—	134	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	67	1	no	—
285	C₄×C₈	—	67	1	no	—
286	C₄×C₈	—	67	1	no	—
287	C₄×C₈	—	67	1	no	—
288	C₄×C₈	—	134	2	no	—
289	C₄×C₈	—	134	1	no	—
290	C₂⁴	—	268	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×Q₈	C₂×C₂²⋊D₆₇	1	2	yes	—
296	C₂×D₄	—	268	1	no	—
297	C₂×D₄	—	268	1	no	—
298	C₂×D₄	—	268	1	no	—
299	C₂×D₄	—	268	2	no	—
300	C₂×D₄	—	268	2	no	—
301	C₂×D₄	—	536	2	no	—
302	C₂×D₄	—	536	2	no	—
303	C₂×D₄	—	134	2	no	—
304	C₂×D₄	—	134	2	no	—
305	C₂×D₄	—	134	2	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₂²×C₄	#11761627	1	1	yes	characteristic
310	SD₁₆	—	536	4	no	—
311	M₁₆	—	268	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	67	1	no	—
314	C₂×C₈	—	67	1	no	—
315	C₂×C₈	—	67	1	no	—
316	C₂×C₈	—	67	1	no	—
317	C₂×C₈	—	134	2	no	—
318	C₂×C₈	—	134	2	no	—
319	C₂×C₈	—	134	2	no	—
320	C₂×C₈	—	134	2	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	D₄×D₆₇	1	1	yes	characteristic
334	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
335	C₄²	#11761627	1	1	yes	characteristic
336	C₄²	C₂³×D₆₇	1	1	yes	characteristic
337	C₂³	—	268	2	no	—
338	C₂³	—	536	2	no	—
339	C₂³	—	536	2	no	—
340	C₂³	—	536	2	no	—
341	C₂³	#14524015	1	1	yes	characteristic
342	Q₈	—	2	4	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	8	2	no	—
346	D₄	—	268	2	no	—
347	D₄	—	268	2	no	—
348	D₄	—	268	4	no	—
349	D₄	—	536	2	no	—
350	D₄	—	536	2	no	—
351	D₄	—	536	2	no	—
352	D₄	—	536	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	#14524015	1	1	yes	characteristic
360	C₂×C₄	#14524015	1	1	yes	characteristic
361	C₂×C₄	#14524015	1	1	yes	characteristic
362	C₂×C₄	#14524066	1	2	yes	—
363	C₂×C₄	#14524075	1	2	yes	—
364	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
367	C₈	—	4	2	no	—
368	C₈	—	268	2	no	—
369	C₈	—	268	2	no	—
370	C₈	—	134	1	no	—
371	C₈	—	134	1	no	—
372	C₈	—	134	1	no	—
373	C₈	—	134	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	536	2	no	—
377	C₂²	—	536	2	no	—
378	C₂²	—	536	2	no	—
379	C₂²	—	1072	4	no	—
380	C₂²	#16863620	1	2	yes	—
381	C₂²	C₂×#14524015	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	—	8	2	no	—
387	C₄	#16863620	1	2	yes	—
388	C₄	#16863664	1	2	yes	—
389	C₄	C₂×#14524066	1	1	yes	characteristic
390	C₄	C₂×#14524075	1	1	yes	characteristic
391	C₄	#16864204	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	1072	2	no	—
394	C₂	C₂×#16863620	1	1	yes	characteristic
395	C₂	#22569850	1	1	yes	characteristic
396	C₂	C₂×#16863664	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524015
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3230	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#14524040	C₂²
hypercenter	#3230	D₆₇
hypocenter	C₆₇	#19212
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3230

Complements

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

N	H ≅ G/N	type
3. #22569144	C₂	semidirect
4. #22559751	C₂	semidirect
5. #22559627	C₂	semidirect
6. #22558955	C₂	semidirect
7. #22558890	C₂	semidirect
9. C₆₇×#3230	C₂	semidirect
20. #16863527	C₂²	semidirect
22. Q₈×C₆₇⋊C₈	C₂²	semidirect
30. #16863148	C₂²	semidirect
31. #16863146	C₂²	semidirect
32. #16863145	C₂²	semidirect
39. #477×C₆₇	C₂²	semidirect
199. #3230	D₆₇	semidirect
208. C₆₇	#19212	semidirect
221. #477	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22571147	2
C₂×#16863744	2
#22569144	2
#22559751	2
#22559627	2
#22558955	2
#22558890	2
#22558876	2
C₆₇×#3230	2
#19212	67