Subgroups — #30841109

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 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×#16863596	C₂	1	1	yes	maximal, characteristic
3	#22569290	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559627	C₂	1	4	yes	maximal, complemented
5	#22559559	C₂	1	4	yes	maximal, complemented
6	#22558981	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558878	C₂	1	1	yes	maximal, characteristic
8	#22558866	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11761615	—	2	1	no	—
11	C₂×#14524040	C₂²	1	1	yes	characteristic
12	#16863770	C₂²	1	2	yes	—
13	#16863744	C₂²	1	2	yes	—
14	#16863596	—	2	2	no	—
15	#16863596	—	2	2	no	—
16	#16863596	C₂²	1	4	yes	—
17	#16863546	—	2	2	no	—
18	Q₈×C₆₇⋊C₈	—	2	2	no	—
19	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	complemented
20	#16863449	—	2	2	no	—
21	#16863449	C₂²	1	1	yes	characteristic
22	#16863449	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×#14523963	—	2	1	no	—
25	#16863146	C₂²	1	4	yes	complemented
26	#16863145	C₂²	1	2	yes	complemented
27	#16863142	C₂²	1	2	yes	complemented
28	#16863139	C₂²	1	4	yes	—
29	#533×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	#479×C₆₇	—	2	2	no	—
32	#477×C₆₇	C₂²	1	2	yes	complemented
33	#475×C₆₇	C₂²	1	2	yes	complemented
34	#428×C₆₇	—	2	1	no	—
35	#423×C₆₇	C₂²	1	1	yes	characteristic
36	#419×C₆₇	—	2	1	no	—
37	C₂²×D₂₆₈	—	2	1	no	—
38	C₂²×D₂₆₈	—	2	1	no	—
39	C₂²×D₂₆₈	—	2	1	no	—
40	C₂×#11761615	—	2	4	no	—
41	C₂×#11761615	—	2	4	no	—
42	C₂×#11761615	—	4	4	no	—
43	C₂²×C₆₇⋊C₈	—	2	1	no	—
44	#14524040	—	2	1	no	—
45	#14524040	—	2	1	no	—
46	#14524040	—	2	1	no	—
47	#14524040	C₂³	1	2	yes	—
48	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
51	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
52	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
53	#14523963	—	2	4	no	—
54	#14523963	—	2	4	no	—
55	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
57	C₄×C₆₇⋊C₈	—	2	4	no	—
58	C₄×C₆₇⋊C₈	—	2	1	no	—
59	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
60	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂×M₁₆×C₆₇	—	2	1	no	—
62	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
64	#125×C₆₇	—	2	1	no	—
65	C₂²⋊Q₈×C₆₇	—	2	1	no	—
66	C₂²⋊Q₈×C₆₇	—	2	1	no	—
67	C₄×Q₈×C₆₇	—	2	1	no	—
68	C₄×Q₈×C₆₇	—	2	1	no	—
69	C₄×Q₈×C₆₇	—	2	1	no	—
70	C₄×Q₈×C₆₇	—	2	1	no	—
71	C₄×Q₈×C₆₇	D₄	1	2	yes	—
72	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
73	#117×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	—	2	2	no	—
79	#106×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	1	no	—
81	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	C₂³×D₆₇	—	4	1	no	—
85	C₂×D₂₆₈	—	2	2	no	—
86	C₂×D₂₆₈	—	2	2	no	—
87	C₂×D₂₆₈	—	2	2	no	—
88	C₂×D₂₆₈	—	4	2	no	—
89	C₂×D₂₆₈	—	4	2	no	—
90	C₂×D₂₆₈	—	4	1	no	—
91	C₂×D₂₆₈	—	4	1	no	—
92	C₂×D₂₆₈	—	4	1	no	—
93	C₂×D₂₆₈	—	8	2	no	—
94	C₂×D₂₆₈	—	8	2	no	—
95	#11761615	—	4	8	no	—
96	#11761615	—	8	4	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	—	4	2	no	—
100	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
101	C₂×Q₈×C₆₇	—	2	1	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	complemented
106	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
112	M₁₆×C₆₇	—	4	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²⋊C₄×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	4	1	no	—
128	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	#121	1	1	yes	characteristic
133	C₂²×D₆₇	—	4	2	no	—
134	C₂²×D₆₇	—	8	2	no	—
135	C₂²×D₆₇	—	8	2	no	—
136	C₂²×D₆₇	—	8	2	no	—
137	Q₈×C₆₇	—	2	2	no	—
138	Q₈×C₆₇	—	2	4	no	—
139	Q₈×C₆₇	—	4	2	no	—
140	Q₈×C₆₇	—	8	1	no	—
141	Q₈×C₆₇	C₂×SD₁₆	1	4	yes	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	8	1	no	—
150	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₆₇	#135	1	2	yes	—
155	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
158	D₂₆₈	—	4	2	no	—
159	D₂₆₈	—	4	2	no	—
160	D₂₆₈	—	4	4	no	—
161	D₂₆₈	—	8	2	no	—
162	D₂₆₈	—	8	2	no	—
163	D₂₆₈	—	8	2	no	—
164	D₂₆₈	—	8	2	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₆₇⋊C₈	—	2	4	no	—
167	C₆₇⋊C₈	—	4	2	no	—
168	C₆₇⋊C₈	—	4	2	no	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	#465	1	2	yes	—
172	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
173	C₂×D₆₇	—	8	2	no	—
174	C₂×D₆₇	—	8	2	no	—
175	C₂×D₆₇	—	8	2	no	—
176	C₂×D₆₇	—	16	4	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	4	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	—	8	1	no	—
183	C₄×C₆₇	—	8	1	no	—
184	C₄×C₆₇	#465	1	2	yes	—
185	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
186	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
187	C₄×C₆₇	#577	1	2	yes	—
188	#19396	—	67	1	no	maximal, Sylow
189	C₂×C₆₇	—	2	2	no	—
190	C₂×C₆₇	#3043	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#465	1	1	yes	characteristic
192	C₂×C₆₇	#3068	1	1	yes	characteristic
193	D₆₇	—	16	2	no	—
194	#3208	—	67	1	no	—
195	#3078	D₆₇	1	1	yes	characteristic, complemented
196	C₂×#460	—	67	1	no	—
197	#2982	—	67	1	no	—
198	#1678	—	67	4	no	—
199	#1666	—	67	4	no	—
200	#1570	—	67	1	no	—
201	#1545	—	67	1	no	—
202	#1542	—	67	1	no	—
203	C₆₇	#19396	1	1	yes	characteristic, complemented, Sylow
204	C₂²×SD₁₆	—	134	1	no	—
205	#533	C₂×D₆₇	1	1	yes	characteristic
206	C₂×#127	—	67	1	no	—
207	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
208	#492	—	67	2	no	—
209	#486	—	67	2	no	—
210	#479	—	2	2	no	—
211	#477	C₂×D₆₇	1	2	yes	complemented
212	#475	C₂×D₆₇	1	2	yes	complemented
213	#460	—	67	4	no	—
214	#460	—	134	2	no	—
215	#460	—	134	2	no	—
216	#450	—	134	2	no	—
217	C₈×Q₈	—	67	4	no	—
218	C₈×Q₈	—	134	2	no	—
219	#431	—	67	1	no	—
220	#431	—	67	1	no	—
221	#431	—	134	2	no	—
222	#428	—	2	1	no	—
223	#423	C₂×D₆₇	1	1	yes	characteristic
224	C₂×C₄⋊C₈	—	67	1	no	—
225	#419	—	2	1	no	—
226	C₂×#102	—	134	1	no	—
227	#334	—	67	4	no	—
228	#333	—	67	2	no	—
229	#332	—	67	2	no	—
230	#331	—	67	4	no	—
231	C₂²×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
232	C₂²×D₄	—	134	1	no	—
233	C₂²×D₄	—	134	1	no	—
234	C₂²×D₄	—	134	1	no	—
235	C₂×SD₁₆	—	268	4	no	—
236	C₂×SD₁₆	—	134	4	no	—
237	C₂×SD₁₆	—	134	4	no	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂²×C₈	—	134	1	no	—
240	C₄⋊Q₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₄⋊Q₈	C₂²×D₆₇	1	1	yes	characteristic
242	#127	—	67	2	no	—
243	#127	—	134	1	no	—
244	#127	—	134	1	no	—
245	#127	—	134	1	no	—
246	#125	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	C₂²⋊D₆₇	1	2	yes	—
254	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
255	#117	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
260	#107	—	2	2	no	—
261	#106	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	67	2	no	—
264	C₄⋊C₈	—	67	2	no	—
265	C₄⋊C₈	—	67	2	no	—
266	C₄⋊C₈	—	134	4	no	—
267	C₄⋊C₈	—	134	1	no	—
268	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
269	#103	—	2	2	no	—
270	#103	—	2	2	no	—
271	#102	—	134	4	no	—
272	#102	—	134	4	no	—
273	C₂²⋊C₈	—	67	2	no	—
274	C₂²⋊C₈	—	134	1	no	—
275	C₄×C₈	—	67	2	no	—
276	C₄×C₈	—	134	4	no	—
277	C₄×C₈	—	134	1	no	—
278	C₂⁴	—	268	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	#11761615	1	4	yes	complemented
284	C₂×Q₈	C₂×C₂²⋊D₆₇	1	2	yes	—
285	C₂×D₄	—	268	1	no	—
286	C₂×D₄	—	268	1	no	—
287	C₂×D₄	—	268	1	no	—
288	C₂×D₄	—	268	2	no	—
289	C₂×D₄	—	268	2	no	—
290	C₂×D₄	—	536	2	no	—
291	C₂×D₄	—	536	2	no	—
292	C₂×D₄	—	134	2	no	—
293	C₂×D₄	—	134	2	no	—
294	C₂×D₄	—	134	2	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
299	C₂²×C₄	#11761627	1	1	yes	characteristic
300	SD₁₆	—	268	8	no	—
301	SD₁₆	—	536	4	no	—
302	M₁₆	—	4	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	268	2	no	—
305	C₂×C₈	—	67	4	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	—	134	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
318	C₂²⋊C₄	—	4	1	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	D₄×D₆₇	1	1	yes	characteristic
323	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
324	C₄²	#11761627	1	1	yes	characteristic
325	C₄²	C₂³×D₆₇	1	1	yes	characteristic
326	C₂³	—	268	2	no	—
327	C₂³	—	536	2	no	—
328	C₂³	—	536	2	no	—
329	C₂³	—	536	2	no	—
330	C₂³	#14524015	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	8	1	no	—
335	Q₈	C₂×#11761615	1	4	yes	—
336	D₄	—	268	2	no	—
337	D₄	—	268	2	no	—
338	D₄	—	268	4	no	—
339	D₄	—	536	2	no	—
340	D₄	—	536	2	no	—
341	D₄	—	536	2	no	—
342	D₄	—	536	2	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	#14524015	1	1	yes	characteristic
352	C₂×C₄	#14524015	1	1	yes	characteristic
353	C₂×C₄	#14524015	1	1	yes	characteristic
354	C₂×C₄	C₂×#11761615	1	2	yes	—
355	C₂×C₄	#14524063	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	268	2	no	—
361	C₈	—	268	2	no	—
362	C₈	—	134	4	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	536	2	no	—
366	C₂²	—	536	2	no	—
367	C₂²	—	536	2	no	—
368	C₂²	—	1072	4	no	—
369	C₂²	#16863627	1	2	yes	—
370	C₂²	C₂×#14524015	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	4	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	#16863627	1	2	yes	—
379	C₄	C₂²×#11761615	1	1	yes	characteristic
380	C₄	C₂×#14524063	1	1	yes	characteristic
381	C₄	#16864205	1	2	yes	—
382	C₂	—	2	2	no	—
383	C₂	—	1072	2	no	—
384	C₂	#22569732	1	1	yes	characteristic
385	C₂	C₂×#16863627	1	1	yes	characteristic
386	C₂	#22569939	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524015
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3078	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#14524040	C₂²
hypercenter	#3078	D₆₇
hypocenter	C₆₇	#19396
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3078

Complements

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

N	H ≅ G/N	type
3. #22569290	C₂	semidirect
4. #22559627	C₂	semidirect
5. #22559559	C₂	semidirect
6. #22558981	C₂	semidirect
8. #22558866	C₂	semidirect
9. C₆₇×#3078	C₂	semidirect
19. Q₈×C₆₇⋊C₈	C₂²	semidirect
25. #16863146	C₂²	semidirect
26. #16863145	C₂²	semidirect
27. #16863142	C₂²	semidirect
32. #477×C₆₇	C₂²	semidirect
33. #475×C₆₇	C₂²	semidirect
105. C₂×Q₈×C₆₇	SD₁₆	semidirect
195. #3078	D₆₇	semidirect
203. C₆₇	#19396	semidirect
211. #477	C₂×D₆₇	semidirect
212. #475	C₂×D₆₇	semidirect
283. C₂×Q₈	#11761615	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×#16863596	2
#22569290	2
#22559627	2
#22559559	2
#22558981	2
#22558878	2
#22558866	2
C₆₇×#3078	2
#19396	67