Subgroups — #36711149

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#26031995	C₂	1	2	yes	maximal, complemented
4	#26022513	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022343	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022338	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022315	C₂	1	2	yes	maximal, complemented
8	#26022194	C₂	1	2	yes	maximal, complemented
9	#26022171	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	2	yes	—
13	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
14	#19152257	—	2	2	no	—
15	#19152030	—	2	4	no	—
16	C₂×Dic₇₁₂	—	2	4	no	—
17	C₂×Dic₇₁₂	C₂²	1	2	yes	—
18	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
19	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×#15667957	C₂²	1	2	yes	—
21	C₂×#15667957	C₂²	1	2	yes	—
22	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
23	#19151982	C₂²	1	2	yes	—
24	#19151839	C₂²	1	2	yes	—
25	#19151834	C₂²	1	2	yes	—
26	#19151821	—	2	2	no	—
27	#19151762	—	2	2	no	—
28	#19151739	—	2	2	no	—
29	C₄×Dic₃₅₆	C₂²	1	2	yes	—
30	#19151424	C₂²	1	1	yes	characteristic
31	#19151403	C₂²	1	2	yes	—
32	#19151403	C₂²	1	2	yes	—
33	#19151400	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	—	2	2	no	—
42	C₂²×Dic₁₇₈	—	2	2	no	—
43	#15668034	—	2	4	no	—
44	#15668034	—	4	2	no	—
45	C₂×Dic₃₅₆	—	2	4	no	—
46	C₂×Dic₃₅₆	—	2	2	no	—
47	C₂×Dic₃₅₆	—	4	2	no	—
48	C₂×Dic₃₅₆	C₂³	1	2	yes	—
49	C₂×Dic₃₅₆	C₂³	1	2	yes	—
50	C₂×#13083563	—	2	2	no	—
51	C₂×#13083563	—	2	2	no	—
52	C₂×D₇₁₂	C₂³	1	2	yes	—
53	#15668000	—	2	2	no	—
54	#15667998	—	4	2	no	—
55	#15667991	—	2	2	no	—
56	#15667990	—	4	2	no	—
57	#15667986	—	2	2	no	—
58	#15667984	—	4	2	no	—
59	C₄×Dic₁₇₈	—	2	2	no	—
60	#15667972	—	4	2	no	—
61	Dic₇₁₂	—	2	4	no	—
62	Dic₇₁₂	—	2	2	no	—
63	Dic₇₁₂	D₄	1	4	yes	complemented
64	#15667957	—	2	4	no	—
65	#15667957	—	4	2	no	—
66	D₁₄₂₄	—	2	2	no	—
67	#15667946	C₂³	1	2	yes	—
68	#15667937	—	2	2	no	—
69	#15667937	—	2	2	no	—
70	#15667934	—	2	2	no	—
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	—	2	2	no	—
75	C₂×C₁₆×C₈₉	—	2	2	no	—
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
81	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
82	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
83	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
84	#13083574	—	4	4	no	—
85	#13083574	—	8	2	no	—
86	C₂×Dic₁₇₈	—	2	2	no	—
87	C₂×Dic₁₇₈	—	2	2	no	—
88	C₂×Dic₁₇₈	—	4	4	no	—
89	C₂×Dic₁₇₈	—	4	2	no	—
90	C₂×Dic₁₇₈	—	8	2	no	—
91	C₂×D₃₅₆	—	2	2	no	—
92	C₂×C₂²⋊D₈₉	—	4	2	no	—
93	C₂×C₂²⋊D₈₉	—	4	2	no	—
94	C₂²×Dic₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	—	4	2	no	—
96	Dic₃₅₆	—	2	2	no	—
97	Dic₃₅₆	—	2	2	no	—
98	Dic₃₅₆	—	2	2	no	—
99	Dic₃₅₆	—	4	4	no	—
100	Dic₃₅₆	C₂×D₄	1	4	yes	—
101	#13083563	—	4	4	no	—
102	#13083563	—	8	2	no	—
103	D₇₁₂	—	2	2	no	—
104	D₇₁₂	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	2	no	—
106	#13083553	—	4	2	no	—
107	#13083553	—	4	2	no	—
108	C₂²⋊Dic₈₉	—	4	2	no	—
109	#13083551	—	4	2	no	—
110	#13083551	—	4	2	no	—
111	#13083551	—	4	2	no	—
112	C₄×Dic₈₉	—	4	2	no	—
113	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	—	2	1	no	—
117	C₂×C₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	—	2	1	no	—
119	C₂×C₈×C₈₉	—	2	1	no	—
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₈₉	D₈	1	1	yes	characteristic
126	C₂²⋊C₄×C₈₉	D₈	1	1	yes	characteristic
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₈₉	—	2	1	no	—
129	C₁₆×C₈₉	—	2	1	no	—
130	C₁₆×C₈₉	—	2	1	no	—
131	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
132	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₈₉	—	4	2	no	—
135	D₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₈₉	—	8	4	no	—
146	C₂²⋊D₈₉	—	8	4	no	—
147	C₂×Dic₈₉	—	4	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	8	4	no	—
151	D₃₅₆	—	4	2	no	—
152	D₃₅₆	—	4	2	no	—
153	C₄×D₈₉	—	8	4	no	—
154	Dic₁₇₈	—	2	4	no	—
155	Dic₁₇₈	—	4	2	no	—
156	Dic₁₇₈	—	4	2	no	—
157	Dic₁₇₈	—	4	2	no	—
158	Dic₁₇₈	—	8	4	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	2	1	no	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	#508	1	2	yes	complemented
170	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₈₉	—	8	2	no	—
172	C₂×D₈₉	—	8	2	no	—
173	C₂×D₈₉	—	8	2	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	D₄²	1	1	yes	characteristic
179	C₄×C₈₉	#576	1	1	yes	characteristic
180	Dic₈₉	—	4	4	no	—
181	Dic₈₉	—	8	2	no	—
182	Dic₈₉	—	8	2	no	—
183	#33851	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	—	2	2	no	—
185	C₂×C₈₉	—	4	1	no	—
186	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
187	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
188	C₂×C₈₉	#3417	1	1	yes	characteristic
189	D₈₉	—	16	2	no	—
190	C₂×#508	—	89	1	no	—
191	C₂²×Q₃₂	—	89	1	no	—
192	#3285	—	89	2	no	—
193	#2249	—	89	1	no	—
194	#2216	—	89	1	no	—
195	#2215	—	89	1	no	—
196	#2211	—	89	2	no	—
197	#2189	—	89	2	no	—
198	#2185	—	89	2	no	—
199	C₄×Q₃₂	—	89	1	no	—
200	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
201	C₈₉	#33851	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	89	2	no	—
203	C₂²×Q₁₆	—	89	2	no	—
204	#548	—	178	2	no	—
205	#508	—	178	4	no	—
206	C₂×Q₃₂	—	89	1	no	—
207	C₂×Q₃₂	—	89	2	no	—
208	C₂×Q₃₂	—	89	1	no	—
209	C₂×Q₃₂	—	178	4	no	—
210	C₂×SD₃₂	—	89	2	no	—
211	C₂×SD₃₂	—	89	2	no	—
212	C₂×D₁₆	—	89	1	no	—
213	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
214	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
215	#495	—	89	2	no	—
216	#467	—	89	2	no	—
217	#466	—	89	2	no	—
218	#464	—	178	2	no	—
219	#452	—	178	2	no	—
220	#448	—	178	2	no	—
221	C₄×Q₁₆	—	89	2	no	—
222	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
223	#366	—	89	1	no	—
224	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
225	#358	—	89	2	no	—
226	#358	—	89	2	no	—
227	#357	—	89	2	no	—
228	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
229	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
230	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
231	C₂×#41	—	178	2	no	—
232	C₂²×Q₈	—	178	2	no	—
233	#135	—	356	2	no	—
234	#135	—	178	4	no	—
235	C₂×Q₁₆	—	89	2	no	—
236	C₂×Q₁₆	—	89	2	no	—
237	C₂×Q₁₆	—	356	2	no	—
238	C₂×Q₁₆	—	178	4	no	—
239	C₂×Q₁₆	—	178	2	no	—
240	C₂×SD₁₆	—	178	2	no	—
241	C₂×SD₁₆	—	178	2	no	—
242	C₂×D₈	—	89	2	no	—
243	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	#124	—	356	2	no	—
246	#124	—	178	2	no	—
247	C₂²⋊Q₈	—	356	2	no	—
248	C₂²⋊Q₈	—	178	2	no	—
249	#121	—	356	2	no	—
250	#121	—	178	2	no	—
251	C₄×Q₈	—	178	2	no	—
252	C₄×D₄	—	356	2	no	—
253	C₄×D₄	D₃₅₆	1	1	yes	characteristic
254	Q₃₂	—	89	4	no	—
255	Q₃₂	—	178	4	no	—
256	Q₃₂	—	178	2	no	—
257	SD₃₂	—	356	2	no	—
258	SD₃₂	—	178	4	no	—
259	D₁₆	—	178	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	2	2	no	—
262	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
266	#107	—	89	2	no	—
267	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
268	#103	—	178	2	no	—
269	#103	—	178	2	no	—
270	#102	—	178	2	no	—
271	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
272	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	#41	—	356	4	no	—
275	#41	—	712	2	no	—
276	C₂×Q₈	—	356	4	no	—
277	C₂×Q₈	—	356	2	no	—
278	C₂×Q₈	—	178	2	no	—
279	C₂×Q₈	—	178	2	no	—
280	C₂×Q₈	—	712	2	no	—
281	C₂×D₄	—	356	2	no	—
282	C₂×D₄	—	356	2	no	—
283	C₂×D₄	—	178	2	no	—
284	C₂×D₄	D₇₁₂	1	1	yes	characteristic
285	C₂²×C₄	—	356	2	no	—
286	C₂²×C₄	—	356	2	no	—
287	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
289	Q₁₆	—	89	4	no	—
290	Q₁₆	—	356	4	no	—
291	Q₁₆	—	178	2	no	—
292	Q₁₆	—	178	2	no	—
293	Q₁₆	—	178	2	no	—
294	SD₁₆	—	356	4	no	—
295	SD₁₆	—	712	2	no	—
296	D₈	—	178	2	no	—
297	D₈	—	178	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
303	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
304	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
305	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
306	C₄⋊C₄	—	356	2	no	—
307	C₄⋊C₄	—	356	2	no	—
308	C₄⋊C₄	—	178	2	no	—
309	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
310	C₂²⋊C₄	—	356	2	no	—
311	C₂²⋊C₄	—	356	2	no	—
312	C₂²⋊C₄	—	356	2	no	—
313	C₂²⋊C₄	—	356	2	no	—
314	C₂²⋊C₄	D₇₁₂	1	1	yes	characteristic
315	C₂²⋊C₄	D₇₁₂	1	1	yes	characteristic
316	C₄²	—	356	2	no	—
317	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	D₄×D₈₉	1	2	yes	—
322	C₂³	—	356	2	no	—
323	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
324	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
325	Q₈	—	356	2	no	—
326	Q₈	—	356	2	no	—
327	Q₈	—	356	2	no	—
328	Q₈	—	178	4	no	—
329	Q₈	—	712	4	no	—
330	D₄	—	2	2	no	—
331	D₄	—	356	2	no	—
332	D₄	—	356	2	no	—
333	D₄	—	712	4	no	—
334	D₄	—	712	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	—	356	2	no	—
342	C₂×C₄	—	712	4	no	—
343	C₂×C₄	—	712	4	no	—
344	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
345	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
346	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
347	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
353	C₈	#15668058	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	712	2	no	—
360	C₂²	—	712	2	no	—
361	C₂²	—	712	2	no	—
362	C₂²	#19152030	1	2	yes	complemented
363	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	356	4	no	—
369	C₄	—	712	2	no	—
370	C₄	—	712	2	no	—
371	C₄	D₄×D₃₅₆	1	1	yes	characteristic
372	C₄	#19152403	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	1	no	—
375	C₂	—	1424	2	no	—
376	C₂	D₄×D₇₁₂	1	1	yes	characteristic
377	C₂	C₂×#19152030	1	1	yes	characteristic
378	C₂	#26033405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33851
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19152030	C₂	semidirect
2. C₂²×Dic₇₁₂	C₂	semidirect
3. #26031995	C₂	semidirect
4. #26022513	C₂	semidirect
5. #26022343	C₂	semidirect
6. #26022338	C₂	semidirect
7. #26022315	C₂	semidirect
8. #26022194	C₂	semidirect
9. #26022171	C₂	semidirect
10. C₄×Dic₇₁₂	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
63. Dic₇₁₂	D₄	semidirect
169. C₂²×C₈₉	#508	semidirect
200. D₄×C₁₆	D₈₉	semidirect
201. C₈₉	#33851	semidirect
362. C₂²	#19152030	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152030	2
C₂²×Dic₇₁₂	2
#26031995	2
#26022513	2
#26022343	2
#26022338	2
#26022315	2
#26022194	2
#26022171	2
C₄×Dic₇₁₂	2
D₄×C₁₆×C₈₉	2
#33851	89