Subgroups — #26031348

Order 11392.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152219	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152167	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151987	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151982	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151958	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151945	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151821	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151799	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151794	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151762	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151751	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151745	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151734	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083563	C₂²	1	1	yes	characteristic
21	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
22	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
25	C₂×D₇₁₂	C₂²	1	1	yes	characteristic, complemented
26	#15668012	C₂²	1	1	yes	characteristic, complemented
27	#15668009	C₂²	1	1	yes	characteristic
28	#15668009	C₂²	1	1	yes	characteristic
29	#15668007	—	2	1	no	—
30	#15668000	C₂²	1	1	yes	characteristic
31	#15668000	C₂²	1	1	yes	characteristic
32	#15668000	C₂²	1	1	yes	characteristic, complemented
33	#15667998	—	2	1	no	—
34	#15667994	—	2	1	no	—
35	#15667991	C₂²	1	1	yes	characteristic
36	#15667990	—	2	1	no	—
37	#15667987	—	2	1	no	—
38	#15667987	—	2	1	no	—
39	#15667986	C₂²	1	1	yes	characteristic
40	#15667982	—	2	1	no	—
41	#15667982	—	2	1	no	—
42	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
43	C₄×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
44	#15667937	C₂²	1	1	yes	characteristic, complemented
45	#15667937	C₂²	1	1	yes	characteristic, complemented
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667934	C₂²	1	1	yes	characteristic
50	#15667934	C₂²	1	1	yes	characteristic, complemented
51	#15667934	C₂²	1	1	yes	characteristic, complemented
52	#117×C₈₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
54	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#97×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	#13083574	—	4	1	no	—
61	#13083574	—	4	1	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	—	2	1	no	—
70	C₂×D₃₅₆	—	4	1	no	—
71	C₂×D₃₅₆	—	4	1	no	—
72	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
73	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
74	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
75	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	—	2	1	no	—
77	C₂×C₂²⋊D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	—	2	1	no	—
80	Dic₃₅₆	—	4	1	no	—
81	#13083563	—	4	1	no	—
82	#13083563	—	4	1	no	—
83	#13083563	—	4	1	no	—
84	#13083563	—	4	1	no	—
85	D₇₁₂	—	4	1	no	—
86	D₇₁₂	—	4	1	no	—
87	D₇₁₂	—	4	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
91	#13083553	—	2	1	no	—
92	#13083553	—	2	1	no	—
93	C₂²⋊Dic₈₉	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	#13083551	—	2	1	no	—
101	#13083551	—	2	1	no	—
102	C₄×Dic₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
111	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂³	1	1	yes	characteristic
113	C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	1	no	—
116	C₂²×D₈₉	—	2	1	no	—
117	C₂²×D₈₉	—	2	1	no	—
118	C₂²×D₈₉	—	2	1	no	—
119	C₂²×D₈₉	—	4	2	no	—
120	C₂²×D₈₉	—	8	1	no	—
121	C₂²×D₈₉	—	8	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂²⋊D₈₉	—	8	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	2	1	no	—
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	D₃₅₆	—	2	1	no	—
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	—	4	2	no	—
145	D₃₅₆	—	8	1	no	—
146	D₃₅₆	—	8	1	no	—
147	D₃₅₆	—	8	1	no	—
148	D₃₅₆	—	8	1	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	8	1	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	—	2	1	no	—
153	Dic₁₇₈	—	2	1	no	—
154	Dic₁₇₈	—	2	1	no	—
155	Dic₁₇₈	—	8	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	2	1	no	—
158	C₈×C₈₉	—	2	1	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₂×D₈₉	—	4	1	no	—
172	C₂×D₈₉	—	4	1	no	—
173	C₂×D₈₉	—	4	1	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	8	4	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
184	Dic₈₉	—	4	1	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	C₂×C₈₉	—	4	1	no	—
189	C₂×C₈₉	#535	1	1	yes	characteristic
190	C₂×C₈₉	#576	1	1	yes	characteristic
191	C₂×C₈₉	#577	1	1	yes	characteristic
192	D₈₉	—	8	1	no	—
193	D₈₉	—	8	1	no	—
194	D₈₉	—	8	1	no	—
195	D₈₉	—	8	1	no	—
196	#3224	—	89	1	no	maximal, Sylow
197	C₈₉	#3224	1	1	yes	characteristic, complemented, Sylow
198	#542	—	89	1	no	—
199	#535	—	89	1	no	—
200	#496	—	89	1	no	—
201	#495	—	89	1	no	—
202	#489	—	89	1	no	—
203	#486	—	89	1	no	—
204	#464	—	89	1	no	—
205	#463	—	89	1	no	—
206	#460	—	89	1	no	—
207	#459	—	89	1	no	—
208	#452	—	89	1	no	—
209	#450	—	89	1	no	—
210	#449	—	89	1	no	—
211	#447	—	89	1	no	—
212	#433	D₈₉	1	1	yes	characteristic, complemented
213	C₂×#41	—	89	1	no	—
214	C₂²×D₄	—	89	1	no	—
215	C₂×Q₁₆	—	89	1	no	—
216	C₂×SD₁₆	—	89	1	no	—
217	C₂×SD₁₆	—	89	1	no	—
218	C₂×SD₁₆	—	89	1	no	—
219	C₂×SD₁₆	—	89	1	no	—
220	C₂×D₈	—	89	1	no	—
221	C₂×D₈	—	89	1	no	—
222	C₂×D₈	—	89	1	no	—
223	C₄⋊Q₈	—	89	1	no	—
224	#127	—	89	1	no	—
225	#127	—	89	1	no	—
226	#126	—	178	1	no	—
227	#124	—	89	1	no	—
228	#124	—	89	1	no	—
229	#124	—	89	1	no	—
230	#124	—	178	1	no	—
231	#123	—	178	1	no	—
232	C₂²⋊Q₈	—	89	1	no	—
233	C₂²⋊Q₈	—	178	1	no	—
234	#121	—	89	1	no	—
235	#121	—	178	1	no	—
236	#121	—	178	1	no	—
237	#120	—	178	1	no	—
238	#120	—	178	1	no	—
239	C₄×Q₈	—	89	1	no	—
240	C₄×D₄	—	89	1	no	—
241	#117	C₂×D₈₉	1	1	yes	characteristic
242	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
243	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
244	#103	—	89	1	no	—
245	#103	—	89	1	no	—
246	#102	—	89	1	no	—
247	#102	—	89	1	no	—
248	#102	—	89	1	no	—
249	#102	—	89	1	no	—
250	#102	—	89	1	no	—
251	#102	—	89	1	no	—
252	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
253	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
254	#97	C₂×D₈₉	1	1	yes	characteristic, complemented
255	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
256	C₂⁴	—	178	1	no	—
257	#41	—	356	1	no	—
258	#41	—	356	1	no	—
259	C₂×Q₈	—	89	1	no	—
260	C₂×Q₈	—	89	1	no	—
261	C₂×Q₈	—	178	1	no	—
262	C₂×D₄	—	89	1	no	—
263	C₂×D₄	—	89	1	no	—
264	C₂×D₄	—	89	1	no	—
265	C₂×D₄	—	89	1	no	—
266	C₂×D₄	—	356	1	no	—
267	C₂×D₄	—	356	1	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	—	178	1	no	—
270	C₂×D₄	—	178	1	no	—
271	C₂×D₄	—	178	1	no	—
272	C₂×D₄	—	178	1	no	—
273	C₂×D₄	—	178	1	no	—
274	C₂×D₄	—	178	1	no	—
275	C₂²×C₄	—	178	1	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
278	Q₁₆	—	356	1	no	—
279	SD₁₆	—	356	1	no	—
280	SD₁₆	—	356	1	no	—
281	SD₁₆	—	356	1	no	—
282	SD₁₆	—	356	1	no	—
283	D₈	—	356	1	no	—
284	D₈	—	356	1	no	—
285	D₈	—	356	1	no	—
286	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	89	1	no	—
291	C₄⋊C₄	—	89	1	no	—
292	C₄⋊C₄	—	178	1	no	—
293	C₄⋊C₄	—	178	1	no	—
294	C₄⋊C₄	—	178	1	no	—
295	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
296	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
297	C₂²⋊C₄	—	178	1	no	—
298	C₂²⋊C₄	—	178	1	no	—
299	C₂²⋊C₄	—	178	1	no	—
300	C₂²⋊C₄	—	178	1	no	—
301	C₂²⋊C₄	—	178	1	no	—
302	C₂²⋊C₄	—	178	1	no	—
303	C₂²⋊C₄	—	178	1	no	—
304	C₂²⋊C₄	—	178	1	no	—
305	C₂²⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
307	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
308	C₄²	—	178	1	no	—
309	C₄²	C₂²×D₈₉	1	1	yes	characteristic
310	C₄²	C₂²×D₈₉	1	1	yes	characteristic
311	C₂³	—	356	2	no	—
312	C₂³	—	178	1	no	—
313	C₂³	—	178	1	no	—
314	C₂³	—	178	1	no	—
315	C₂³	—	178	1	no	—
316	C₂³	—	712	1	no	—
317	C₂³	—	712	1	no	—
318	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
319	Q₈	—	178	1	no	—
320	Q₈	—	178	1	no	—
321	Q₈	—	178	1	no	—
322	Q₈	—	178	1	no	—
323	Q₈	—	712	1	no	—
324	D₄	—	356	2	no	—
325	D₄	—	356	2	no	—
326	D₄	—	178	1	no	—
327	D₄	—	178	1	no	—
328	D₄	—	178	1	no	—
329	D₄	—	178	1	no	—
330	D₄	—	178	1	no	—
331	D₄	—	178	1	no	—
332	D₄	—	178	1	no	—
333	D₄	—	178	1	no	—
334	D₄	—	712	1	no	—
335	D₄	—	712	1	no	—
336	D₄	—	712	1	no	—
337	D₄	—	712	1	no	—
338	D₄	—	712	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	356	2	no	—
342	C₂×C₄	—	178	1	no	—
343	C₂×C₄	—	178	1	no	—
344	C₂×C₄	—	178	1	no	—
345	C₂×C₄	—	178	1	no	—
346	C₂×C₄	—	712	1	no	—
347	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
348	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
349	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
350	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
351	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
352	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	356	1	no	—
361	C₂²	—	356	1	no	—
362	C₂²	—	356	1	no	—
363	C₂²	—	356	1	no	—
364	C₂²	—	356	1	no	—
365	C₂²	—	356	1	no	—
366	C₂²	—	356	1	no	—
367	C₂²	—	356	1	no	—
368	C₂²	—	356	1	no	—
369	C₂²	—	356	1	no	—
370	C₂²	—	356	1	no	—
371	C₂²	—	356	1	no	—
372	C₂²	—	712	4	no	—
373	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	356	1	no	—
380	C₄	—	356	1	no	—
381	C₄	—	356	1	no	—
382	C₄	—	356	1	no	—
383	C₄	#15668058	1	1	yes	characteristic
384	C₄	#15668058	1	1	yes	characteristic
385	C₂	—	4	1	no	—
386	C₂	—	712	1	no	—
387	C₂	—	712	1	no	—
388	C₂	—	712	1	no	—
389	C₂	—	712	1	no	—
390	C₂	#19152167	1	1	yes	characteristic
391	C₂	#19152403	1	1	yes	characteristic
392	C₂	#19152410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#433×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#433	D₈₉
hypocenter	C₈₉	#3224
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #19152219	C₂	semidirect
2. #19152167	C₂	semidirect
3. #19151987	C₂	semidirect
4. #19151982	C₂	semidirect
5. #19151958	C₂	semidirect
6. #19151945	C₂	semidirect
7. #19151821	C₂	semidirect
8. #19151814	C₂	semidirect
9. #19151799	C₂	semidirect
10. #19151794	C₂	semidirect
11. #19151762	C₂	semidirect
12. #19151751	C₂	semidirect
13. #19151745	C₂	semidirect
14. #19151734	C₂	semidirect
15. #433×C₈₉	C₂	semidirect
18. C₂×Dic₃₅₆	C₂²	semidirect
21. C₂×#13083563	C₂²	semidirect
22. C₂×#13083563	C₂²	semidirect
25. C₂×D₇₁₂	C₂²	semidirect
26. #15668012	C₂²	semidirect
32. #15668000	C₂²	semidirect
42. C₄×Dic₁₇₈	C₂²	semidirect
43. C₄×D₃₅₆	C₂²	semidirect
44. #15667937	C₂²	semidirect
45. #15667937	C₂²	semidirect
50. #15667934	C₂²	semidirect
51. #15667934	C₂²	semidirect
53. C₄⋊C₈×C₈₉	C₂²	semidirect
54. C₄⋊C₈×C₈₉	C₂²	semidirect
57. #97×C₈₉	C₂²	semidirect
58. C₄×C₈×C₈₉	C₂²	semidirect
197. C₈₉	#3224	semidirect
212. #433	D₈₉	semidirect
242. C₄⋊C₈	C₂×D₈₉	semidirect
243. C₄⋊C₈	C₂×D₈₉	semidirect
254. #97	C₂×D₈₉	semidirect
255. C₄×C₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152219	2
#19152167	2
#19151987	2
#19151982	2
#19151958	2
#19151945	2
#19151821	2
#19151814	2
#19151799	2
#19151794	2
#19151762	2
#19151751	2
#19151745	2
#19151734	2
#433×C₈₉	2
#3224	89