Subgroups — #26029705

Order 11392.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668034	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15668032	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152144	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152066	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151823	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151819	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151812	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151797	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151792	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#15667941	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151592	C₂	1	1	yes	maximal, characteristic, complemented
15	#416×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083574	C₂²	1	1	yes	characteristic
18	#15668034	—	2	4	no	—
19	#15668032	—	2	4	no	—
20	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
21	C₂×#13083564	C₂²	1	1	yes	characteristic
22	C₂×#13083563	C₂²	1	1	yes	characteristic
23	C₂×#13083563	C₂²	1	1	yes	characteristic
24	C₂×#13083562	C₂²	1	1	yes	characteristic
25	C₂×#13083561	C₂²	1	1	yes	characteristic
26	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
27	C₂×#13083559	C₂²	1	1	yes	characteristic
28	C₂²×#10623494	C₂²	1	1	yes	characteristic
29	#15668010	C₂²	1	1	yes	characteristic
30	#15668008	C₂²	1	1	yes	characteristic
31	#15668001	C₂²	1	1	yes	characteristic
32	#15667999	C₂²	1	1	yes	characteristic
33	#15667984	—	2	1	no	—
34	#15667984	—	2	1	no	—
35	#15667977	C₂²	1	1	yes	characteristic
36	#15667977	C₂²	1	1	yes	characteristic
37	#15667974	C₂²	1	1	yes	characteristic
38	#15667974	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊D₈₉	—	2	1	no	—
40	C₄×C₂²⋊D₈₉	—	2	1	no	—
41	#15667972	—	2	1	no	—
42	#15667972	—	2	1	no	—
43	#15667969	—	2	1	no	—
44	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
45	C₄²×D₈₉	—	2	1	no	—
46	#15667941	C₂²	1	1	yes	characteristic
47	#15667941	C₂²	1	1	yes	characteristic
48	#15667941	C₂²	1	1	yes	characteristic
49	#15667941	C₂²	1	1	yes	characteristic
50	#15667935	C₂²	1	1	yes	characteristic
51	#15667935	C₂²	1	1	yes	characteristic
52	#15667932	C₂²	1	1	yes	characteristic
53	#15667932	C₂²	1	1	yes	characteristic
54	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
56	#117×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#102×C₈₉	C₂²	1	1	yes	characteristic
60	#102×C₈₉	C₂²	1	1	yes	characteristic
61	#13083574	—	2	2	no	—
62	#13083574	—	2	2	no	—
63	#13083574	—	2	2	no	—
64	#13083574	D₄	1	4	yes	complemented
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
74	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	Dic₃₅₆	—	2	2	no	—
80	#13083564	—	2	2	no	—
81	#13083563	—	2	2	no	—
82	#13083563	—	2	2	no	—
83	#13083562	—	2	2	no	—
84	#13083561	—	2	2	no	—
85	D₇₁₂	—	2	2	no	—
86	#13083559	—	2	2	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	C₂³	1	1	yes	characteristic
89	C₂×#10623494	C₂³	1	1	yes	characteristic
90	#13083553	—	2	1	no	—
91	#13083553	—	2	1	no	—
92	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	C₂²⋊Dic₈₉	—	2	1	no	—
95	C₂²⋊Dic₈₉	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	C₄×Dic₈₉	—	2	1	no	—
101	C₄×Dic₈₉	—	2	1	no	—
102	C₄×Dic₈₉	D₄	1	1	yes	characteristic
103	C₄×Dic₈₉	D₄	1	1	yes	characteristic
104	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
105	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
106	#41×C₈₉	—	2	2	no	—
107	#41×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×D₄×C₈₉	—	2	1	no	—
110	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
118	C₂²⋊C₄×C₈₉	—	2	1	no	—
119	C₄²×C₈₉	—	2	1	no	—
120	C₂³×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂²×D₈₉	—	2	1	no	—
123	C₂²×D₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	2	1	no	—
125	Q₈×C₈₉	—	2	1	no	—
126	D₄×C₈₉	—	2	1	no	—
127	D₄×C₈₉	—	2	1	no	—
128	D₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#41	1	2	yes	—
136	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	2	4	no	—
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂²⋊D₈₉	—	4	2	no	—
140	C₂²⋊D₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
146	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
147	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
148	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
149	D₃₅₆	—	2	1	no	—
150	D₃₅₆	—	2	1	no	—
151	D₃₅₆	—	2	1	no	—
152	D₃₅₆	C₂×D₄	1	2	yes	—
153	C₄×D₈₉	—	2	4	no	—
154	C₄×D₈₉	—	4	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₄×D₈₉	—	4	2	no	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	2	1	no	—
160	Dic₁₇₈	C₂×D₄	1	2	yes	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	2	1	no	—
167	C₂²×C₈₉	—	4	1	no	—
168	C₂²×C₈₉	—	4	1	no	—
169	C₂²×C₈₉	—	4	1	no	—
170	C₂²×C₈₉	#121	1	2	yes	—
171	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
172	C₂×D₈₉	—	2	2	no	—
173	C₂×D₈₉	—	4	1	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	4	1	no	—
177	C₂×D₈₉	—	4	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	#121	1	2	yes	—
182	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	2	1	no	—
187	Dic₈₉	—	2	1	no	—
188	Dic₈₉	—	4	1	no	—
189	Dic₈₉	#135	1	2	yes	complemented
190	C₂×C₈₉	—	2	2	no	—
191	C₂×C₈₉	—	8	1	no	—
192	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
193	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
194	C₂×C₈₉	#575	1	1	yes	characteristic
195	D₈₉	—	4	2	no	—
196	D₈₉	—	8	1	no	—
197	#3035	—	89	1	no	maximal, Sylow
198	C₈₉	#3035	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	89	1	no	—
200	C₂×#135	—	89	1	no	—
201	#532	—	89	1	no	—
202	C₄×#41	—	89	1	no	—
203	#464	—	89	1	no	—
204	#464	—	89	1	no	—
205	#463	—	89	1	no	—
206	#463	—	89	1	no	—
207	#462	—	89	1	no	—
208	#461	—	89	1	no	—
209	#460	—	89	1	no	—
210	#459	—	89	1	no	—
211	C₂×C₄⋊C₈	—	89	1	no	—
212	#416	—	89	1	no	—
213	#416	D₈₉	1	1	yes	characteristic, complemented
214	C₂×#41	—	89	1	no	—
215	C₂×#41	—	89	1	no	—
216	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
217	#135	—	178	4	no	—
218	#135	—	178	4	no	—
219	C₂×Q₁₆	—	89	1	no	—
220	C₂×Q₁₆	—	89	1	no	—
221	C₂×SD₁₆	—	89	1	no	—
222	C₂×SD₁₆	—	89	1	no	—
223	C₂×SD₁₆	—	89	1	no	—
224	C₂×SD₁₆	—	89	1	no	—
225	C₂×D₈	—	89	1	no	—
226	C₂×D₈	—	89	1	no	—
227	C₂²×C₈	—	89	1	no	—
228	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	89	1	no	—
230	#127	—	89	1	no	—
231	#124	—	89	1	no	—
232	#124	—	89	1	no	—
233	#121	—	178	1	no	—
234	#121	—	178	1	no	—
235	C₄×Q₈	—	89	1	no	—
236	C₄×Q₈	—	89	1	no	—
237	C₄×D₄	—	89	1	no	—
238	C₄×D₄	—	89	1	no	—
239	C₄×D₄	—	178	1	no	—
240	C₄×D₄	—	178	1	no	—
241	C₄×D₄	—	178	1	no	—
242	C₄×D₄	—	178	1	no	—
243	#117	—	178	1	no	—
244	#117	C₂×D₈₉	1	1	yes	characteristic
245	C₂×C₄²	—	89	1	no	—
246	C₂×C₄²	—	178	1	no	—
247	C₄⋊C₈	—	89	1	no	—
248	C₄⋊C₈	—	89	1	no	—
249	C₄⋊C₈	—	89	1	no	—
250	C₄⋊C₈	—	89	1	no	—
251	#103	—	89	1	no	—
252	#103	—	89	1	no	—
253	#103	C₂×D₈₉	1	1	yes	characteristic
254	#103	C₂×D₈₉	1	1	yes	characteristic
255	#102	—	89	1	no	—
256	#102	—	89	1	no	—
257	#102	C₂×D₈₉	1	1	yes	characteristic
258	#102	C₂×D₈₉	1	1	yes	characteristic
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	#41	—	89	4	no	—
262	#41	—	178	2	no	—
263	#41	—	178	2	no	—
264	#41	—	178	2	no	—
265	C₂×Q₈	—	89	1	no	—
266	C₂×Q₈	—	89	1	no	—
267	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	89	1	no	—
270	C₂×D₄	—	89	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	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	89	1	no	—
278	C₂²×C₄	—	89	1	no	—
279	C₂²×C₄	—	178	1	no	—
280	C₂²×C₄	—	178	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
284	Q₁₆	—	178	2	no	—
285	Q₁₆	—	178	2	no	—
286	SD₁₆	—	178	2	no	—
287	SD₁₆	—	178	2	no	—
288	SD₁₆	—	178	2	no	—
289	SD₁₆	—	178	2	no	—
290	D₈	—	178	2	no	—
291	D₈	—	178	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	89	1	no	—
294	C₂×C₈	—	89	1	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
298	C₄⋊C₄	—	178	1	no	—
299	C₄⋊C₄	—	178	1	no	—
300	C₄⋊C₄	—	178	1	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
303	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	—	178	1	no	—
307	C₂²⋊C₄	—	178	1	no	—
308	C₂²⋊C₄	—	178	1	no	—
309	C₂²⋊C₄	—	178	1	no	—
310	C₂²⋊C₄	—	178	1	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	89	1	no	—
313	C₄²	—	89	1	no	—
314	C₄²	—	89	1	no	—
315	C₄²	—	89	1	no	—
316	C₄²	—	178	1	no	—
317	C₄²	—	178	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	178	1	no	—
320	C₂³	—	178	1	no	—
321	C₂³	D₄×D₈₉	1	1	yes	characteristic
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	89	2	no	—
325	Q₈	—	178	1	no	—
326	Q₈	—	178	1	no	—
327	Q₈	—	178	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	4	2	no	—
331	D₄	—	89	2	no	—
332	D₄	—	356	2	no	—
333	D₄	—	356	2	no	—
334	D₄	—	356	2	no	—
335	D₄	—	178	4	no	—
336	D₄	—	178	1	no	—
337	D₄	—	178	1	no	—
338	D₄	—	178	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	356	1	no	—
351	C₂×C₄	—	178	2	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	178	4	no	—
355	C₂×C₄	D₄×D₈₉	1	2	yes	—
356	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
357	C₂×C₄	#13083574	1	2	yes	—
358	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	178	1	no	—
362	C₈	—	178	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	356	1	no	—
369	C₂²	—	356	1	no	—
370	C₂²	—	356	1	no	—
371	C₂²	—	356	1	no	—
372	C₂²	—	356	1	no	—
373	C₂²	—	178	2	no	—
374	C₂²	#15667985	1	2	yes	—
375	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	4	1	no	—
379	C₄	—	89	2	no	—
380	C₄	—	356	1	no	—
381	C₄	—	178	1	no	—
382	C₄	—	178	1	no	—
383	C₄	—	178	2	no	—
384	C₄	—	178	1	no	—
385	C₄	#15667985	1	2	yes	—
386	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
387	C₄	C₂×#13083574	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	8	1	no	—
390	C₂	—	356	2	no	—
391	C₂	—	712	1	no	—
392	C₂	C₂×#15667985	1	1	yes	characteristic
393	C₂	#135×D₈₉	1	1	yes	characteristic
394	C₂	#19152395	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#416×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#416	D₈₉
hypocenter	C₈₉	#3035
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. C₂×#15668034	C₂	semidirect
2. C₂×#15668032	C₂	semidirect
3. #19152144	C₂	semidirect
4. #19152066	C₂	semidirect
5. #19151823	C₂	semidirect
6. #19151819	C₂	semidirect
7. #19151816	C₂	semidirect
8. #19151812	C₂	semidirect
9. #19151807	C₂	semidirect
10. #19151802	C₂	semidirect
11. #19151797	C₂	semidirect
12. #19151792	C₂	semidirect
13. C₂×#15667941	C₂	semidirect
14. #19151592	C₂	semidirect
15. #416×C₈₉	C₂	semidirect
64. #13083574	D₄	semidirect
189. Dic₈₉	#135	semidirect
198. C₈₉	#3035	semidirect
213. #416	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668034	2
C₂×#15668032	2
#19152144	2
#19152066	2
#19151823	2
#19151819	2
#19151816	2
#19151812	2
#19151807	2
#19151802	2
#19151797	2
#19151792	2
C₂×#15667941	2
#19151592	2
#416×C₈₉	2
#3035	89