Subgroups — #26030438

Order 11392.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 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	#19152072	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151940	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151923	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151889	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151879	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151751	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151734	C₂	1	1	yes	maximal, characteristic
11	#19151711	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13083563	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
21	#15668012	C₂²	1	1	yes	characteristic, complemented
22	#15668007	—	2	1	no	—
23	#15668000	C₂²	1	1	yes	characteristic, complemented
24	#15667998	—	2	1	no	—
25	#15667994	—	2	1	no	—
26	#15667991	C₂²	1	1	yes	characteristic
27	#15667990	—	2	1	no	—
28	#15667986	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
32	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
33	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
34	C₄×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
35	#15667974	—	2	1	no	—
36	#15667974	—	2	1	no	—
37	#15667971	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×D₈₉	—	2	1	no	—
39	C₂²⋊C₄×D₈₉	—	2	1	no	—
40	#15667946	C₂²	1	1	yes	characteristic, complemented
41	#15667946	C₂²	1	1	yes	characteristic, complemented
42	#15667944	C₂²	1	1	yes	characteristic, complemented
43	#15667944	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	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	#13083563	—	2	2	no	—
75	#13083563	—	2	2	no	—
76	D₇₁₂	—	2	2	no	—
77	D₇₁₂	—	2	2	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	1	no	—
111	C₂²×D₈₉	—	2	4	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	8	1	no	—
114	C₂²×D₈₉	—	8	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	8	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	D₃₅₆	—	2	4	no	—
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	—	2	1	no	—
132	D₃₅₆	—	2	1	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	#41	1	2	yes	—
135	D₃₅₆	#41	1	2	yes	—
136	C₄×D₈₉	—	8	1	no	—
137	C₄×D₈₉	—	8	1	no	—
138	C₄×D₈₉	—	8	1	no	—
139	C₄×D₈₉	—	8	1	no	—
140	C₄×D₈₉	—	8	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	8	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	—	4	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	8	4	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	—	4	1	no	—
174	C₂×C₈₉	—	4	1	no	—
175	C₂×C₈₉	#525	1	1	yes	characteristic
176	C₂×C₈₉	#576	1	1	yes	characteristic
177	C₂×C₈₉	#577	1	1	yes	characteristic
178	D₈₉	—	4	2	no	—
179	D₈₉	—	4	2	no	—
180	D₈₉	—	8	1	no	—
181	#3118	—	89	1	no	maximal, Sylow
182	C₈₉	#3118	1	1	yes	characteristic, complemented, Sylow
183	#542	—	89	1	no	—
184	#518	—	89	1	no	—
185	#485	—	89	1	no	—
186	#482	—	89	1	no	—
187	#476	—	89	1	no	—
188	#474	—	89	1	no	—
189	#463	—	89	1	no	—
190	#463	—	89	1	no	—
191	#450	—	89	1	no	—
192	#447	—	89	1	no	—
193	#442	—	89	1	no	—
194	#440	—	89	1	no	—
195	C₄×SD₁₆	—	89	1	no	—
196	C₄×D₈	—	89	1	no	—
197	#433	D₈₉	1	1	yes	characteristic, complemented
198	C₂²×D₄	—	89	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×SD₁₆	—	89	1	no	—
201	C₂×D₈	—	89	1	no	—
202	C₂×D₈	—	89	1	no	—
203	C₄⋊Q₈	—	89	1	no	—
204	#126	—	178	1	no	—
205	#124	—	89	1	no	—
206	#124	—	178	1	no	—
207	#123	—	178	1	no	—
208	C₂²⋊Q₈	—	89	1	no	—
209	C₂²⋊Q₈	—	178	1	no	—
210	#121	—	89	1	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	178	1	no	—
218	C₄×D₄	—	178	1	no	—
219	#117	—	89	1	no	—
220	#117	C₂×D₈₉	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	178	1	no	—
222	C₂×C₂²⋊C₄	—	178	1	no	—
223	#107	—	89	1	no	—
224	#107	—	89	1	no	—
225	#106	—	89	1	no	—
226	#106	—	89	1	no	—
227	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
228	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
238	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
239	#97	C₂×D₈₉	1	1	yes	characteristic, complemented
240	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
241	C₂⁴	—	178	1	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×Q₈	—	178	1	no	—
244	C₂×D₄	—	89	1	no	—
245	C₂×D₄	—	89	1	no	—
246	C₂×D₄	—	89	1	no	—
247	C₂×D₄	—	178	2	no	—
248	C₂×D₄	—	178	2	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
257	SD₁₆	—	178	2	no	—
258	SD₁₆	—	178	2	no	—
259	D₈	—	178	2	no	—
260	D₈	—	178	2	no	—
261	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
273	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	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	—	178	1	no	—
284	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
285	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
286	C₄²	—	178	1	no	—
287	C₄²	—	178	1	no	—
288	C₄²	C₂²×D₈₉	1	1	yes	characteristic
289	C₄²	C₂²×D₈₉	1	1	yes	characteristic
290	C₂³	—	178	4	no	—
291	C₂³	—	178	1	no	—
292	C₂³	—	178	1	no	—
293	C₂³	—	178	1	no	—
294	C₂³	—	712	1	no	—
295	C₂³	—	712	1	no	—
296	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
297	Q₈	—	178	1	no	—
298	Q₈	—	178	1	no	—
299	Q₈	—	712	1	no	—
300	D₄	—	89	2	no	—
301	D₄	—	89	2	no	—
302	D₄	—	178	4	no	—
303	D₄	—	178	1	no	—
304	D₄	—	178	1	no	—
305	D₄	—	178	1	no	—
306	D₄	—	178	1	no	—
307	D₄	—	712	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	—	178	1	no	—
315	C₂×C₄	—	712	1	no	—
316	C₂×C₄	—	712	1	no	—
317	C₂×C₄	—	712	1	no	—
318	C₂×C₄	—	712	1	no	—
319	C₂×C₄	—	712	1	no	—
320	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
321	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
322	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
323	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
324	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	356	1	no	—
334	C₂²	—	356	1	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	356	1	no	—
338	C₂²	—	356	1	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	178	2	no	—
341	C₂²	—	178	2	no	—
342	C₂²	—	712	4	no	—
343	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	356	1	no	—
351	C₄	—	356	1	no	—
352	C₄	—	356	1	no	—
353	C₄	—	356	1	no	—
354	C₄	#41×D₈₉	1	1	yes	characteristic
355	C₄	#41×D₈₉	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	356	2	no	—
358	C₂	—	356	2	no	—
359	C₂	—	712	1	no	—
360	C₂	#19152110	1	1	yes	characteristic
361	C₂	#19152403	1	1	yes	characteristic
362	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
prime radical	#19151734	C₂
hypercenter	#433	D₈₉
hypocenter	C₈₉	#3118
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. #19152072	C₂	semidirect
3. #19151940	C₂	semidirect
4. #19151923	C₂	semidirect
5. #19151889	C₂	semidirect
6. #19151879	C₂	semidirect
7. #19151814	C₂	semidirect
8. #19151814	C₂	semidirect
9. #19151751	C₂	semidirect
11. #19151711	C₂	semidirect
12. #19151700	C₂	semidirect
13. C₄×#13083563	C₂	semidirect
14. C₄×D₇₁₂	C₂	semidirect
15. #433×C₈₉	C₂	semidirect
21. #15668012	C₂²	semidirect
23. #15668000	C₂²	semidirect
29. C₄×Dic₁₇₈	C₂²	semidirect
34. C₄×D₃₅₆	C₂²	semidirect
40. #15667946	C₂²	semidirect
41. #15667946	C₂²	semidirect
42. #15667944	C₂²	semidirect
43. #15667944	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
182. C₈₉	#3118	semidirect
197. #433	D₈₉	semidirect
227. C₄⋊C₈	C₂×D₈₉	semidirect
228. C₄⋊C₈	C₂×D₈₉	semidirect
239. #97	C₂×D₈₉	semidirect
240. 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
#19152072	2
#19151940	2
#19151923	2
#19151889	2
#19151879	2
#19151814	2
#19151751	2
#19151734	2
#19151711	2
#19151700	2
C₄×#13083563	2
C₄×D₇₁₂	2
#433×C₈₉	2
#3118	89