Subgroups — #26030920

Order 11392.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 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
2	#19152210	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152007	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151992	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151953	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151949	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151856	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151844	C₂	1	1	yes	maximal, characteristic
9	#19151839	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151827	C₂	1	1	yes	maximal, characteristic
11	#19151706	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13083563	C₂	1	1	yes	maximal, characteristic, complemented
15	#405×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	#15668012	C₂²	1	1	yes	characteristic
21	#15668012	C₂²	1	1	yes	characteristic
22	#15668007	—	2	1	no	—
23	#15668005	—	2	1	no	—
24	#15668005	—	2	1	no	—
25	#15668004	C₂²	1	1	yes	characteristic
26	#15668002	—	2	1	no	—
27	#15668002	—	2	1	no	—
28	#15668000	C₂²	1	1	yes	characteristic
29	#15667998	—	2	1	no	—
30	#15667994	—	2	1	no	—
31	#15667991	C₂²	1	1	yes	characteristic
32	#15667991	C₂²	1	1	yes	characteristic
33	#15667991	C₂²	1	1	yes	characteristic
34	#15667990	—	2	1	no	—
35	#15667986	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
38	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
39	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
40	#15667946	C₂²	1	1	yes	characteristic
41	#15667946	C₂²	1	1	yes	characteristic
42	#15667944	C₂²	1	1	yes	characteristic
43	#15667944	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667937	C₂²	1	1	yes	characteristic
48	#15667937	C₂²	1	1	yes	characteristic
49	#15667937	C₂²	1	1	yes	characteristic
50	#15667934	C₂²	1	1	yes	characteristic
51	#15667934	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	#117×C₈₉	C₂²	1	1	yes	characteristic
55	#97×C₈₉	C₂²	1	1	yes	characteristic
56	#97×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	Dic₃₅₆	—	2	2	no	—
67	Dic₃₅₆	—	2	2	no	—
68	#13083563	—	2	2	no	—
69	#13083563	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
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	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₉	—	8	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	D₃₅₆	—	2	1	no	—
127	D₃₅₆	—	2	1	no	—
128	C₄×D₈₉	—	8	1	no	—
129	Dic₁₇₈	—	2	1	no	—
130	Dic₁₇₈	—	2	1	no	—
131	Dic₁₇₈	—	2	1	no	—
132	Dic₁₇₈	—	2	1	no	—
133	Dic₁₇₈	—	2	1	no	—
134	Dic₁₇₈	—	2	1	no	—
135	Dic₁₇₈	—	8	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	#41	1	2	yes	—
139	C₈×C₈₉	#41	1	2	yes	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₈₉	—	4	1	no	—
145	C₂×D₈₉	—	4	1	no	—
146	C₂×D₈₉	—	4	1	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	Dic₈₉	—	4	1	no	—
155	Dic₈₉	—	4	1	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	C₂×C₈₉	—	4	1	no	—
162	C₂×C₈₉	#532	1	1	yes	characteristic
163	C₂×C₈₉	#577	1	1	yes	characteristic
164	C₂×C₈₉	#578	1	1	yes	characteristic
165	D₈₉	—	8	1	no	—
166	#3178	—	89	1	no	maximal, Sylow
167	C₈₉	#3178	1	1	yes	characteristic, complemented, Sylow
168	#542	—	89	1	no	—
169	#541	—	89	1	no	—
170	#501	—	89	1	no	—
171	#497	—	89	1	no	—
172	#488	—	89	1	no	—
173	#487	—	89	1	no	—
174	#470	—	89	1	no	—
175	#468	—	89	1	no	—
176	#467	—	89	1	no	—
177	#465	—	89	1	no	—
178	#441	—	89	1	no	—
179	#440	—	89	1	no	—
180	C₄×Q₁₆	—	89	1	no	—
181	C₄×SD₁₆	—	89	1	no	—
182	#405	D₈₉	1	1	yes	characteristic, complemented
183	C₂×Q₁₆	—	89	1	no	—
184	C₂×Q₁₆	—	89	1	no	—
185	C₂×SD₁₆	—	89	1	no	—
186	C₂×SD₁₆	—	89	1	no	—
187	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
188	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
189	C₄⋊Q₈	—	89	1	no	—
190	C₄⋊Q₈	—	89	1	no	—
191	#126	—	178	1	no	—
192	#126	—	178	1	no	—
193	#126	—	178	1	no	—
194	#125	—	89	1	no	—
195	#125	—	178	1	no	—
196	#125	—	178	1	no	—
197	#124	—	89	1	no	—
198	#124	—	178	1	no	—
199	#123	—	178	1	no	—
200	C₂²⋊Q₈	—	89	1	no	—
201	C₂²⋊Q₈	—	89	1	no	—
202	C₂²⋊Q₈	—	89	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	#121	—	89	1	no	—
205	C₄×Q₈	—	89	1	no	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×D₄	—	89	1	no	—
209	#117	C₂×D₈₉	1	1	yes	characteristic
210	#107	—	89	1	no	—
211	#107	—	89	1	no	—
212	#106	—	89	1	no	—
213	#106	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#102	—	89	1	no	—
221	#102	—	89	1	no	—
222	#97	C₂×D₈₉	1	1	yes	characteristic
223	#97	C₂×D₈₉	1	1	yes	characteristic
224	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
225	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
226	C₂×Q₈	—	89	1	no	—
227	C₂×Q₈	—	89	1	no	—
228	C₂×Q₈	—	89	1	no	—
229	C₂×Q₈	—	178	1	no	—
230	C₂×D₄	—	89	1	no	—
231	C₂×D₄	—	178	1	no	—
232	C₂²×C₄	—	178	1	no	—
233	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
234	Q₁₆	—	178	2	no	—
235	Q₁₆	—	178	2	no	—
236	SD₁₆	—	178	2	no	—
237	SD₁₆	—	178	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄⋊C₄	—	89	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	89	1	no	—
247	C₄⋊C₄	—	89	1	no	—
248	C₄⋊C₄	—	178	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
260	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
261	C₂²⋊C₄	—	178	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₂²⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	178	1	no	—
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
269	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
270	C₄²	—	178	1	no	—
271	C₄²	—	178	1	no	—
272	C₄²	—	178	1	no	—
273	C₄²	C₂²×D₈₉	1	1	yes	characteristic
274	C₄²	C₂²×D₈₉	1	1	yes	characteristic
275	C₂³	—	178	1	no	—
276	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
277	Q₈	—	178	1	no	—
278	Q₈	—	178	1	no	—
279	Q₈	—	178	1	no	—
280	Q₈	—	178	1	no	—
281	Q₈	—	178	1	no	—
282	Q₈	—	178	1	no	—
283	Q₈	—	712	1	no	—
284	D₄	—	178	1	no	—
285	D₄	—	178	1	no	—
286	D₄	—	712	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	—	178	1	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	712	1	no	—
297	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
298	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
299	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
300	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	#13083574	1	2	yes	—
306	C₈	#13083574	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	356	1	no	—
311	C₂²	—	356	1	no	—
312	C₂²	—	356	1	no	—
313	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	356	1	no	—
320	C₄	—	356	1	no	—
321	C₄	—	356	1	no	—
322	C₄	—	356	1	no	—
323	C₄	—	356	1	no	—
324	C₄	—	356	1	no	—
325	C₄	—	356	1	no	—
326	C₄	C₂×#13083574	1	1	yes	characteristic
327	C₄	C₂×#13083574	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	712	1	no	—
330	C₂	#19152148	1	1	yes	characteristic
331	C₂	#19152410	1	1	yes	characteristic
332	C₂	#19152415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15667986	C₂²
hypercenter	#405	D₈₉
hypocenter	C₈₉	#3178
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #19152210	C₂	semidirect
3. #19152007	C₂	semidirect
4. #19151992	C₂	semidirect
5. #19151953	C₂	semidirect
6. #19151949	C₂	semidirect
7. #19151856	C₂	semidirect
9. #19151839	C₂	semidirect
11. #19151706	C₂	semidirect
12. #19151700	C₂	semidirect
13. C₄×Dic₃₅₆	C₂	semidirect
14. C₄×#13083563	C₂	semidirect
15. #405×C₈₉	C₂	semidirect
167. C₈₉	#3178	semidirect
182. #405	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
#19152210	2
#19152007	2
#19151992	2
#19151953	2
#19151949	2
#19151856	2
#19151844	2
#19151839	2
#19151827	2
#19151706	2
#19151700	2
C₄×Dic₃₅₆	2
C₄×#13083563	2
#405×C₈₉	2
#3178	89