Subgroups — #26033286

Order 11392.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152326	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152317	C₂	1	1	yes	maximal, characteristic
3	#19151993	C₂	1	1	yes	maximal, characteristic
4	#19151989	C₂	1	1	yes	maximal, characteristic
5	#19151976	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151819	C₂	1	1	yes	maximal, characteristic
7	#19151809	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151801	C₂	1	1	yes	maximal, characteristic
11	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151702	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151689	C₂	1	1	yes	maximal, characteristic
15	#464×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083564	C₂²	1	1	yes	characteristic
21	C₂×#13083562	C₂²	1	1	yes	characteristic
22	C₂×#13083562	C₂²	1	1	yes	characteristic
23	#15668012	C₂²	1	1	yes	characteristic
24	#15668010	C₂²	1	1	yes	characteristic
25	#15668010	C₂²	1	1	yes	characteristic
26	#15668010	C₂²	1	1	yes	characteristic
27	#15668005	—	2	1	no	—
28	#15668005	—	2	1	no	—
29	#15668004	—	2	1	no	—
30	#15668002	—	2	1	no	—
31	#15668002	—	2	1	no	—
32	#15667999	C₂²	1	1	yes	characteristic
33	#15667989	—	2	1	no	—
34	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₇₈	—	2	1	no	—
36	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
38	#15667977	C₂²	1	1	yes	characteristic
39	#15667977	C₂²	1	1	yes	characteristic
40	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
41	#15667968	—	2	1	no	—
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667923	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	#124×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
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	C₂²×Dic₈₉	—	2	1	no	—
64	Dic₃₅₆	—	4	1	no	—
65	Dic₃₅₆	—	4	1	no	—
66	#13083564	—	2	2	no	—
67	#13083564	—	2	2	no	—
68	#13083564	—	4	1	no	—
69	#13083562	—	2	2	no	—
70	#13083562	—	4	1	no	—
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	—	2	1	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	D₄	1	1	yes	characteristic
88	#13083553	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₉	—	4	1	no	—
102	SD₁₆×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	—	2	1	no	—
113	Q₈×C₈₉	—	2	1	no	—
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	D₄×C₈₉	—	2	1	no	—
117	D₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	—	4	2	no	—
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	Dic₁₇₈	—	2	2	no	—
134	Dic₁₇₈	—	2	1	no	—
135	Dic₁₇₈	—	2	1	no	—
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	—	2	1	no	—
139	Dic₁₇₈	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	#41	1	2	yes	—
144	C₂²×C₈₉	—	4	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	2	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	#137	1	2	yes	—
163	C₂×C₈₉	—	8	1	no	—
164	C₂×C₈₉	#550	1	1	yes	characteristic
165	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
166	C₂×C₈₉	#578	1	1	yes	characteristic
167	#3398	—	89	1	no	maximal, Sylow
168	C₈₉	#3398	1	1	yes	characteristic, complemented, Sylow
169	#557	—	89	1	no	—
170	#556	—	89	1	no	—
171	#497	—	89	1	no	—
172	#497	—	89	1	no	—
173	#494	—	89	1	no	—
174	#464	—	89	1	no	—
175	#464	D₈₉	1	1	yes	characteristic
176	#462	—	89	1	no	—
177	#462	—	89	1	no	—
178	#462	—	89	1	no	—
179	#461	—	89	1	no	—
180	#446	—	89	1	no	—
181	#441	—	89	1	no	—
182	#441	—	89	1	no	—
183	C₄×SD₁₆	—	89	1	no	—
184	C₂×Q₁₆	—	89	1	no	—
185	C₂×Q₁₆	—	89	1	no	—
186	C₂×Q₁₆	—	89	1	no	—
187	C₂×Q₁₆	—	89	1	no	—
188	C₂×Q₁₆	—	89	1	no	—
189	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	89	1	no	—
191	C₂×SD₁₆	—	89	1	no	—
192	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
193	C₄⋊Q₈	—	89	1	no	—
194	C₄⋊Q₈	—	89	1	no	—
195	C₄⋊Q₈	—	89	1	no	—
196	C₄⋊Q₈	—	89	1	no	—
197	#126	—	178	1	no	—
198	#126	—	178	1	no	—
199	#125	—	178	1	no	—
200	#125	—	178	1	no	—
201	#125	—	178	1	no	—
202	#124	—	89	1	no	—
203	#124	C₂×D₈₉	1	1	yes	characteristic
204	C₂²⋊Q₈	—	178	1	no	—
205	C₄×Q₈	—	89	1	no	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	—	178	1	no	—
211	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
212	C₄×D₄	—	89	1	no	—
213	#117	—	178	1	no	—
214	#106	—	89	1	no	—
215	C₄⋊C₈	—	89	1	no	—
216	C₄⋊C₈	—	89	1	no	—
217	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#103	—	89	1	no	—
221	#103	—	89	1	no	—
222	#103	C₂×D₈₉	1	1	yes	characteristic
223	#102	C₂×D₈₉	1	1	yes	characteristic
224	#97	—	89	1	no	—
225	#97	—	89	1	no	—
226	C₄×C₈	—	89	1	no	—
227	C₂×Q₈	—	89	1	no	—
228	C₂×Q₈	—	89	1	no	—
229	C₂×Q₈	—	89	1	no	—
230	C₂×Q₈	—	89	1	no	—
231	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
234	C₂²×C₄	—	178	1	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	356	1	no	—
237	Q₁₆	—	356	1	no	—
238	Q₁₆	—	356	1	no	—
239	Q₁₆	—	178	2	no	—
240	Q₁₆	—	178	2	no	—
241	SD₁₆	—	4	1	no	—
242	SD₁₆	—	356	1	no	—
243	SD₁₆	—	178	2	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	—	89	1	no	—
246	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	89	1	no	—
251	C₄⋊C₄	—	89	1	no	—
252	C₄⋊C₄	—	89	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	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	89	1	no	—
273	C₄²	—	89	1	no	—
274	C₄²	—	178	1	no	—
275	C₄²	—	178	1	no	—
276	C₄²	—	178	1	no	—
277	C₄²	—	178	1	no	—
278	C₄²	C₂²×D₈₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	178	2	no	—
285	Q₈	—	178	1	no	—
286	Q₈	—	178	1	no	—
287	Q₈	—	178	1	no	—
288	Q₈	—	178	1	no	—
289	Q₈	—	178	1	no	—
290	Q₈	—	178	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	89	1	no	—
297	C₂×C₄	—	89	1	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	89	2	no	—
311	C₈	—	178	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	89	2	no	—
322	C₄	—	356	1	no	—
323	C₄	—	356	1	no	—
324	C₄	—	356	1	no	—
325	C₄	—	356	1	no	—
326	C₄	—	178	1	no	—
327	C₄	—	178	1	no	—
328	C₄	—	178	1	no	—
329	C₄	#41×D₈₉	1	1	yes	characteristic
330	C₄	#15668057	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	#19152266	1	1	yes	characteristic
333	C₂	#137×D₈₉	1	1	yes	characteristic
334	C₂	#19152412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#464×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₄×C₈₉	C₂³
hypercenter	#464	D₈₉
hypocenter	C₈₉	#3398
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19152326	C₂	semidirect
5. #19151976	C₂	semidirect
7. #19151809	C₂	semidirect
8. #19151807	C₂	semidirect
9. #19151806	C₂	semidirect
11. #19151727	C₂	semidirect
12. #19151703	C₂	semidirect
13. #19151702	C₂	semidirect
168. C₈₉	#3398	semidirect

Maximal subgroups

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

subgroup	index
#19152326	2
#19152317	2
#19151993	2
#19151989	2
#19151976	2
#19151819	2
#19151809	2
#19151807	2
#19151806	2
#19151801	2
#19151727	2
#19151703	2
#19151702	2
#19151689	2
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