Subgroups — #26034059

Order 11392.

964 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152328	C₂	1	3	yes	maximal, complemented
3	#19152326	C₂	1	3	yes	maximal, complemented
4	#19152325	C₂	1	3	yes	maximal, complemented
5	#19152325	C₂	1	3	yes	maximal, complemented
6	#19152324	C₂	1	1	yes	maximal, characteristic, complemented
7	#19152324	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152228	C₂	1	3	yes	maximal
9	#19152225	C₂	1	3	yes	maximal
10	#19152154	C₂	1	3	yes	maximal
11	#19152152	C₂	1	3	yes	maximal
12	#19152074	C₂	1	1	yes	maximal, characteristic
13	#19152074	C₂	1	1	yes	maximal, characteristic
14	C₄×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
15	#557×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	#15668012	C₂²	1	3	yes	—
18	#15668011	C₂²	1	3	yes	—
19	#15668011	C₂²	1	3	yes	—
20	#15668011	C₂²	1	3	yes	—
21	#15668011	C₂²	1	3	yes	—
22	#15668011	C₂²	1	3	yes	—
23	#15668011	C₂²	1	6	yes	—
24	#15668010	C₂²	1	3	yes	—
25	#15668010	C₂²	1	6	yes	—
26	#15668004	C₂²	1	3	yes	—
27	#15668004	C₂²	1	3	yes	—
28	#15668003	C₂²	1	3	yes	—
29	#15668003	C₂²	1	3	yes	—
30	#15668002	C₂²	1	3	yes	—
31	#15668002	C₂²	1	3	yes	—
32	#15668002	C₂²	1	6	yes	—
33	#15668002	C₂²	1	6	yes	—
34	#15667992	C₂²	1	3	yes	—
35	#15667990	C₂²	1	1	yes	characteristic
36	#15667990	C₂²	1	3	yes	—
37	#15667990	C₂²	1	6	yes	—
38	#15667988	C₂²	1	3	yes	—
39	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
40	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
41	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
42	C₄×Dic₁₇₈	C₂²	1	3	yes	—
43	C₄×Dic₁₇₈	C₂²	1	3	yes	—
44	C₄×Dic₁₇₈	C₂²	1	3	yes	—
45	C₄×Dic₁₇₈	C₂²	1	3	yes	—
46	C₄×Dic₁₇₈	C₂²	1	3	yes	—
47	C₄×Dic₁₇₈	C₂²	1	3	yes	—
48	#15667977	C₂²	1	6	yes	—
49	#15667977	C₂²	1	6	yes	—
50	#15667977	C₂²	1	3	yes	—
51	#15667977	C₂²	1	3	yes	—
52	#15667977	C₂²	1	3	yes	—
53	#15667970	C₂²	1	3	yes	—
54	#15667970	C₂²	1	3	yes	—
55	#15667969	C₂²	1	3	yes	—
56	#15667969	C₂²	1	3	yes	—
57	C₄⋊C₄×D₈₉	C₂²	1	3	yes	—
58	C₄²×D₈₉	C₂²	1	3	yes	—
59	C₄⋊Q₈×C₈₉	C₂²	1	3	yes	—
60	#125×C₈₉	C₂²	1	3	yes	—
61	#125×C₈₉	C₂²	1	3	yes	—
62	C₄×Q₈×C₈₉	C₂²	1	3	yes	—
63	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
64	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
65	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
66	C₂×Dic₁₇₈	C₂³	1	3	yes	—
67	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₇₈	C₂³	1	6	yes	—
69	C₂×Dic₁₇₈	C₂³	1	3	yes	—
70	C₂×Dic₁₇₈	C₂³	1	3	yes	—
71	Q₈×D₈₉	Q₈	1	8	yes	complemented
72	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₈₉	C₂³	1	3	yes	—
74	C₂×C₄×D₈₉	C₂³	1	3	yes	—
75	C₄⋊Dic₈₉	C₂³	1	3	yes	—
76	C₄⋊Dic₈₉	C₂³	1	3	yes	—
77	C₄⋊Dic₈₉	C₂³	1	3	yes	—
78	C₄⋊Dic₈₉	C₂³	1	3	yes	—
79	C₄⋊Dic₈₉	C₂³	1	3	yes	—
80	C₄⋊Dic₈₉	C₂³	1	3	yes	—
81	#13083553	C₂³	1	3	yes	—
82	#13083553	C₂³	1	3	yes	—
83	#13083553	C₂³	1	3	yes	—
84	#13083553	C₂³	1	3	yes	—
85	#13083553	C₂³	1	3	yes	—
86	#13083553	C₂³	1	3	yes	—
87	#13083553	C₂³	1	6	yes	—
88	#13083553	C₂³	1	6	yes	—
89	#13083553	C₂³	1	6	yes	—
90	#13083553	C₂³	1	1	yes	characteristic
91	#13083553	C₂³	1	1	yes	characteristic
92	#13083553	C₂³	1	3	yes	—
93	#13083553	C₂³	1	3	yes	—
94	#13083551	C₂³	1	1	yes	characteristic
95	#13083551	C₂³	1	1	yes	characteristic
96	#13083551	C₂³	1	3	yes	—
97	#13083551	C₂³	1	3	yes	—
98	C₄×Dic₈₉	C₂³	1	3	yes	—
99	C₄×Dic₈₉	C₂³	1	3	yes	—
100	C₄×Dic₈₉	C₂³	1	6	yes	—
101	C₄×Dic₈₉	C₂³	1	6	yes	—
102	C₄×Dic₈₉	C₂³	1	3	yes	—
103	C₄×Dic₈₉	C₂³	1	3	yes	—
104	C₄×Dic₈₉	C₂³	1	3	yes	—
105	C₂×Q₈×C₈₉	C₂³	1	3	yes	—
106	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
108	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
109	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
110	C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
111	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
112	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
114	C₄²×C₈₉	C₂³	1	3	yes	—
115	C₄²×C₈₉	C₂³	1	3	yes	—
116	C₄²×C₈₉	C₂³	1	3	yes	—
117	C₂²×D₈₉	C₂⁴	1	1	yes	characteristic
118	Q₈×C₈₉	—	4	3	no	—
119	Q₈×C₈₉	C₂×Q₈	1	4	yes	—
120	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
124	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
125	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
126	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
127	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂⁴	1	3	yes	—
129	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂⁴	1	3	yes	—
131	C₂×Dic₈₉	C₂⁴	1	3	yes	—
132	C₂×Dic₈₉	C₂⁴	1	3	yes	—
133	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	4	3	no	—
136	C₄×D₈₉	C₂×Q₈	1	12	yes	—
137	Dic₁₇₈	—	2	6	no	—
138	Dic₁₇₈	—	2	2	no	—
139	Dic₁₇₈	—	4	3	no	—
140	Dic₁₇₈	—	4	6	no	—
141	Dic₁₇₈	C₂×Q₈	1	12	yes	—
142	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
143	C₂×D₈₉	—	2	1	no	—
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	C₂²×Q₈	1	2	yes	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	2	3	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	3	no	—
150	C₄×C₈₉	—	2	3	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	C₂²×Q₈	1	6	yes	—
153	Dic₈₉	—	2	3	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	3	no	—
157	Dic₈₉	—	2	3	no	—
158	Dic₈₉	C₂²×Q₈	1	6	yes	—
159	Dic₈₉	2₋¹⁺⁴	1	2	yes	—
160	C₂×C₈₉	C₂³×Q₈	1	1	yes	characteristic
161	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
162	C₂×C₈₉	#585	1	1	yes	characteristic
163	D₈₉	—	2	2	no	—
164	#3502	—	89	1	no	maximal, Sylow
165	C₈₉	#3502	1	1	yes	characteristic, complemented, Sylow
166	Q₈²	—	89	3	no	—
167	Q₈²	—	89	1	no	—
168	#557	—	89	3	no	—
169	#557	—	89	3	no	—
170	#557	—	89	1	no	—
171	#557	—	89	1	no	—
172	#557	—	89	3	no	—
173	#557	D₈₉	1	1	yes	characteristic, complemented
174	#544	—	89	3	no	—
175	#544	—	89	3	no	—
176	#533	—	89	3	no	—
177	#533	—	89	3	no	—
178	#519	—	89	1	no	—
179	#519	—	89	1	no	—
180	C₂×C₄×Q₈	—	89	1	no	—
181	C₂²×Q₈	—	89	1	no	—
182	C₄⋊Q₈	—	89	3	no	—
183	C₄⋊Q₈	—	89	6	no	—
184	C₄⋊Q₈	—	89	3	no	—
185	C₄⋊Q₈	—	89	3	no	—
186	C₄⋊Q₈	—	89	3	no	—
187	C₄⋊Q₈	—	89	6	no	—
188	C₄⋊Q₈	—	89	3	no	—
189	C₄⋊Q₈	—	89	3	no	—
190	C₄⋊Q₈	—	89	3	no	—
191	C₄⋊Q₈	C₂×D₈₉	1	3	yes	—
192	#125	—	89	3	no	—
193	#125	—	89	3	no	—
194	#125	—	89	6	no	—
195	#125	—	89	6	no	—
196	#125	—	89	3	no	—
197	#125	—	89	3	no	—
198	#125	—	89	3	no	—
199	#125	—	89	3	no	—
200	#125	C₂×D₈₉	1	3	yes	—
201	#125	C₂×D₈₉	1	3	yes	—
202	C₂²⋊Q₈	—	89	3	no	—
203	C₂²⋊Q₈	—	89	3	no	—
204	C₂²⋊Q₈	—	89	1	no	—
205	C₂²⋊Q₈	—	89	6	no	—
206	C₂²⋊Q₈	—	89	3	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	—	89	3	no	—
211	C₄×Q₈	—	89	3	no	—
212	C₄×Q₈	—	89	3	no	—
213	C₄×Q₈	—	89	6	no	—
214	C₄×Q₈	—	89	6	no	—
215	C₄×Q₈	—	89	3	no	—
216	C₄×Q₈	—	89	3	no	—
217	C₄×Q₈	—	89	3	no	—
218	C₄×Q₈	—	89	3	no	—
219	C₄×Q₈	—	89	3	no	—
220	C₄×Q₈	—	89	3	no	—
221	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
222	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
223	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
224	C₄×Q₈	C₂×D₈₉	1	3	yes	—
225	#117	—	89	3	no	—
226	#117	—	89	3	no	—
227	#117	—	89	3	no	—
228	#117	—	89	3	no	—
229	C₂×C₄⋊C₄	—	89	3	no	—
230	C₂×C₄²	—	89	3	no	—
231	C₂×Q₈	—	89	8	no	—
232	C₂×Q₈	—	89	3	no	—
233	C₂×Q₈	—	89	3	no	—
234	C₂×Q₈	—	89	6	no	—
235	C₂×Q₈	—	89	3	no	—
236	C₂×Q₈	—	89	1	no	—
237	C₂×Q₈	C₂²×D₈₉	1	3	yes	—
238	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₂²×C₄	—	89	3	no	—
240	C₂²×C₄	—	89	3	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₄⋊C₄	—	89	6	no	—
243	C₄⋊C₄	—	89	3	no	—
244	C₄⋊C₄	—	89	3	no	—
245	C₄⋊C₄	—	89	3	no	—
246	C₄⋊C₄	—	89	3	no	—
247	C₄⋊C₄	—	89	3	no	—
248	C₄⋊C₄	—	89	3	no	—
249	C₄⋊C₄	—	89	3	no	—
250	C₄⋊C₄	—	89	3	no	—
251	C₄⋊C₄	—	89	1	no	—
252	C₄⋊C₄	—	89	3	no	—
253	C₄⋊C₄	—	89	1	no	—
254	C₄⋊C₄	—	89	6	no	—
255	C₄⋊C₄	—	89	3	no	—
256	C₄⋊C₄	—	89	6	no	—
257	C₄⋊C₄	—	89	3	no	—
258	C₄⋊C₄	—	89	3	no	—
259	C₄⋊C₄	—	89	3	no	—
260	C₄⋊C₄	—	89	3	no	—
261	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
262	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
263	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
264	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
265	C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
266	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
268	C₂²⋊C₄	—	89	3	no	—
269	C₂²⋊C₄	—	89	1	no	—
270	C₂²⋊C₄	—	89	3	no	—
271	C₂²⋊C₄	—	89	1	no	—
272	C₄²	—	89	3	no	—
273	C₄²	—	89	3	no	—
274	C₄²	—	89	3	no	—
275	C₄²	—	89	6	no	—
276	C₄²	—	89	6	no	—
277	C₄²	—	89	3	no	—
278	C₄²	—	89	3	no	—
279	C₄²	C₂²×D₈₉	1	3	yes	—
280	C₄²	C₂²×D₈₉	1	3	yes	—
281	C₄²	C₂²×D₈₉	1	3	yes	—
282	C₂³	—	89	1	no	—
283	Q₈	—	4	3	no	—
284	Q₈	—	89	12	no	—
285	Q₈	—	356	3	no	—
286	Q₈	—	356	6	no	—
287	Q₈	—	178	6	no	—
288	Q₈	—	178	2	no	—
289	Q₈	Q₈×D₈₉	1	4	yes	—
290	C₂×C₄	—	89	3	no	—
291	C₂×C₄	—	89	1	no	—
292	C₂×C₄	—	89	12	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	3	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	89	3	no	—
297	C₂×C₄	—	89	3	no	—
298	C₂×C₄	—	356	3	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₈₉	1	3	yes	—
302	C₂×C₄	C₂³×D₈₉	1	3	yes	—
303	C₂×C₄	C₂³×D₈₉	1	3	yes	—
304	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₉	1	3	yes	—
307	C₂²	—	89	2	no	—
308	C₂²	—	178	1	no	—
309	C₂²	—	178	1	no	—
310	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	3	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	3	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	3	no	—
317	C₄	—	89	6	no	—
318	C₄	—	89	2	no	—
319	C₄	—	178	1	no	—
320	C₄	—	178	3	no	—
321	C₄	—	178	3	no	—
322	C₄	—	178	1	no	—
323	C₄	—	178	3	no	—
324	C₄	C₂×Q₈×D₈₉	1	6	yes	—
325	C₂	—	178	2	no	—
326	C₂	C₂²×Q₈×D₈₉	1	1	yes	characteristic
327	C₂	2₋¹⁺⁴×D₈₉	1	1	yes	characteristic
328	C₂	#19152434	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#557×C₈₉	C₂
prime radical	C₂²×D₈₉	C₂⁴
hypercenter	#557	D₈₉
hypocenter	C₈₉	#3502
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₇₈	C₂	semidirect
2. #19152328	C₂	semidirect
3. #19152326	C₂	semidirect
4. #19152325	C₂	semidirect
5. #19152325	C₂	semidirect
6. #19152324	C₂	semidirect
7. #19152324	C₂	semidirect
15. #557×C₈₉	C₂	semidirect
71. Q₈×D₈₉	Q₈	semidirect
165. C₈₉	#3502	semidirect
173. #557	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₁₇₈	2
#19152328	2
#19152326	2
#19152325	2
#19152324	2
#19152228	2
#19152225	2
#19152154	2
#19152152	2
#19152074	2
C₄×Q₈×D₈₉	2
#557×C₈₉	2
#3502	89