Subgroups — #26034070

Order 11392.

1060 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#19152324	C₂	1	2	yes	maximal, complemented
3	#19152309	C₂	1	3	yes	maximal, complemented
4	#19152308	C₂	1	3	yes	maximal, complemented
5	#19152263	C₂	1	6	yes	maximal, complemented
6	#19152225	C₂	1	6	yes	maximal, complemented
7	#19152152	C₂	1	3	yes	maximal, complemented
8	C₄⋊Q₈×D₈₉	C₂	1	3	yes	maximal, complemented
9	#19152080	C₂	1	2	yes	maximal
10	C₄×#13083575	C₂	1	1	yes	maximal, characteristic
11	Q₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083575	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₈₉	C₂²	1	6	yes	—
14	#15668012	C₂²	1	3	yes	—
15	#15668011	C₂²	1	3	yes	—
16	#15668011	C₂²	1	6	yes	—
17	#15668011	C₂²	1	6	yes	—
18	#15668010	C₂²	1	6	yes	—
19	#15668010	C₂²	1	1	yes	characteristic
20	#15668004	C₂²	1	6	yes	—
21	#15668003	C₂²	1	6	yes	—
22	#15668002	C₂²	1	6	yes	—
23	#15667992	C₂²	1	3	yes	—
24	#15667990	C₂²	1	6	yes	—
25	#15667990	C₂²	1	12	yes	—
26	#15667988	C₂²	1	3	yes	—
27	#15667988	C₂²	1	12	yes	—
28	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
29	Q₈×Dic₈₉	C₂²	1	2	yes	—
30	C₄×Dic₁₇₈	C₂²	1	6	yes	—
31	C₄×Dic₁₇₈	C₂²	1	3	yes	—
32	C₄×D₃₅₆	C₂²	1	6	yes	—
33	C₄×D₃₅₆	C₂²	1	3	yes	—
34	#15667974	C₂²	1	3	yes	—
35	#15667974	C₂²	1	6	yes	—
36	#15667970	C₂²	1	3	yes	—
37	#15667969	C₂²	1	6	yes	—
38	C₄⋊C₄×D₈₉	C₂²	1	6	yes	—
39	C₄⋊C₄×D₈₉	C₂²	1	6	yes	—
40	C₄²×D₈₉	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₈₉	C₂²	1	3	yes	—
42	C₄⋊Q₈×C₈₉	C₂²	1	6	yes	—
43	C₄×Q₈×C₈₉	C₂²	1	3	yes	—
44	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
45	C₄×Q₈×C₈₉	C₂²	1	2	yes	—
46	#13083575	Q₈	1	8	yes	complemented
47	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₇₈	C₂³	1	6	yes	—
49	C₂×Dic₁₇₈	C₂³	1	6	yes	—
50	Q₈×D₈₉	—	4	12	no	—
51	C₂×D₃₅₆	C₂³	1	3	yes	—
52	C₂×C₄×D₈₉	C₂³	1	3	yes	—
53	C₂×C₄×D₈₉	C₂³	1	3	yes	—
54	C₂×C₄×D₈₉	C₂³	1	12	yes	—
55	C₂×C₄×D₈₉	C₂³	1	3	yes	—
56	C₄⋊Dic₈₉	C₂³	1	3	yes	—
57	C₄⋊Dic₈₉	C₂³	1	6	yes	—
58	C₄⋊Dic₈₉	C₂³	1	6	yes	—
59	C₄⋊Dic₈₉	C₂³	1	3	yes	—
60	#13083553	C₂³	1	2	yes	—
61	#13083553	C₂³	1	6	yes	—
62	#13083553	C₂³	1	2	yes	—
63	#13083553	C₂³	1	6	yes	—
64	#13083553	C₂³	1	6	yes	—
65	#13083553	C₂³	1	12	yes	—
66	#13083551	C₂³	1	6	yes	—
67	#13083551	C₂³	1	6	yes	—
68	#13083551	C₂³	1	6	yes	—
69	C₄×Dic₈₉	C₂³	1	6	yes	—
70	C₄×Dic₈₉	C₂³	1	3	yes	—
71	C₄×Dic₈₉	C₂³	1	3	yes	—
72	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₈₉	C₂³	1	6	yes	—
75	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
77	C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
78	C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
79	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
80	C₄²×C₈₉	C₂³	1	6	yes	—
81	C₄²×C₈₉	C₂³	1	3	yes	—
82	C₂²×D₈₉	C₂⁴	1	3	yes	—
83	Q₈×C₈₉	—	2	2	no	—
84	Q₈×C₈₉	—	4	6	no	—
85	Q₈×C₈₉	C₂×Q₈	1	4	yes	complemented
86	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
87	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
88	C₂×C₄×C₈₉	C₂⁴	1	6	yes	—
89	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
90	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₈₉	C₂⁴	1	2	yes	—
92	C₂×Dic₈₉	C₂⁴	1	3	yes	—
93	C₂×Dic₈₉	C₂⁴	1	6	yes	—
94	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
96	D₃₅₆	C₂×Q₈	1	12	yes	complemented
97	C₄×D₈₉	—	2	6	no	—
98	C₄×D₈₉	—	2	6	no	—
99	C₄×D₈₉	—	4	12	no	—
100	C₄×D₈₉	C₂×Q₈	1	12	yes	complemented
101	Dic₁₇₈	—	2	2	no	—
102	Dic₁₇₈	—	2	12	no	—
103	Dic₁₇₈	—	4	6	no	—
104	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
105	C₂×D₈₉	—	2	3	no	—
106	C₂×D₈₉	—	2	3	no	—
107	C₂×D₈₉	C₂²×Q₈	1	6	yes	—
108	C₄×C₈₉	—	2	6	no	—
109	C₄×C₈₉	—	2	3	no	—
110	C₄×C₈₉	—	2	2	no	—
111	C₄×C₈₉	C₂²×Q₈	1	6	yes	—
112	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
113	Dic₈₉	—	2	6	no	—
114	Dic₈₉	—	2	2	no	—
115	Dic₈₉	—	2	3	no	—
116	Dic₈₉	C₂²×Q₈	1	2	yes	—
117	Dic₈₉	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₈₉	C₂³×Q₈	1	1	yes	characteristic
119	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
121	D₈₉	—	2	6	no	—
122	#3503	—	89	1	no	maximal, Sylow
123	C₈₉	#3503	1	1	yes	characteristic, complemented, Sylow
124	Q₈²	—	89	1	no	—
125	Q₈²	D₈₉	1	1	yes	characteristic, complemented
126	#557	—	89	2	no	—
127	#554	—	89	3	no	—
128	#554	—	89	3	no	—
129	D₄×Q₈	—	89	6	no	—
130	#544	—	89	6	no	—
131	#533	—	89	3	no	—
132	C₂×C₄⋊Q₈	—	89	3	no	—
133	#520	—	89	2	no	—
134	C₄×#41	—	89	1	no	—
135	C₂×#41	—	89	1	no	—
136	C₂²×Q₈	—	89	6	no	—
137	C₄⋊Q₈	—	89	6	no	—
138	C₄⋊Q₈	—	89	1	no	—
139	C₄⋊Q₈	—	89	3	no	—
140	C₄⋊Q₈	—	89	6	no	—
141	C₄⋊Q₈	—	89	6	no	—
142	C₄⋊Q₈	—	89	3	no	—
143	C₄⋊Q₈	C₂×D₈₉	1	3	yes	—
144	C₄⋊Q₈	C₂×D₈₉	1	6	yes	—
145	#125	—	89	6	no	—
146	#125	—	89	6	no	—
147	#125	—	89	6	no	—
148	C₂²⋊Q₈	—	89	3	no	—
149	C₂²⋊Q₈	—	89	12	no	—
150	C₂²⋊Q₈	—	89	6	no	—
151	C₂²⋊Q₈	—	89	3	no	—
152	C₂²⋊Q₈	—	89	12	no	—
153	C₄×Q₈	—	89	1	no	—
154	C₄×Q₈	—	89	2	no	—
155	C₄×Q₈	—	89	6	no	—
156	C₄×Q₈	—	89	3	no	—
157	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
158	C₄×Q₈	C₂×D₈₉	1	2	yes	—
159	C₄×Q₈	C₂×D₈₉	1	3	yes	—
160	C₄×D₄	—	89	3	no	—
161	C₄×D₄	—	89	6	no	—
162	C₄×D₄	—	89	3	no	—
163	C₄×D₄	—	89	6	no	—
164	#117	—	89	3	no	—
165	#117	—	89	6	no	—
166	C₂×C₄⋊C₄	—	89	6	no	—
167	C₂×C₄⋊C₄	—	89	6	no	—
168	C₂×C₄²	—	89	3	no	—
169	#41	—	89	8	no	—
170	C₂×Q₈	—	89	6	no	—
171	C₂×Q₈	—	89	1	no	—
172	C₂×Q₈	—	89	6	no	—
173	C₂×Q₈	—	356	12	no	—
174	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
175	C₂×Q₈	C₂²×D₈₉	1	6	yes	—
176	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
177	C₂×D₄	—	89	3	no	—
178	C₂²×C₄	—	89	3	no	—
179	C₂²×C₄	—	89	12	no	—
180	C₂²×C₄	—	89	3	no	—
181	C₂²×C₄	—	89	3	no	—
182	C₄⋊C₄	—	89	6	no	—
183	C₄⋊C₄	—	89	3	no	—
184	C₄⋊C₄	—	89	6	no	—
185	C₄⋊C₄	—	89	12	no	—
186	C₄⋊C₄	—	89	2	no	—
187	C₄⋊C₄	—	89	3	no	—
188	C₄⋊C₄	—	89	2	no	—
189	C₄⋊C₄	—	89	6	no	—
190	C₄⋊C₄	—	89	6	no	—
191	C₄⋊C₄	—	89	6	no	—
192	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
193	C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
194	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
195	C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
196	C₂²⋊C₄	—	89	6	no	—
197	C₂²⋊C₄	—	89	6	no	—
198	C₂²⋊C₄	—	89	6	no	—
199	C₄²	—	89	1	no	—
200	C₄²	—	89	3	no	—
201	C₄²	—	89	3	no	—
202	C₄²	—	89	6	no	—
203	C₄²	C₂²×D₈₉	1	3	yes	—
204	C₄²	C₂²×D₈₉	1	6	yes	—
205	C₂³	—	89	3	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	6	no	—
208	Q₈	—	356	6	no	—
209	Q₈	—	178	12	no	—
210	Q₈	—	178	2	no	—
211	Q₈	Q₈×D₈₉	1	4	yes	complemented
212	D₄	—	89	12	no	—
213	C₂×C₄	—	89	1	no	—
214	C₂×C₄	—	89	1	no	—
215	C₂×C₄	—	89	12	no	—
216	C₂×C₄	—	89	2	no	—
217	C₂×C₄	—	89	3	no	—
218	C₂×C₄	—	89	6	no	—
219	C₂×C₄	—	356	12	no	—
220	C₂×C₄	—	178	6	no	—
221	C₂×C₄	—	178	6	no	—
222	C₂×C₄	C₂³×D₈₉	1	3	yes	—
223	C₂×C₄	C₂³×D₈₉	1	2	yes	—
224	C₂×C₄	C₂³×D₈₉	1	6	yes	—
225	C₂×C₄	C₂³×D₈₉	1	3	yes	—
226	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
227	C₂²	—	89	6	no	—
228	C₂²	—	178	3	no	—
229	C₂²	—	178	3	no	—
230	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
231	C₄	—	2	6	no	—
232	C₄	—	2	3	no	—
233	C₄	—	2	2	no	—
234	C₄	—	89	2	no	—
235	C₄	—	89	2	no	—
236	C₄	—	178	6	no	—
237	C₄	—	178	3	no	—
238	C₄	—	178	2	no	—
239	C₄	C₂×Q₈×D₈₉	1	6	yes	—
240	C₄	#15668060	1	2	yes	—
241	C₂	—	178	6	no	—
242	C₂	C₂²×Q₈×D₈₉	1	1	yes	characteristic
243	C₂	H(5,𝔽₂)×D₈₉	1	1	yes	characteristic
244	C₂	C₂×#15668060	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	Q₈²×C₈₉	C₂
prime radical	C₂×#13083575	C₂²
hypercenter	Q₈²	D₈₉
hypocenter	C₈₉	#3503
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₇₈	C₂	semidirect
2. #19152324	C₂	semidirect
3. #19152309	C₂	semidirect
4. #19152308	C₂	semidirect
5. #19152263	C₂	semidirect
6. #19152225	C₂	semidirect
7. #19152152	C₂	semidirect
8. C₄⋊Q₈×D₈₉	C₂	semidirect
11. Q₈²×C₈₉	C₂	semidirect
46. #13083575	Q₈	semidirect
85. Q₈×C₈₉	C₂×Q₈	semidirect
96. D₃₅₆	C₂×Q₈	semidirect
100. C₄×D₈₉	C₂×Q₈	semidirect
123. C₈₉	#3503	semidirect
125. Q₈²	D₈₉	semidirect
211. Q₈	Q₈×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
#19152324	2
#19152309	2
#19152308	2
#19152263	2
#19152225	2
#19152152	2
C₄⋊Q₈×D₈₉	2
#19152080	2
C₄×#13083575	2
Q₈²×C₈₉	2
#3503	89