Subgroups — #26025622

Order 11392.

484 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#19151531	C₂	1	4	yes	maximal
3	#19151505	C₂	1	2	yes	maximal
4	#19151504	C₂	1	2	yes	maximal
5	#19151496	C₂	1	2	yes	maximal
6	#19151494	C₂	1	2	yes	maximal
7	#19151468	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₇₈	—	2	4	no	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
15	C₂×#13083553	C₂²	1	2	yes	—
16	C₂×#13083553	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
18	#15667919	C₂²	1	4	yes	—
19	#15667919	C₂²	1	4	yes	—
20	#15667919	C₂²	1	4	yes	—
21	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₈₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₇₈	—	2	2	no	—
28	C₂×Dic₁₇₈	—	2	2	no	—
29	C₂×Dic₁₇₈	Q₈	1	4	yes	complemented
30	C₂²×Dic₈₉	C₂³	1	4	yes	—
31	C₂²×Dic₈₉	C₂³	1	2	yes	—
32	C₂²×Dic₈₉	C₂³	1	2	yes	—
33	C₄⋊Dic₈₉	—	2	2	no	—
34	C₄⋊Dic₈₉	—	2	4	no	—
35	C₄⋊Dic₈₉	—	2	4	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	#13083553	—	2	4	no	—
38	#13083553	—	4	4	no	—
39	C₄×Dic₈₉	—	2	4	no	—
40	C₂×Q₈×C₈₉	—	2	2	no	—
41	C₂×Q₈×C₈₉	—	2	2	no	—
42	C₂×Q₈×C₈₉	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₈₉	—	2	4	no	—
49	C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	—	2	4	no	—
51	C₄²×C₈₉	D₄	1	4	yes	—
52	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
53	Q₈×C₈₉	—	2	8	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	#41	1	2	yes	—
68	C₂×C₄×C₈₉	#41	1	2	yes	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	—	2	2	no	—
72	C₂×Dic₈₉	—	2	4	no	—
73	C₂×Dic₈₉	—	2	4	no	—
74	C₂×Dic₈₉	—	2	2	no	—
75	C₂×Dic₈₉	—	2	4	no	—
76	C₂×Dic₈₉	—	2	2	no	—
77	C₂×Dic₈₉	C₂×Q₈	1	4	yes	—
78	Dic₁₇₈	—	2	8	no	—
79	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
83	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
84	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
85	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
86	C₄×C₈₉	—	2	4	no	—
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	—	4	2	no	—
89	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
90	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
91	Dic₈₉	—	2	4	no	—
92	Dic₈₉	—	4	2	no	—
93	Dic₈₉	—	4	4	no	—
94	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₈₉	#537	1	1	yes	characteristic
97	C₂×C₈₉	#557	1	1	yes	characteristic
98	C₂×C₈₉	#557	1	1	yes	characteristic
99	C₂×C₈₉	Q₈²	1	1	yes	characteristic
100	C₂×C₈₉	#559	1	1	yes	characteristic
101	#2582	—	89	1	no	maximal, Sylow
102	C₈₉	#2582	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄×Q₈	—	89	1	no	—
104	C₂×C₄×Q₈	D₈₉	1	1	yes	characteristic
105	#398	—	89	4	no	—
106	#391	—	89	2	no	—
107	#391	—	89	2	no	—
108	#389	—	89	2	no	—
109	#389	—	89	2	no	—
110	#384	—	89	1	no	—
111	C₂²×Q₈	—	89	1	no	—
112	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
113	C₄×Q₈	—	2	4	no	—
114	C₄×Q₈	—	178	4	no	—
115	C₂×C₄⋊C₄	—	89	2	no	—
116	C₂×C₄⋊C₄	—	89	4	no	—
117	C₂×C₄⋊C₄	—	89	1	no	—
118	C₂×C₄⋊C₄	—	89	2	no	—
119	C₂×C₄⋊C₄	—	89	2	no	—
120	C₂×C₄⋊C₄	—	89	2	no	—
121	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
123	C₂×C₄²	—	89	2	no	—
124	C₂×C₄²	C₂×D₈₉	1	2	yes	—
125	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
126	#95	—	89	4	no	—
127	#95	—	89	4	no	—
128	#95	—	89	4	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	89	4	no	—
132	C₂×Q₈	—	178	2	no	—
133	C₂×Q₈	—	178	2	no	—
134	C₂×Q₈	Dic₁₇₈	1	4	yes	complemented
135	C₂²×C₄	—	89	2	no	—
136	C₂²×C₄	—	89	4	no	—
137	C₂²×C₄	—	89	2	no	—
138	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
139	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
141	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	356	4	no	—
146	C₄⋊C₄	—	356	2	no	—
147	C₄⋊C₄	—	178	4	no	—
148	C₄⋊C₄	—	178	2	no	—
149	C₄⋊C₄	—	178	4	no	—
150	C₄⋊C₄	—	178	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	178	4	no	—
153	C₄²	C₂²⋊D₈₉	1	4	yes	—
154	C₂³	C₂³×D₈₉	1	1	yes	characteristic
155	Q₈	—	2	8	no	—
156	Q₈	—	178	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	89	4	no	—
165	C₂×C₄	—	178	2	no	—
166	C₂×C₄	—	178	4	no	—
167	C₂×C₄	—	178	4	no	—
168	C₂×C₄	—	178	2	no	—
169	C₂×C₄	—	178	4	no	—
170	C₂×C₄	—	178	2	no	—
171	C₂×C₄	—	178	2	no	—
172	C₂×C₄	—	178	2	no	—
173	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
176	C₂×C₄	Q₈×D₈₉	1	2	yes	—
177	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
178	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
179	C₂×C₄	#13083574	1	2	yes	—
180	C₂×C₄	#13083575	1	2	yes	—
181	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
183	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
184	C₂²	C₂×#13083574	1	1	yes	characteristic
185	C₂²	C₂×#13083575	1	1	yes	characteristic
186	C₂²	#15668058	1	1	yes	characteristic
187	C₂²	#15668059	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	4	2	no	—
191	C₄	—	356	2	no	—
192	C₄	—	356	4	no	—
193	C₄	—	178	4	no	—
194	C₄	#15667988	1	4	yes	—
195	C₄	#15667991	1	4	yes	—
196	C₂	C₂×#15667988	1	1	yes	characteristic
197	C₂	C₂×#15667991	1	1	yes	characteristic
198	C₂	#19152176	1	1	yes	characteristic
199	C₂	#19152326	1	1	yes	characteristic
200	C₂	#19152327	1	1	yes	characteristic
201	C₂	Q₈×Dic₁₇₈	1	1	yes	characteristic
202	C₂	#19152333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×C₄×Q₈×C₈₉	C₂
hypercenter	C₂×C₄×Q₈	D₈₉
hypocenter	C₈₉	#2582
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
29. C₂×Dic₁₇₈	Q₈	semidirect
42. C₂×Q₈×C₈₉	Q₈	semidirect
102. C₈₉	#2582	semidirect
134. C₂×Q₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇₈	2
#19151531	2
#19151505	2
#19151504	2
#19151496	2
#19151494	2
#19151468	2
C₂×C₄×Q₈×C₈₉	2
#2582	89