Subgroups — #26029328

Order 11392.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151676	C₂	1	2	yes	maximal, complemented
2	#19151670	C₂	1	4	yes	maximal, complemented
3	D₄×#10623494	C₂	1	2	yes	maximal, complemented
4	#19151655	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151651	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151565	C₂	1	4	yes	maximal, complemented
7	#532×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083558	C₂²	1	4	yes	—
9	C₂²×#10623494	C₂²	1	4	yes	—
10	C₄⋊#10623494	C₂²	1	2	yes	—
11	C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂²⋊#10623494	C₂²	1	2	yes	—
13	C₂²⋊#10623494	C₂²	1	2	yes	—
14	C₂²⋊#10623494	C₂²	1	4	yes	—
15	C₂²⋊#10623494	C₂²	1	4	yes	—
16	#15667923	C₂²	1	2	yes	—
17	C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₂×#41×C₈₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
21	#127×C₈₉	C₄	1	1	yes	characteristic
22	#124×C₈₉	C₄	1	2	yes	—
23	#121×C₈₉	C₄	1	2	yes	—
24	#121×C₈₉	C₄	1	2	yes	—
25	C₄×D₄×C₈₉	C₂²	1	2	yes	—
26	C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
28	#13083558	—	4	4	no	—
29	C₂×#10623494	—	4	4	no	—
30	C₂×#10623494	C₂³	1	2	yes	—
31	C₂×#10623494	C₂³	1	4	yes	—
32	C₂×#10623494	C₂³	1	2	yes	—
33	#41×C₈₉	—	4	4	no	—
34	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₈₉	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
47	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂³×C₈₉	C₂²×C₄	1	4	yes	—
52	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₈₉	—	4	2	no	—
54	D₄×C₈₉	—	2	4	no	—
55	D₄×C₈₉	—	4	2	no	—
56	D₄×C₈₉	—	4	2	no	—
57	D₄×C₈₉	—	4	4	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	4	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	4	1	no	—
63	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
70	#10623494	—	2	2	no	—
71	#10623494	—	2	4	no	—
72	#10623494	—	2	2	no	—
73	C₂²×C₈₉	—	2	1	no	—
74	C₂²×C₈₉	—	2	4	no	—
75	C₂²×C₈₉	—	2	1	no	—
76	C₂²×C₈₉	—	2	4	no	—
77	C₂²×C₈₉	—	2	1	no	—
78	C₂²×C₈₉	—	2	4	no	—
79	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₈₉	—	2	1	no	—
81	C₄×C₈₉	—	2	4	no	—
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	#131	1	2	yes	—
86	C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₈₉	—	4	1	no	—
88	C₂×C₈₉	—	4	4	no	—
89	C₂×C₈₉	#518	1	1	yes	characteristic
90	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
91	C₂×C₈₉	#568	1	1	yes	characteristic
92	#2981	—	89	1	no	maximal, Sylow
93	C₈₉	#2981	1	1	yes	characteristic, complemented, Sylow
94	#532	D₈₉	1	1	yes	characteristic
95	#436	—	89	2	no	—
96	#435	—	89	4	no	—
97	C₈×D₄	—	89	2	no	—
98	#432	—	89	1	no	—
99	#431	—	89	1	no	—
100	#408	—	89	4	no	—
101	C₂×#41	C₂×D₈₉	1	2	yes	—
102	C₂×M₁₆	—	89	4	no	—
103	C₂²×C₈	—	89	4	no	—
104	C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
105	#127	Dic₈₉	1	1	yes	characteristic
106	#124	Dic₈₉	1	2	yes	—
107	#121	Dic₈₉	1	2	yes	—
108	#121	Dic₈₉	1	2	yes	—
109	C₄×D₄	C₂×D₈₉	1	2	yes	—
110	C₄×D₄	C₂×D₈₉	1	2	yes	—
111	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
112	C₄⋊C₈	—	89	2	no	—
113	C₄⋊C₈	—	89	2	no	—
114	C₂²⋊C₈	—	89	2	no	—
115	C₂²⋊C₈	—	89	2	no	—
116	C₂²⋊C₈	—	89	4	no	—
117	C₂²⋊C₈	—	89	4	no	—
118	#97	—	89	2	no	—
119	C₄×C₈	—	89	1	no	—
120	C₄×C₈	—	89	1	no	—
121	#41	—	4	4	no	—
122	C₂×Q₈	C₂×Dic₈₉	1	2	yes	—
123	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
124	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
125	C₂×D₄	C₂×Dic₈₉	1	4	yes	—
126	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
127	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
131	M₁₆	—	356	4	no	—
132	C₂×C₈	—	89	2	no	—
133	C₂×C₈	—	89	2	no	—
134	C₂×C₈	—	89	4	no	—
135	C₂×C₈	—	356	4	no	—
136	C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
137	C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
138	C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
139	C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
140	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
141	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
142	C₄²	C₂²×D₈₉	1	1	yes	characteristic
143	C₄²	C₂²×D₈₉	1	1	yes	characteristic
144	C₂³	C₂²×Dic₈₉	1	4	yes	—
145	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₈₉	1	4	yes	—
161	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
163	C₈	—	178	2	no	—
164	C₈	—	178	4	no	—
165	C₈	—	178	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#15668019	1	2	yes	—
179	C₄	#15668057	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#19152073	1	1	yes	characteristic
183	C₂	C₂×#15668019	1	1	yes	characteristic
184	C₂	#19152365	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting, prime radical	#532×C₈₉	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#532	D₈₉
hypocenter	C₈₉	#2981
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #19151676	C₂	semidirect
2. #19151670	C₂	semidirect
3. D₄×#10623494	C₂	semidirect
4. #19151655	C₂	semidirect
5. #19151651	C₂	semidirect
6. #19151565	C₂	semidirect
93. C₈₉	#2981	semidirect

Maximal subgroups

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

subgroup	index
#19151676	2
#19151670	2
D₄×#10623494	2
#19151655	2
#19151651	2
#19151565	2
#532×C₈₉	2
#2981	89