Subgroups — #36466017

Order 22784.

242 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	#26019000	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
3	#26018326	C₂	1	2	yes	maximal, complemented
4	#26018324	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	—	2	1	no	—
7	C₂×C₄⋊#10623494	—	2	1	no	—
8	#19151351	—	2	1	no	—
9	#19151351	C₂²	1	1	yes	characteristic
10	#19151351	C₂²	1	2	yes	—
11	#19151351	C₂²	1	1	yes	characteristic
12	#19151349	—	2	1	no	—
13	C₂×M₃₂×C₈₉	C₄	1	1	yes	characteristic
14	C₂×M₃₂×C₈₉	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#346×C₈₉	C₄	1	2	yes	—
17	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂²×#10623494	—	4	1	no	—
19	C₄⋊#10623494	—	2	2	no	—
20	C₄⋊#10623494	—	2	1	no	—
21	C₄⋊#10623494	—	2	1	no	—
22	C₄⋊#10623494	—	4	1	no	—
23	C₂²⋊#10623494	—	4	1	no	—
24	C₄×#10623494	—	4	1	no	—
25	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	M₃₂×C₈₉	—	2	2	no	—
29	M₃₂×C₈₉	C₈	1	4	yes	complemented
30	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×#10623494	—	4	2	no	—
37	C₂×#10623494	—	4	1	no	—
38	C₂×#10623494	—	4	1	no	—
39	C₂×#10623494	—	8	2	no	—
40	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
43	C₂×C₈×C₈₉	—	2	2	no	—
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
48	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₈₉	D₈	1	1	yes	characteristic
52	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
53	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₁₆×C₈₉	—	2	2	no	—
55	C₁₆×C₈₉	C₂×C₈	1	4	yes	complemented
56	C₂³×C₈₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	—	4	1	no	—
60	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#108	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₈₉	—	2	1	no	—
69	C₈×C₈₉	—	2	1	no	—
70	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
71	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
73	#10623494	—	8	1	no	—
74	#10623494	—	8	2	no	—
75	#10623494	—	8	1	no	—
76	C₂²×C₈₉	—	2	1	no	—
77	C₂²×C₈₉	—	2	1	no	—
78	C₂²×C₈₉	#335	1	2	yes	—
79	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
80	C₄×C₈₉	—	2	1	no	—
81	C₄×C₈₉	—	2	1	no	—
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	#335	1	2	yes	—
84	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
86	C₄×C₈₉	#509	1	1	yes	characteristic
87	C₄×C₈₉	#510	1	1	yes	characteristic
88	#12009	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	—	2	2	no	—
90	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
91	C₂×C₈₉	#2159	1	1	yes	characteristic
92	C₂×C₈₉	#2161	1	1	yes	characteristic
93	C₄×M₃₂	D₈₉	1	1	yes	characteristic
94	#1564	—	89	1	no	—
95	C₂×#335	—	89	1	no	—
96	#1372	—	89	2	no	—
97	#1371	—	89	2	no	—
98	C₈₉	#12009	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	Dic₈₉	1	1	yes	characteristic
100	C₂×M₃₂	Dic₈₉	1	1	yes	characteristic
101	#431	—	178	1	no	—
102	C₂×C₄⋊C₈	—	178	1	no	—
103	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
104	#346	Dic₈₉	1	2	yes	—
105	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
106	#335	—	89	2	no	—
107	#335	—	89	1	no	—
108	#335	—	89	1	no	—
109	#335	—	178	1	no	—
110	#334	—	178	1	no	—
111	C₂²×C₈	—	356	1	no	—
112	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
114	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	#10623494	1	4	yes	complemented
117	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
118	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
119	C₄⋊C₈	—	356	1	no	—
120	C₄⋊C₈	—	178	1	no	—
121	C₄⋊C₈	—	178	1	no	—
122	C₄⋊C₈	—	178	2	no	—
123	C₂²⋊C₈	—	356	1	no	—
124	C₄×C₈	—	356	1	no	—
125	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
126	C₄×C₈	D₃₅₆	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
130	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
131	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	356	1	no	—
134	C₂×C₈	—	356	1	no	—
135	C₂×C₈	—	356	2	no	—
136	C₂×C₈	—	712	2	no	—
137	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
139	C₂×C₈	C₂×#10623494	1	2	yes	—
140	C₂×C₈	#13083558	1	2	yes	—
141	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
143	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
144	C₄²	D₇₁₂	1	1	yes	characteristic
145	C₄²	Dic₃₅₆	1	1	yes	characteristic
146	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	C₂×#10623494	1	4	yes	complemented
149	C₂³	#15667946	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊#10623494	1	2	yes	—
154	C₂×C₄	#15667946	1	1	yes	characteristic
155	C₂×C₄	#15667946	1	1	yes	characteristic
156	C₂×C₄	#15667946	1	1	yes	characteristic
157	C₂×C₄	#15667948	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
159	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	712	1	no	—
164	C₈	—	712	2	no	—
165	C₈	—	712	1	no	—
166	C₈	C₄⋊#10623494	1	2	yes	—
167	C₈	C₂²×#10623494	1	1	yes	characteristic
168	C₈	C₂×#13083558	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#19151351	1	2	yes	—
172	C₂²	C₂×#15667946	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#19151351	1	2	yes	—
177	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
178	C₄	C₂×#15667948	1	1	yes	characteristic
179	C₄	#19152036	1	1	yes	characteristic
180	C₄	#19152041	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#19151351	1	1	yes	characteristic
183	C₂	#26022040	1	1	yes	characteristic
184	C₂	#26022047	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	#107
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#12009
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #26018326	C₂	semidirect
4. #26018324	C₂	semidirect
29. M₃₂×C₈₉	C₈	semidirect
55. C₁₆×C₈₉	C₂×C₈	semidirect
98. C₈₉	#12009	semidirect
116. M₃₂	#10623494	semidirect
148. C₁₆	C₂×#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
C₂×#19151351	2
#26018326	2
#26018324	2
C₄×M₃₂×C₈₉	2
#12009	89