Subgroups — #36441313

Order 22784.

202 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151419	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
3	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	—	2	1	no	—
5	C₂²×#13083549	C₄	1	1	yes	characteristic
6	C₂²×#13083549	C₄	1	1	yes	characteristic
7	#19151419	—	2	1	no	—
8	#19151419	C₄	1	2	yes	—
9	#19151374	—	2	1	no	—
10	#19151374	—	2	1	no	—
11	C₂×M₃₂×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#364×C₈₉	—	2	1	no	—
14	#364×C₈₉	C₄	1	2	yes	—
15	#15667952	—	2	2	no	—
16	#15667952	—	4	1	no	—
17	C₂×#13083549	—	2	1	no	—
18	C₂×#13083549	—	2	1	no	—
19	C₂×#13083549	—	2	1	no	—
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	4	1	no	—
22	C₂×#13083549	C₈	1	2	yes	—
23	C₂×#13083549	C₈	1	2	yes	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
27	M₃₂×C₈₉	—	2	2	no	—
28	M₃₂×C₈₉	—	4	1	no	—
29	C₂×C₁₆×C₈₉	—	4	1	no	—
30	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
34	#13083549	—	2	2	no	—
35	#13083549	—	2	2	no	—
36	#13083549	—	4	2	no	—
37	C₂²×C₄×C₈₉	—	2	1	no	—
38	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
51	C₄²×C₈₉	—	2	1	no	—
52	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₈₉	—	4	2	no	—
55	C₂³×C₈₉	#108	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	4	2	no	—
59	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#108	1	1	yes	characteristic
65	C₈×C₈₉	C₄×C₈	1	1	yes	characteristic
66	C₈×C₈₉	#97	1	1	yes	characteristic
67	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
68	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₈₉	#102	1	1	yes	characteristic
70	C₈×C₈₉	#102	1	1	yes	characteristic
71	C₈×C₈₉	#103	1	1	yes	characteristic
72	C₈×C₈₉	#103	1	1	yes	characteristic
73	C₂²×C₈₉	—	2	2	no	—
74	C₂²×C₈₉	#334	1	1	yes	characteristic
75	C₂²×C₈₉	#335	1	1	yes	characteristic
76	C₂²×C₈₉	#341	1	1	yes	characteristic
77	C₄×C₈₉	—	4	1	no	—
78	C₄×C₈₉	—	4	1	no	—
79	C₄×C₈₉	#334	1	1	yes	characteristic
80	C₄×C₈₉	#335	1	1	yes	characteristic
81	C₄×C₈₉	#336	1	1	yes	characteristic
82	C₄×C₈₉	#340	1	1	yes	characteristic
83	#7161	—	89	1	no	maximal, Sylow
84	C₂×C₈₉	—	2	1	no	—
85	C₂×C₈₉	—	2	1	no	—
86	C₂×C₈₉	#1277	1	1	yes	characteristic
87	C₂×C₈₉	#1369	1	2	yes	—
88	C₂×#364	—	89	1	no	—
89	C₂×#364	D₈₉	1	1	yes	characteristic
90	C₂×#346	—	89	1	no	—
91	C₈₉	#7161	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	2	1	no	—
93	C₂×M₃₂	—	178	1	no	—
94	C₂²×C₁₆	—	89	1	no	—
95	C₂²×C₁₆	—	89	1	no	—
96	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
97	#364	—	2	1	no	—
98	#364	—	89	2	no	—
99	#364	—	178	1	no	—
100	#364	Dic₈₉	1	2	yes	—
101	#346	—	178	1	no	—
102	#346	—	178	1	no	—
103	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
105	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	4	1	no	—
108	M₃₂	—	356	1	no	—
109	M₃₂	—	178	2	no	—
110	C₂×C₁₆	—	4	1	no	—
111	C₂×C₁₆	—	89	2	no	—
112	C₂×C₁₆	—	89	2	no	—
113	C₂×C₁₆	—	356	1	no	—
114	C₂×C₁₆	—	178	1	no	—
115	C₂×C₁₆	—	178	1	no	—
116	C₂×C₁₆	—	178	1	no	—
117	C₂×C₁₆	—	178	1	no	—
118	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
119	C₄×C₈	D₃₅₆	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#13083553	1	1	yes	characteristic
124	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
125	C₂×C₈	#13083551	1	1	yes	characteristic
126	C₂×C₈	#13083551	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
128	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
129	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
130	C₂×C₈	#13083557	1	1	yes	characteristic
131	C₂×C₈	#13083557	1	1	yes	characteristic
132	C₂×C₈	#13083559	1	1	yes	characteristic
133	C₂×C₈	#13083561	1	1	yes	characteristic
134	C₂×C₈	#13083562	1	1	yes	characteristic
135	C₂×C₈	#13083564	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	#13083553	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	C₁₆	—	356	2	no	—
141	C₁₆	—	178	2	no	—
142	C₁₆	—	178	2	no	—
143	C₂³	#15667947	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	#15667919	1	1	yes	characteristic
148	C₂×C₄	#15667941	1	1	yes	characteristic
149	C₂×C₄	#15667941	1	1	yes	characteristic
150	C₂×C₄	#15667943	1	1	yes	characteristic
151	C₂×C₄	#15667945	1	1	yes	characteristic
152	C₂×C₄	#15667947	1	1	yes	characteristic
153	C₈	C₈×Dic₈₉	1	1	yes	characteristic
154	C₈	#15667922	1	1	yes	characteristic
155	C₈	#15667924	1	1	yes	characteristic
156	C₈	#15667924	1	1	yes	characteristic
157	C₈	#15667932	1	1	yes	characteristic
158	C₈	#15667933	1	1	yes	characteristic
159	C₈	#15667935	1	1	yes	characteristic
160	C₈	#15667936	1	1	yes	characteristic
161	C₂²	—	2	2	no	—
162	C₂²	#19151348	1	1	yes	characteristic
163	C₂²	#19151350	1	1	yes	characteristic
164	C₂²	#19151362	1	1	yes	characteristic
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	#19151348	1	1	yes	characteristic
168	C₄	#19151350	1	1	yes	characteristic
169	C₄	#19151352	1	1	yes	characteristic
170	C₄	#19151360	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#26018086	1	1	yes	characteristic
174	C₂	#26018319	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667943
commutator	C₈×C₈₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	#334
prime radical	C₂³×C₈₉	#108
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#7161
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
91. C₈₉	#7161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151419	2
C₂×#19151374	2
C₂×#364×C₈₉	2
#7161	89