Subgroups — #36465985

Order 22784.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	#26018317	C₂	1	4	yes	maximal, complemented
4	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₄	1	1	yes	characteristic
6	C₂×#15667952	C₄	1	1	yes	characteristic
7	#19151651	—	2	1	no	—
8	#19151374	C₂²	1	2	yes	—
9	C₄×#13083549	C₄	1	2	yes	—
10	#19151351	—	2	1	no	—
11	#19151349	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#334×C₈₉	—	2	1	no	—
15	#334×C₈₉	C₂²	1	2	yes	—
16	#15667952	—	2	2	no	—
17	#15667952	C₈	1	4	yes	complemented
18	C₂×#13083549	C₂×C₄	1	2	yes	—
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₂²⋊#10623494	—	4	1	no	—
22	C₄×#10623494	—	4	1	no	—
23	C₂²×C₈×C₈₉	—	4	1	no	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	2	no	—
28	C₄⋊C₈×C₈₉	—	4	1	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×#10623494	—	4	2	no	—
34	#13083549	—	2	2	no	—
35	#13083549	C₂×C₈	1	4	yes	complemented
36	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	—	8	2	no	—
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
46	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
50	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
51	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₈₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#108	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	—	2	1	no	—
65	C₈×C₈₉	—	8	2	no	—
66	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
67	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
68	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
69	#10623494	—	8	2	no	—
70	C₂²×C₈₉	—	2	1	no	—
71	C₂²×C₈₉	—	2	1	no	—
72	C₂²×C₈₉	#334	1	2	yes	—
73	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	#334	1	2	yes	—
78	C₄×C₈₉	#365	1	2	yes	—
79	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
81	#12000	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	2	no	—
83	C₂×C₈₉	C₂×#334	1	1	yes	characteristic
84	C₂×C₈₉	C₂×#365	1	1	yes	characteristic
85	C₂×C₈₉	#2155	1	1	yes	characteristic
86	C₄×M₃₂	—	89	1	no	—
87	#1564	—	89	1	no	—
88	C₂×#334	D₈₉	1	1	yes	characteristic
89	#1368	—	89	4	no	—
90	C₈₉	#12000	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	89	1	no	—
92	C₂×M₃₂	—	89	1	no	—
93	#431	—	178	1	no	—
94	C₂×C₄⋊C₈	—	2	1	no	—
95	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
96	#346	—	89	2	no	—
97	C₄×C₁₆	—	89	2	no	—
98	#335	—	178	1	no	—
99	#334	—	2	1	no	—
100	#334	—	89	2	no	—
101	#334	C₂×D₈₉	1	2	yes	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
106	M₃₂	—	89	4	no	—
107	M₃₂	—	178	2	no	—
108	C₂×C₁₆	—	89	2	no	—
109	C₂×C₁₆	—	89	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	178	2	no	—
113	C₂²⋊C₈	—	356	1	no	—
114	C₄×C₈	—	356	1	no	—
115	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
116	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
119	C₂²×C₄	#13083553	1	1	yes	characteristic
120	C₂²×C₄	#13083561	1	1	yes	characteristic
121	C₂²×C₄	#13083562	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	8	2	no	—
125	C₂×C₈	—	356	2	no	—
126	C₂×C₈	#13083553	1	1	yes	characteristic
127	C₂×C₈	#13083553	1	1	yes	characteristic
128	C₂×C₈	C₈×D₈₉	1	2	yes	—
129	C₂×C₈	#13083557	1	2	yes	—
130	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
132	C₄²	#13083553	1	1	yes	characteristic
133	C₄²	#13083561	1	1	yes	characteristic
134	C₄²	#13083562	1	1	yes	characteristic
135	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
136	C₁₆	—	89	4	no	—
137	C₁₆	—	178	2	no	—
138	C₂³	#15667943	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	#15667941	1	2	yes	—
142	C₂×C₄	#15667943	1	1	yes	characteristic
143	C₂×C₄	#15667943	1	1	yes	characteristic
144	C₂×C₄	#15667943	1	1	yes	characteristic
145	C₂×C₄	#15667947	1	2	yes	—
146	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
147	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
148	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	712	2	no	—
153	C₈	#15667941	1	2	yes	—
154	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
155	C₈	C₂×#13083557	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#19151348	1	2	yes	—
159	C₂²	C₂×#15667943	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#19151348	1	2	yes	—
164	C₄	#19151420	1	2	yes	—
165	C₄	C₂×#15667941	1	1	yes	characteristic
166	C₄	C₂×#15667947	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#19151348	1	1	yes	characteristic
169	C₂	C₂×#19151420	1	1	yes	characteristic
170	C₂	#26022026	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667943
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×C₈₉	#106
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#12000
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #26018317	C₂	semidirect
17. #15667952	C₈	semidirect
35. #13083549	C₂×C₈	semidirect
90. C₈₉	#12000	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667952	2
#26019000	2
#26018317	2
C₂×#334×C₈₉	2
#12000	89