Subgroups — #11796050

Order 1152.

330 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#63371	C₂	1	2	yes	maximal, complemented
3	#10588230	C₂	1	4	yes	maximal, complemented
4	#72007	—	3	1	no	maximal
5	C₂×C₉×#41	C₄	1	1	yes	characteristic
6	C₉×#131	—	2	2	no	—
7	C₉×#131	—	2	2	no	—
8	C₂×C₉×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₉×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₄	1	2	yes	—
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	#63371	—	2	2	no	—
13	#63371	C₂²	1	4	yes	—
14	C₂×C₉⋊C₁₆	—	2	2	no	—
15	C₂×C₃×#131	D₃	1	1	yes	characteristic
16	C₂×#812	—	3	2	no	—
17	#4553	—	3	4	no	—
18	C₉×#41	—	2	2	no	—
19	C₉×#41	—	2	2	no	—
20	C₂×Q₈×C₉	C₈	1	1	yes	characteristic
21	C₂×D₄×C₉	C₈	1	2	yes	—
22	C₂×D₄×C₉	C₈	1	1	yes	characteristic
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₉×M₁₆	—	2	2	no	—
26	C₉×M₁₆	—	2	4	no	—
27	C₂×C₈×C₉	—	2	2	no	—
28	C₂×C₈×C₉	—	2	2	no	—
29	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₉	D₄	1	2	yes	—
32	C₂×C₈×C₉	D₄	1	2	yes	—
33	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
34	C₉⋊C₁₆	—	2	4	no	—
35	#2115	—	9	1	no	Sylow
36	C₂×C₃×#41	Dic₃	1	1	yes	characteristic
37	C₃×#131	—	2	2	no	—
38	C₃×#131	—	2	2	no	—
39	C₂×C₃×M₁₆	Dic₃	1	1	yes	characteristic
40	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
41	C₂²×C₃×C₈	Dic₃	1	2	yes	—
42	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
43	#812	—	3	4	no	—
44	#812	—	6	2	no	—
45	C₂×C₃⋊C₁₆	—	6	2	no	—
46	C₂³×C₉	C₂×C₈	1	2	yes	—
47	C₂³×C₉	C₂×C₈	1	1	yes	characteristic
48	Q₈×C₉	—	2	2	no	—
49	D₄×C₉	—	2	4	no	—
50	D₄×C₉	—	2	2	no	—
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
55	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×C₉	M₁₆	1	2	yes	—
57	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₉	—	2	2	no	—
59	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₈×C₉	C₂×D₄	1	2	yes	—
62	C₂×#131	D₉	1	1	yes	characteristic
63	C₂×M₃₂	—	9	2	no	—
64	#349	—	9	4	no	—
65	C₃×#41	—	2	2	no	—
66	C₃×#41	—	2	2	no	—
67	C₂×C₃×Q₈	C₃⋊C₈	1	1	yes	characteristic
68	C₂×C₃×D₄	C₃⋊C₈	1	2	yes	—
69	C₂×C₃×D₄	C₃⋊C₈	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
71	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
72	C₃×M₁₆	—	2	2	no	—
73	C₃×M₁₆	—	2	4	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
77	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₃×C₈	#67	1	2	yes	—
79	C₂×C₃×C₈	#67	1	2	yes	—
80	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
81	C₃⋊C₁₆	—	6	4	no	—
82	C₂²×C₉	—	2	2	no	—
83	C₂²×C₉	—	4	1	no	—
84	C₂²×C₉	—	4	2	no	—
85	C₂²×C₉	C₂²⋊C₈	1	2	yes	—
86	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
87	C₄×C₉	—	2	2	no	—
88	C₄×C₉	C₂²⋊C₈	1	2	yes	—
89	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂×#41	Dic₉	1	1	yes	characteristic
92	#131	—	2	2	no	—
93	#131	—	2	2	no	—
94	C₂×M₁₆	Dic₉	1	1	yes	characteristic
95	C₂×M₁₆	C₂×D₉	1	2	yes	—
96	C₂²×C₈	Dic₉	1	2	yes	—
97	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
98	M₃₂	—	9	4	no	—
99	M₃₂	—	18	2	no	—
100	C₂×C₁₆	—	18	2	no	—
101	C₂³×C₃	C₂×C₃⋊C₈	1	2	yes	—
102	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
103	C₃×Q₈	—	2	2	no	—
104	C₃×D₄	—	2	4	no	—
105	C₃×D₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
109	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
110	C₂×C₃×C₄	#208	1	2	yes	—
111	C₂×C₃×C₄	#217	1	2	yes	—
112	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
113	C₃×C₈	—	2	2	no	—
114	C₃×C₈	#217	1	1	yes	characteristic
115	C₃×C₈	#217	1	1	yes	characteristic
116	C₃×C₈	C₂×#67	1	2	yes	—
117	C₂×C₉	—	2	1	no	—
118	C₂×C₉	—	2	2	no	—
119	C₂×C₉	—	4	2	no	—
120	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	C₉⋊C₈	1	1	yes	characteristic
124	C₂×D₄	C₉⋊C₈	1	2	yes	—
125	C₂×D₄	C₉⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×Dic₉	1	2	yes	—
127	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	C₂×Dic₉	1	2	yes	—
133	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
134	C₂×C₈	#611	1	2	yes	—
135	C₂×C₈	#611	1	2	yes	—
136	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
137	C₁₆	—	18	4	no	—
138	C₂²×C₃	—	2	2	no	—
139	C₂²×C₃	—	4	1	no	—
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	#830	1	2	yes	—
142	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	#830	1	2	yes	—
145	C₃×C₄	C₂×#208	1	1	yes	characteristic
146	C₃×C₄	C₂×#217	1	1	yes	characteristic
147	C₉	#2115	1	1	yes	characteristic, complemented, Sylow
148	C₂³	C₂×C₉⋊C₈	1	2	yes	—
149	C₂³	C₂×C₉⋊C₈	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
156	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
157	C₂×C₄	#3667	1	2	yes	—
158	C₂×C₄	#3676	1	2	yes	—
159	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	#3676	1	1	yes	characteristic
162	C₈	#3676	1	1	yes	characteristic
163	C₈	C₂×#611	1	2	yes	—
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	—	2	2	no	—
166	C₂×C₃	—	4	2	no	—
167	C₂×C₃	C₂×#830	1	1	yes	characteristic
168	C₂²	—	2	2	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	2	no	—
171	C₂²	#63389	1	2	yes	—
172	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	#63389	1	2	yes	—
175	C₄	C₂×#3667	1	1	yes	characteristic
176	C₄	C₂×#3676	1	1	yes	characteristic
177	C₃	#72007	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	2	no	—
180	C₂	—	4	2	no	—
181	C₂	C₂×#63389	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#3676
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂×C₃	C₂×#830
prime radical	C₂×C₃×#41	Dic₃
cosocle	C₂×C₈×C₉	C₂³
hypercenter	C₂×#131	D₉
hypocenter	C₉	#2115
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#63371	C₂	semidirect
3. #10588230	C₂	semidirect
147. C₉	#2115	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
C₂×#63371	2
#10588230	2
#72007	3