Subgroups — #11797123

Order 1152.

432 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#63646	C₂	1	1	yes	maximal, characteristic
3	#10592805	C₂	1	1	yes	maximal, characteristic
4	#10589379	C₂	1	4	yes	maximal, complemented
5	#71427	—	3	4	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	#64135	—	2	1	no	—
8	C₂×#3748	—	2	1	no	—
9	C₃²×M₃₂	C₄	1	2	yes	complemented
10	C₃²×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃²×C₁₆	C₄	1	2	yes	complemented
12	#63646	—	2	1	no	—
13	#63646	C₂²	1	2	yes	—
14	#63645	—	2	1	no	—
15	#63644	C₂²	1	2	yes	—
16	C₂×C₃×M₃₂	D₃	1	4	yes	—
17	C₂×#819	—	3	4	no	—
18	#5106	—	3	4	no	—
19	#4513	—	3	16	no	—
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
25	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
27	#3757	—	4	1	no	—
28	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
29	C₄×C₃⋊Dic₃	—	4	1	no	—
30	#3748	—	2	2	no	—
31	#3748	—	4	1	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
33	C₃²×C₁₆	C₂×C₄	1	4	yes	—
34	#2155	—	9	1	no	Sylow
35	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
36	#926	—	6	4	no	—
37	C₂×#208	—	6	4	no	—
38	C₃×M₃₂	Dic₃	1	8	yes	complemented
39	C₃×M₃₂	C₂×D₃	1	8	yes	—
40	C₂×C₃×C₁₆	Dic₃	1	8	yes	complemented
41	#819	—	3	8	no	—
42	#819	—	6	4	no	—
43	#818	—	6	4	no	—
44	#817	—	3	8	no	—
45	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
49	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
50	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃⋊Dic₃	—	4	2	no	—
53	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₃⋊(C₃⋊C₈)	—	4	2	no	—
58	C₂×M₃₂	C₃⋊D₃	1	1	yes	characteristic
59	C₂×#108	—	9	1	no	—
60	#427	—	9	1	no	—
61	#365	—	9	4	no	—
62	C₂²×C₃×C₄	D₁₂	1	4	yes	—
63	C₂×C₃×C₈	Dic₆	1	4	yes	—
64	C₂×C₃×C₈	Dic₆	1	4	yes	—
65	C₂×C₃×C₈	D₁₂	1	4	yes	—
66	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
67	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
68	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
69	#217	—	12	4	no	—
70	C₄⋊Dic₃	—	6	8	no	—
71	C₄×Dic₃	—	12	4	no	—
72	#208	—	6	8	no	—
73	#208	—	12	4	no	—
74	C₂×C₃⋊C₈	—	12	4	no	—
75	C₃×C₁₆	C₂×Dic₃	1	16	yes	—
76	C₂²×C₃²	—	4	1	no	—
77	C₂²×C₃²	#106	1	1	yes	characteristic
78	C₂²×C₃²	#106	1	1	yes	characteristic
79	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
80	C₃²×C₄	#106	1	1	yes	characteristic
81	C₃²×C₄	#106	1	1	yes	characteristic
82	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
84	C₃⋊Dic₃	—	8	2	no	—
85	C₂×M₁₆	—	18	1	no	—
86	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
87	#117	—	18	1	no	—
88	M₃₂	C₃⋊Dic₃	1	2	yes	complemented
89	M₃₂	C₂×C₃⋊D₃	1	2	yes	—
90	C₂×C₁₆	C₃⋊Dic₃	1	2	yes	complemented
91	#108	—	9	2	no	—
92	#108	—	18	1	no	—
93	#107	—	18	1	no	—
94	#106	—	9	2	no	—
95	C₂³×C₃	#204	1	4	yes	—
96	C₂×C₃×C₄	#204	1	4	yes	—
97	C₂×C₃×C₄	#204	1	4	yes	—
98	C₂×C₃×C₄	#204	1	4	yes	—
99	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
100	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
101	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
102	C₂×Dic₃	—	12	8	no	—
103	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
104	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
105	C₃×C₈	C₂×Dic₆	1	4	yes	—
106	C₃×C₈	C₂²×Dic₃	1	4	yes	—
107	C₃⋊C₈	—	12	8	no	—
108	C₂×C₃²	—	2	1	no	—
109	C₂×C₃²	—	2	1	no	—
110	C₂×C₃²	—	2	1	no	—
111	C₂×C₃²	C₂×#106	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
113	M₁₆	—	18	2	no	—
114	M₁₆	—	36	1	no	—
115	C₂×C₈	—	36	1	no	—
116	C₂×C₈	#634	1	1	yes	characteristic
117	C₂×C₈	#634	1	1	yes	characteristic
118	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
119	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
120	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
122	C₄⋊C₄	—	18	2	no	—
123	C₂²⋊C₄	—	36	1	no	—
124	C₄²	—	36	1	no	—
125	C₁₆	C₂×C₃⋊Dic₃	1	4	yes	—
126	C₂²×C₃	—	4	4	no	—
127	C₂²×C₃	#817	1	4	yes	—
128	C₂²×C₃	#817	1	4	yes	—
129	C₂²×C₃	C₂×#204	1	4	yes	—
130	C₃×C₄	#817	1	4	yes	—
131	C₃×C₄	#817	1	4	yes	—
132	C₃×C₄	C₂×#204	1	4	yes	—
133	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
134	Dic₃	—	24	8	no	—
135	C₃²	#2155	1	1	yes	characteristic, complemented, Sylow
136	C₂³	#3744	1	1	yes	characteristic
137	C₂×C₄	—	36	2	no	—
138	C₂×C₄	#3744	1	1	yes	characteristic
139	C₂×C₄	#3744	1	1	yes	characteristic
140	C₂×C₄	#3744	1	1	yes	characteristic
141	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
142	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₈	—	36	2	no	—
145	C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
146	C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
147	C₈	C₂×#634	1	1	yes	characteristic
148	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
149	C₂×C₃	—	2	4	no	—
150	C₂×C₃	—	2	4	no	—
151	C₂×C₃	—	2	4	no	—
152	C₂×C₃	C₂×#817	1	4	yes	—
153	C₂²	—	4	1	no	—
154	C₂²	#63644	1	1	yes	characteristic
155	C₂²	#63644	1	1	yes	characteristic
156	C₂²	C₂×#3744	1	1	yes	characteristic
157	C₄	—	72	2	no	—
158	C₄	#63644	1	1	yes	characteristic
159	C₄	#63644	1	1	yes	characteristic
160	C₄	C₂×#3744	1	1	yes	characteristic
161	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
162	C₃	#71427	1	4	yes	complemented
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	C₂×#63644	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63644
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×M₃₂	C₂
socle	C₂×C₃²	C₂×#106
prime radical	C₂³×C₃²	SD₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×M₃₂	C₃⋊D₃
hypocenter	C₃²	#2155
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10589379	C₂	semidirect
9. C₃²×M₃₂	C₄	semidirect
11. C₂×C₃²×C₁₆	C₄	semidirect
38. C₃×M₃₂	Dic₃	semidirect
40. C₂×C₃×C₁₆	Dic₃	semidirect
88. M₃₂	C₃⋊Dic₃	semidirect
90. C₂×C₁₆	C₃⋊Dic₃	semidirect
135. C₃²	#2155	semidirect
162. C₃	#71427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×M₃₂	2
C₂×#63646	2
#10592805	2
#10589379	2
#71427	3