Subgroups — C₂²×#11763867 #17981020

Order 4352.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763867	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#10587538	C₂	1	3	yes	maximal, complemented
3	C₂³×#63216	C₂	1	3	yes	maximal, complemented
4	C₂²×#10587634	C₂	1	1	yes	maximal, characteristic
5	#11763867	C₂²	1	64	yes	direct product factor
6	C₂×#10587538	C₂²	1	36	yes	complemented
7	C₂²×#63216	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	3	yes	—
9	C₂²×#63216	C₂²	1	36	yes	complemented
10	C₂×#10587634	C₄	1	12	yes	—
11	C₂×#10587634	C₂²	1	12	yes	—
12	C₂²×#63223	C₄	1	3	yes	—
13	C₂³×#3622	C₂²	1	3	yes	—
14	C₂²×#41×C₁₇	C₄	1	1	yes	characteristic
15	C₂×#41×C₁₇	C₈	1	12	yes	—
16	C₂×#41×C₁₇	C₂×C₄	1	12	yes	—
17	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
18	C₂²×D₄×C₁₇	C₈	1	3	yes	—
19	C₂³×C₄×C₁₇	C₂×C₄	1	3	yes	—
20	#10587634	C₂×C₄	1	48	yes	—
21	#10587634	C₂³	1	16	yes	—
22	C₂×#63223	C₂×C₄	1	36	yes	—
23	C₂²×#3622	C₂×C₄	1	3	yes	—
24	C₂²×#3622	C₂×C₄	1	18	yes	—
25	C₂²×#3622	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂³	1	18	yes	—
27	#10587538	C₂³	1	48	yes	complemented
28	C₂×#63216	C₂³	1	6	yes	—
29	C₂×#63216	C₂³	1	18	yes	—
30	C₂×#63216	C₂³	1	48	yes	complemented
31	C₂⁴×C₁₇	C₂×C₈	1	3	yes	—
32	#41×C₁₇	C₂×C₈	1	48	yes	—
33	#41×C₁₇	C₂²×C₄	1	16	yes	—
34	C₂×Q₈×C₁₇	C₂×C₈	1	12	yes	—
35	C₂×D₄×C₁₇	C₂×C₈	1	36	yes	—
36	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₁₇	C₂×C₈	1	18	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	18	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	#63223	C₂²×C₄	1	48	yes	—
41	C₂×#3622	C₂²×C₄	1	3	yes	—
42	C₂×#3622	C₂²×C₄	1	18	yes	—
43	C₂×#3622	C₂²×C₄	1	36	yes	—
44	C₂×#3622	C₂⁴	1	3	yes	—
45	C₂×#3622	C₂⁴	1	12	yes	—
46	#63216	C₂⁴	1	4	yes	—
47	#63216	C₂⁴	1	12	yes	—
48	C₂²×#504	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	—	2	12	no	—
50	C₂³×C₁₇	C₂²×C₈	1	18	yes	—
51	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
52	Q₈×C₁₇	C₂²×C₈	1	16	yes	—
53	D₄×C₁₇	C₂²×C₈	1	48	yes	—
54	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
55	C₂×C₄×C₁₇	C₂²×C₈	1	18	yes	—
56	C₂×C₄×C₁₇	C₂²×C₈	1	36	yes	—
57	C₂×C₄×C₁₇	C₂³×C₄	1	3	yes	—
58	C₂×C₄×C₁₇	C₂³×C₄	1	12	yes	—
59	#3622	C₂³×C₄	1	3	yes	—
60	#3622	C₂³×C₄	1	12	yes	—
61	#3622	C₂⁵	1	1	yes	characteristic
62	C₂²×#131	—	17	1	no	—
63	C₂×#504	—	17	24	no	—
64	C₂²×M₃₂	—	17	3	no	—
65	C₂³×C₁₆	—	17	3	no	—
66	C₂²×C₁₇	—	2	36	no	—
67	C₂²×C₁₇	#504	1	4	yes	complemented
68	C₂²×C₁₇	C₂³×C₈	1	12	yes	—
69	C₂²×C₁₇	C₂³×C₈	1	3	yes	—
70	C₄×C₁₇	C₂³×C₈	1	3	yes	—
71	C₄×C₁₇	C₂³×C₈	1	12	yes	—
72	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
73	C₂²×#41	#595	1	1	yes	characteristic
74	C₂×#131	—	17	12	no	—
75	C₂×#131	—	17	12	no	—
76	C₂²×M₁₆	—	17	3	no	—
77	C₂³×C₈	—	17	3	no	—
78	#504	—	17	64	no	—
79	C₂×M₃₂	—	17	36	no	—
80	C₂²×C₁₆	—	17	1	no	—
81	C₂²×C₁₆	—	17	3	no	—
82	C₂²×C₁₆	—	17	36	no	—
83	C₂×C₁₇	—	2	12	no	—
84	C₂×C₁₇	C₂×#504	1	6	yes	complemented
85	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
86	C₂×#41	#3624	1	12	yes	—
87	C₂×#41	C₂×#595	1	12	yes	—
88	C₂²×Q₈	#3624	1	1	yes	characteristic
89	C₂²×D₄	#3624	1	3	yes	—
90	C₂³×C₄	C₂×#595	1	3	yes	—
91	#131	—	17	48	no	—
92	#131	—	17	16	no	—
93	C₂×M₁₆	—	17	36	no	—
94	C₂²×C₈	—	17	18	no	—
95	C₂²×C₈	—	17	18	no	—
96	C₂²×C₈	—	17	1	no	—
97	C₂²×C₈	—	17	3	no	—
98	M₃₂	—	17	48	no	—
99	C₂×C₁₆	—	17	6	no	—
100	C₂×C₁₆	—	17	18	no	—
101	C₂×C₁₆	—	17	48	no	—
102	C₁₇	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴	C₂×#3624	1	3	yes	—
104	#41	C₂×#3624	1	48	yes	—
105	#41	C₂²×#595	1	16	yes	—
106	C₂×Q₈	C₂×#3624	1	12	yes	—
107	C₂×D₄	C₂×#3624	1	36	yes	—
108	C₂²×C₄	C₂×#3624	1	3	yes	—
109	C₂²×C₄	C₂×#3624	1	18	yes	—
110	C₂²×C₄	C₂²×#595	1	18	yes	—
111	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
112	M₁₆	—	17	48	no	—
113	C₂×C₈	—	17	3	no	—
114	C₂×C₈	—	17	12	no	—
115	C₂×C₈	—	17	3	no	—
116	C₂×C₈	—	17	18	no	—
117	C₂×C₈	—	17	36	no	—
118	C₁₆	—	17	4	no	—
119	C₁₆	—	17	12	no	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×#3624	1	18	yes	—
122	C₂³	C₂²×#3624	1	1	yes	characteristic
123	Q₈	C₂²×#3624	1	16	yes	—
124	D₄	C₂²×#3624	1	48	yes	—
125	C₂×C₄	C₂²×#3624	1	3	yes	—
126	C₂×C₄	C₂²×#3624	1	18	yes	—
127	C₂×C₄	C₂²×#3624	1	36	yes	—
128	C₂×C₄	C₂³×#595	1	3	yes	—
129	C₂×C₄	C₂³×#595	1	12	yes	—
130	C₈	—	17	1	no	—
131	C₈	—	17	3	no	—
132	C₈	—	17	12	no	—
133	C₂²	—	2	36	no	—
134	C₂²	#11763867	1	4	yes	direct product factor
135	C₂²	C₂³×#3624	1	12	yes	—
136	C₂²	C₂³×#3624	1	3	yes	—
137	C₄	C₂³×#3624	1	3	yes	—
138	C₄	C₂³×#3624	1	12	yes	—
139	C₄	C₂⁴×#595	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×#11763867	1	6	yes	direct product factor
142	C₂	C₂⁴×#3624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#595
commutator	C₂×C₁₇	C₂⁴×C₈
Frattini	C₄	C₂⁴×#595
Fitting, prime radical	C₂²×#41×C₁₇	C₄
socle	C₂³×C₁₇	C₂²×C₈
cosocle	#3622	C₂⁵
hypercenter	C₂²×#41	#595
hypocenter	C₁₇	C₂²×#504
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11763867	C₂	direct (N × H)
2. C₂²×#10587538	C₂	semidirect
3. C₂³×#63216	C₂	semidirect
5. #11763867	C₂²	direct (N × H)
6. C₂×#10587538	C₂²	semidirect
9. C₂²×#63216	C₂²	semidirect
27. #10587538	C₂³	semidirect
30. C₂×#63216	C₂³	semidirect
67. C₂²×C₁₇	#504	semidirect
84. C₂×C₁₇	C₂×#504	semidirect
102. C₁₇	C₂²×#504	semidirect
134. C₂²	#11763867	direct (N × H)
141. C₂	C₂×#11763867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763867	2
C₂²×#10587538	2
C₂³×#63216	2
C₂²×#10587634	2
C₂²×#504	17

Subgroups — C22×#11763867 #17981020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11763867 #17981020