Subgroups — C₂²×#11762675 #17192599

Order 4352.

4286 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₁₇	C₂	1	2	yes	maximal
2	C₂×#11762675	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#10587502	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₁₇	C₂²	1	4	yes	—
6	C₂²×C₄⋊Dic₁₇	—	2	16	no	—
7	C₂²×C₄⋊Dic₁₇	C₂²	1	24	yes	—
8	#11762675	C₂²	1	64	yes	direct product factor
9	C₂×#10587502	C₂²	1	48	yes	—
10	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₇	C₄	1	16	yes	complemented
13	C₂²×C₄²×C₁₇	C₂²	1	12	yes	—
14	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₁₇	—	2	16	no	—
16	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	—
17	C₂³×C₄×C₁₇	C₂×C₄	1	8	yes	—
18	C₂³×C₄×C₁₇	D₄	1	8	yes	—
19	C₂³×C₄×C₁₇	Q₈	1	8	yes	—
20	C₂³×C₄×C₁₇	C₂³	1	12	yes	—
21	C₂³×C₄×C₁₇	C₂³	1	6	yes	—
22	C₂³×Dic₁₇	—	2	16	no	—
23	C₂³×Dic₁₇	—	2	32	no	—
24	C₂³×Dic₁₇	C₂³	1	24	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	96	yes	complemented
26	C₂×C₄²×C₁₇	C₂³	1	16	yes	—
27	C₂×C₄⋊Dic₁₇	—	2	96	no	—
28	C₂×C₄⋊Dic₁₇	C₂³	1	32	yes	—
29	#10587502	C₂³	1	64	yes	—
30	C₂⁴×C₁₇	C₂²×C₄	1	4	yes	—
31	C₂⁴×C₁₇	C₂×D₄	1	4	yes	—
32	C₂⁴×C₁₇	C₂×Q₈	1	4	yes	—
33	C₂⁴×C₁₇	#41	1	16	yes	—
34	C₂⁴×C₁₇	C₂⁴	1	3	yes	—
35	C₂²×C₄×C₁₇	—	2	64	no	—
36	C₂²×C₄×C₁₇	—	2	48	no	—
37	C₂²×C₄×C₁₇	C₄⋊C₄	1	64	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	48	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	24	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	24	yes	—
41	C₂²×C₄×C₁₇	C₂×Q₈	1	24	yes	—
42	C₂²×C₄×C₁₇	C₂⁴	1	8	yes	—
43	C₂²×C₄×C₁₇	C₂⁴	1	4	yes	—
44	C₂²×Dic₁₇	—	2	48	no	—
45	C₂²×Dic₁₇	—	2	96	no	—
46	C₂²×Dic₁₇	—	4	64	no	—
47	C₂²×Dic₁₇	C₂⁴	1	16	yes	—
48	C₄²×C₁₇	C₂²×C₄	1	64	yes	complemented
49	C₄⋊Dic₁₇	—	2	64	no	—
50	C₂²×#382	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	C₂×C₄⋊C₄	1	16	yes	—
52	C₂³×C₁₇	#117	1	32	yes	—
53	C₂³×C₁₇	#124	1	32	yes	—
54	C₂³×C₁₇	#125	1	32	yes	—
55	C₂³×C₁₇	C₂³×C₄	1	6	yes	—
56	C₂³×C₁₇	C₂²×D₄	1	6	yes	—
57	C₂³×C₁₇	C₂²×Q₈	1	6	yes	—
58	C₂³×C₁₇	C₂×#41	1	24	yes	—
59	C₂³×C₁₇	C₂⁵	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	96	no	—
61	C₂×C₄×C₁₇	—	2	16	no	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	96	yes	—
63	C₂×C₄×C₁₇	C₂³×C₄	1	16	yes	—
64	C₂×C₄×C₁₇	C₂³×C₄	1	8	yes	—
65	C₂×C₄×C₁₇	C₂²×D₄	1	8	yes	—
66	C₂×C₄×C₁₇	C₂²×Q₈	1	8	yes	—
67	C₂×Dic₁₇	—	2	16	no	—
68	C₂×Dic₁₇	—	2	32	no	—
69	C₂×Dic₁₇	—	4	96	no	—
70	C₂³×C₄⋊C₄	—	17	2	no	—
71	C₂³×C₄²	D₁₇	1	1	yes	characteristic
72	C₂×#382	—	17	24	no	—
73	C₂²×#95	—	17	4	no	—
74	C₂²×C₁₇	#382	1	64	yes	complemented
75	C₂²×C₁₇	C₂²×C₄⋊C₄	1	12	yes	—
76	C₂²×C₁₇	C₂×#117	1	24	yes	—
77	C₂²×C₁₇	C₂×#124	1	24	yes	—
78	C₂²×C₁₇	C₂×#125	1	24	yes	—
79	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
80	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
81	C₂²×C₁₇	C₂³×Q₈	1	1	yes	characteristic
82	C₂²×C₁₇	C₂²×#41	1	4	yes	—
83	C₄×C₁₇	—	2	16	no	—
84	C₄×C₁₇	C₂²×C₄⋊C₄	1	16	yes	—
85	Dic₁₇	—	4	16	no	—
86	C₂⁴×C₄	—	17	4	no	—
87	C₂⁴×C₄	C₂×D₁₇	1	2	yes	—
88	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
89	C₂²×C₄⋊C₄	—	17	24	no	—
90	C₂²×C₄⋊C₄	—	34	16	no	—
91	C₂²×C₄²	Dic₁₇	1	16	yes	complemented
92	C₂²×C₄²	C₂×D₁₇	1	12	yes	—
93	#382	—	17	64	no	—
94	C₂×#95	—	17	48	no	—
95	C₂×C₁₇	C₂×#382	1	24	yes	complemented
96	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₁₇	C₂²×#117	1	2	yes	—
98	C₂×C₁₇	C₂²×#124	1	2	yes	—
99	C₂×C₁₇	C₂²×#125	1	2	yes	—
100	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
101	C₂³×C₄	—	2	16	no	—
102	C₂³×C₄	—	17	24	no	—
103	C₂³×C₄	—	34	16	no	—
104	C₂³×C₄	—	34	32	no	—
105	C₂³×C₄	Dic₃₄	1	8	yes	—
106	C₂³×C₄	D₆₈	1	8	yes	—
107	C₂³×C₄	C₂×Dic₁₇	1	16	yes	—
108	C₂³×C₄	C₂×Dic₁₇	1	8	yes	—
109	C₂³×C₄	C₂²×D₁₇	1	12	yes	—
110	C₂³×C₄	C₂²×D₁₇	1	6	yes	—
111	C₂×C₄⋊C₄	—	17	32	no	—
112	C₂×C₄⋊C₄	—	34	96	no	—
113	C₂×C₄²	C₂×Dic₁₇	1	96	yes	complemented
114	C₂×C₄²	C₂²×D₁₇	1	16	yes	—
115	#95	—	17	64	no	—
116	C₁₇	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴	C₂×Dic₃₄	1	4	yes	—
118	C₂⁴	C₂×D₆₈	1	4	yes	—
119	C₂⁴	#63252	1	16	yes	—
120	C₂⁴	C₂²×Dic₁₇	1	4	yes	—
121	C₂⁴	C₂³×D₁₇	1	3	yes	—
122	C₂²×C₄	—	2	48	no	—
123	C₂²×C₄	—	2	64	no	—
124	C₂²×C₄	—	17	16	no	—
125	C₂²×C₄	—	34	48	no	—
126	C₂²×C₄	—	34	96	no	—
127	C₂²×C₄	—	68	64	no	—
128	C₂²×C₄	C₄⋊Dic₁₇	1	64	yes	—
129	C₂²×C₄	C₂×Dic₃₄	1	24	yes	—
130	C₂²×C₄	C₂×D₆₈	1	24	yes	—
131	C₂²×C₄	C₂²×Dic₁₇	1	48	yes	—
132	C₂²×C₄	C₂²×Dic₁₇	1	24	yes	—
133	C₂²×C₄	C₂³×D₁₇	1	8	yes	—
134	C₂²×C₄	C₂³×D₁₇	1	4	yes	—
135	C₄⋊C₄	—	34	64	no	—
136	C₄²	C₂²×Dic₁₇	1	64	yes	complemented
137	C₂³	#10587556	1	32	yes	—
138	C₂³	#10587561	1	32	yes	—
139	C₂³	C₂×C₄⋊Dic₁₇	1	16	yes	—
140	C₂³	#10587612	1	32	yes	—
141	C₂³	C₂²×Dic₃₄	1	6	yes	—
142	C₂³	C₂²×D₆₈	1	6	yes	—
143	C₂³	C₂×#63252	1	24	yes	—
144	C₂³	C₂³×Dic₁₇	1	6	yes	—
145	C₂³	C₂⁴×D₁₇	1	1	yes	characteristic
146	C₂×C₄	—	2	96	no	—
147	C₂×C₄	—	2	16	no	—
148	C₂×C₄	—	34	16	no	—
149	C₂×C₄	—	34	32	no	—
150	C₂×C₄	—	68	96	no	—
151	C₂×C₄	C₂×C₄⋊Dic₁₇	1	96	yes	—
152	C₂×C₄	C₂²×Dic₃₄	1	8	yes	—
153	C₂×C₄	C₂²×D₆₈	1	8	yes	—
154	C₂×C₄	C₂³×Dic₁₇	1	16	yes	—
155	C₂×C₄	C₂³×Dic₁₇	1	8	yes	—
156	C₂²	#11762675	1	64	yes	direct product factor
157	C₂²	C₂²×C₄⋊Dic₁₇	1	12	yes	—
158	C₂²	C₂×#10587612	1	24	yes	—
159	C₂²	C₂×#10587561	1	24	yes	—
160	C₂²	C₂×#10587556	1	24	yes	—
161	C₂²	C₂⁴×Dic₁₇	1	1	yes	characteristic
162	C₂²	C₂³×D₆₈	1	1	yes	characteristic
163	C₂²	C₂³×Dic₃₄	1	1	yes	characteristic
164	C₂²	C₂²×#63252	1	4	yes	—
165	C₄	—	2	16	no	—
166	C₄	—	68	16	no	—
167	C₄	C₂²×C₄⋊Dic₁₇	1	16	yes	—
168	C₂	C₂×#11762675	1	24	yes	direct product factor
169	C₂	C₂³×C₄⋊Dic₁₇	1	1	yes	characteristic
170	C₂	C₂²×#10587612	1	2	yes	—
171	C₂	C₂²×#10587561	1	2	yes	—
172	C₂	C₂²×#10587556	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₇
Fitting	C₂³×C₄²×C₁₇	C₂
socle, prime radical	C₂⁵×C₁₇	C₂³
cosocle	C₂³×C₁₇	C₂⁵
hypercenter	C₂³×C₄²	D₁₇
hypocenter	C₁₇	C₂²×#382
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#11762675	C₂	direct (N × H)
8. #11762675	C₂²	direct (N × H)
12. C₂²×C₄²×C₁₇	C₄	semidirect
25. C₂×C₄²×C₁₇	C₂×C₄	semidirect
48. C₄²×C₁₇	C₂²×C₄	semidirect
74. C₂²×C₁₇	#382	semidirect
91. C₂²×C₄²	Dic₁₇	semidirect
95. C₂×C₁₇	C₂×#382	semidirect
113. C₂×C₄²	C₂×Dic₁₇	semidirect
116. C₁₇	C₂²×#382	semidirect
136. C₄²	C₂²×Dic₁₇	semidirect
156. C₂²	#11762675	direct (N × H)
168. C₂	C₂×#11762675	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₁₇	2
C₂×#11762675	2
C₂²×#10587502	2
C₂³×C₄²×C₁₇	2
C₂²×#382	17

Subgroups — C22×#11762675 #17192599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762675 #17192599