Subgroups — C₂²×#11762584 #17186949

Order 4352.

942 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63214	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762584	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×#3622	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂²×#63214	C₄	1	8	yes	complemented
8	C₂²×#63214	C₂²	1	6	yes	—
9	C₂×C₈×Dic₁₇	C₄	1	6	yes	—
10	C₂×C₈×Dic₁₇	C₂²	1	6	yes	—
11	#11762584	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₁₇	C₄	1	8	yes	complemented
14	C₂²×C₁₆×C₁₇	C₂²	1	6	yes	—
15	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×Dic₁₇	C₈	1	2	yes	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
21	C₂×C₄×Dic₁₇	C₈	1	6	yes	—
22	C₂×C₄×Dic₁₇	C₂×C₄	1	6	yes	—
23	C₂²×#3622	C₈	1	8	yes	—
24	C₂²×#3622	C₂×C₄	1	6	yes	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	24	yes	complemented
26	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
27	C₈×Dic₁₇	C₂×C₄	1	12	yes	—
28	C₈×Dic₁₇	C₂³	1	4	yes	—
29	C₂×#63214	C₂×C₄	1	24	yes	complemented
30	C₂×#63214	C₂³	1	4	yes	—
31	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	C₄²	1	4	yes	—
33	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
36	C₂²×Dic₁₇	—	2	8	no	—
37	C₂²×Dic₁₇	C₂×C₈	1	12	yes	—
38	C₂×C₈×C₁₇	C₄²	1	12	yes	—
39	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
41	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
42	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂×C₈	1	12	yes	—
44	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
45	C₂×#3622	C₂×C₈	1	24	yes	—
46	C₂×#3622	C₂²×C₄	1	4	yes	—
47	C₁₆×C₁₇	C₂²×C₄	1	8	yes	complemented
48	#63214	C₂²×C₄	1	8	yes	complemented
49	C₂²×#346	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₄×C₈	1	4	yes	—
51	C₂³×C₁₇	M₃₂	1	8	yes	—
52	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
53	C₂×C₄×C₁₇	C₄×C₈	1	12	yes	—
54	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
55	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
56	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
57	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
58	C₂×Dic₁₇	—	2	24	no	—
59	C₂×Dic₁₇	C₂²×C₈	1	8	yes	—
60	C₈×C₁₇	C₂×C₄²	1	6	yes	—
61	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
62	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
63	#3622	C₂²×C₈	1	8	yes	—
64	C₂³×C₁₆	—	17	1	no	—
65	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
66	C₂²×C₄×C₈	—	17	1	no	—
67	C₂×#346	—	17	12	no	—
68	C₂²×C₁₇	#346	1	16	yes	complemented
69	C₂²×C₁₇	C₂×C₄×C₈	1	6	yes	—
70	C₂²×C₁₇	C₂×M₃₂	1	12	yes	—
71	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
72	C₄×C₁₇	C₂×C₄×C₈	1	6	yes	—
73	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
74	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
75	Dic₁₇	—	2	8	no	—
76	C₂³×C₈	—	17	1	no	—
77	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
78	C₂²×C₄²	—	17	1	no	—
79	C₂²×C₁₆	—	17	8	no	—
80	C₂²×C₁₆	—	17	6	no	—
81	C₂²×C₁₆	Dic₁₇	1	8	yes	complemented
82	C₂²×C₁₆	C₂×D₁₇	1	6	yes	—
83	C₂×C₄×C₈	—	17	6	no	—
84	C₂×C₄×C₈	—	17	6	no	—
85	#346	—	17	16	no	—
86	C₂×C₁₇	C₂×#346	1	12	yes	complemented
87	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
88	C₂×C₁₇	C₂²×M₃₂	1	2	yes	—
89	C₂³×C₄	—	17	2	no	—
90	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
91	C₂²×C₈	—	17	6	no	—
92	C₂²×C₈	—	17	8	no	—
93	C₂²×C₈	C₄×D₁₇	1	4	yes	—
94	C₂²×C₈	C₄×D₁₇	1	3	yes	—
95	C₂²×C₈	C₂×Dic₁₇	1	4	yes	—
96	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
97	C₂×C₄²	—	17	6	no	—
98	C₂×C₄²	—	17	6	no	—
99	C₂×C₁₆	—	17	24	no	—
100	C₂×C₁₆	—	17	4	no	—
101	C₂×C₁₆	C₂×Dic₁₇	1	24	yes	complemented
102	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
103	C₄×C₈	—	17	12	no	—
104	C₄×C₈	—	17	4	no	—
105	C₁₇	C₂²×#346	1	1	yes	characteristic, complemented, Sylow
106	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
107	C₂²×C₄	—	17	12	no	—
108	C₂²×C₄	—	34	8	no	—
109	C₂²×C₄	C₈×D₁₇	1	3	yes	—
110	C₂²×C₄	C₈×D₁₇	1	4	yes	—
111	C₂²×C₄	C₄×Dic₁₇	1	4	yes	—
112	C₂²×C₄	C₂×C₄×D₁₇	1	3	yes	—
113	C₂×C₈	—	17	4	no	—
114	C₂×C₈	—	17	24	no	—
115	C₂×C₈	C₄×Dic₁₇	1	12	yes	—
116	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
117	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
118	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
119	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
120	C₄²	—	17	4	no	—
121	C₄²	—	17	12	no	—
122	C₁₆	—	17	8	no	—
123	C₁₆	C₂²×Dic₁₇	1	8	yes	complemented
124	C₂³	#10587469	1	8	yes	—
125	C₂³	C₈×Dic₁₇	1	4	yes	—
126	C₂³	C₂×C₈×D₁₇	1	3	yes	—
127	C₂×C₄	—	17	8	no	—
128	C₂×C₄	—	34	24	no	—
129	C₂×C₄	C₈×Dic₁₇	1	12	yes	—
130	C₂×C₄	C₂×C₈×D₁₇	1	3	yes	—
131	C₂×C₄	C₂×C₈×D₁₇	1	6	yes	—
132	C₂×C₄	C₂×C₄×Dic₁₇	1	6	yes	—
133	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
134	C₈	—	17	8	no	—
135	C₈	C₂×C₄×Dic₁₇	1	6	yes	—
136	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
137	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
138	C₂²	#11762584	1	16	yes	direct product factor
139	C₂²	C₂×C₈×Dic₁₇	1	6	yes	—
140	C₂²	C₂×#10587469	1	12	yes	—
141	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
142	C₄	—	34	8	no	—
143	C₄	C₂×C₈×Dic₁₇	1	6	yes	—
144	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
145	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
146	C₂	C₂×#11762584	1	12	yes	direct product factor
147	C₂	C₂²×C₈×Dic₁₇	1	1	yes	characteristic
148	C₂	C₂²×#10587469	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	C₂²×#346
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#11762584	C₂	direct (N × H)
7. C₂²×#63214	C₄	semidirect
11. #11762584	C₂²	direct (N × H)
13. C₂²×C₁₆×C₁₇	C₄	semidirect
25. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
29. C₂×#63214	C₂×C₄	semidirect
47. C₁₆×C₁₇	C₂²×C₄	semidirect
48. #63214	C₂²×C₄	semidirect
68. C₂²×C₁₇	#346	semidirect
81. C₂²×C₁₆	Dic₁₇	semidirect
86. C₂×C₁₇	C₂×#346	semidirect
101. C₂×C₁₆	C₂×Dic₁₇	semidirect
105. C₁₇	C₂²×#346	semidirect
123. C₁₆	C₂²×Dic₁₇	semidirect
138. C₂²	#11762584	direct (N × H)
146. C₂	C₂×#11762584	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×#63214	2
C₂²×C₈×Dic₁₇	2
C₂×#11762584	2
C₂³×C₁₆×C₁₇	2
C₂²×#346	17

Subgroups — C22×#11762584 #17186949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762584 #17186949