Subgroups — C₂²×#11762627 #17187298

Order 4352.

862 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	C₂³×#63214	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762627	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	—	2	4	no	—
8	C₂²×#63214	C₂²	1	6	yes	—
9	C₂×C₈×Dic₁₇	C₄	1	6	yes	—
10	C₂×C₈×Dic₁₇	C₂²	1	6	yes	—
11	#11762627	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₁₇	—	2	4	no	—
14	C₂²×C₁₆×C₁₇	C₂²	1	6	yes	—
15	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×Dic₁₇	C₈	1	1	yes	characteristic
17	C₂³×Dic₁₇	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₁₇	D₄	1	4	yes	—
20	C₂²×C₈×C₁₇	Q₈	1	4	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
22	C₂×C₄×Dic₁₇	C₈	1	6	yes	—
23	C₂×C₄×Dic₁₇	C₂×C₄	1	6	yes	—
24	C₂²×#3622	—	2	4	no	—
25	C₂²×#3622	C₂×C₄	1	6	yes	—
26	C₂×C₁₆×C₁₇	—	2	12	no	—
27	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
28	C₈×Dic₁₇	C₂×C₄	1	12	yes	—
29	C₈×Dic₁₇	C₂³	1	4	yes	—
30	C₂×#63214	—	2	12	no	—
31	C₂×#63214	C₂³	1	4	yes	—
32	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
34	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
35	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
37	C₂²×Dic₁₇	—	2	4	no	—
38	C₂²×Dic₁₇	C₁₆	1	8	yes	complemented
39	C₂²×Dic₁₇	C₂×C₈	1	6	yes	—
40	C₂²×Dic₁₇	C₂×C₈	1	6	yes	—
41	C₂×C₈×C₁₇	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₁₇	C₂×D₄	1	6	yes	—
44	C₂×C₈×C₁₇	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
46	C₄×Dic₁₇	C₂×C₈	1	12	yes	—
47	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
48	C₂×#3622	—	2	12	no	—
49	C₂×#3622	C₂²×C₄	1	4	yes	—
50	C₁₆×C₁₇	—	2	4	no	—
51	#63214	—	2	4	no	—
52	C₂²×C₄⋊C₁₆	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	C₄⋊C₈	1	4	yes	—
54	C₂³×C₁₇	C₂×C₁₆	1	4	yes	—
55	C₂³×C₁₇	M₃₂	1	4	yes	—
56	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
57	C₂×C₄×C₁₇	C₄⋊C₈	1	12	yes	—
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
60	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
61	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×Dic₁₇	—	2	12	no	—
63	C₂×Dic₁₇	C₂×C₁₆	1	24	yes	complemented
64	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
65	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
66	C₈×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
67	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
68	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
69	#3622	—	2	4	no	—
70	C₂³×C₁₆	—	17	1	no	—
71	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
72	C₂²×C₄×C₈	—	17	1	no	—
73	C₂×C₄⋊C₁₆	—	17	12	no	—
74	C₂²×C₁₇	C₄⋊C₁₆	1	16	yes	complemented
75	C₂²×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
76	C₂²×C₁₇	C₂²×C₁₆	1	6	yes	—
77	C₂²×C₁₇	C₂×M₃₂	1	6	yes	—
78	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
79	C₄×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
80	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
82	Dic₁₇	—	2	4	no	—
83	Dic₁₇	C₂²×C₁₆	1	8	yes	complemented
84	C₂³×C₈	—	17	1	no	—
85	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂²×C₄²	—	17	1	no	—
87	C₂²×C₁₆	—	2	4	no	—
88	C₂²×C₁₆	—	17	6	no	—
89	C₂²×C₁₆	—	34	4	no	—
90	C₂²×C₁₆	C₂×D₁₇	1	6	yes	—
91	C₂×C₄×C₈	—	17	6	no	—
92	C₂×C₄×C₈	—	17	6	no	—
93	C₄⋊C₁₆	—	17	16	no	—
94	C₂×C₁₇	C₂×C₄⋊C₁₆	1	12	yes	complemented
95	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
97	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
98	C₂³×C₄	—	17	1	no	—
99	C₂³×C₄	—	17	1	no	—
100	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
101	C₂²×C₈	—	17	6	no	—
102	C₂²×C₈	—	34	4	no	—
103	C₂²×C₈	Dic₃₄	1	4	yes	—
104	C₂²×C₈	C₄×D₁₇	1	3	yes	—
105	C₂²×C₈	C₂²⋊D₁₇	1	4	yes	—
106	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
107	C₂×C₄²	—	17	6	no	—
108	C₂×C₄²	—	17	6	no	—
109	C₂×C₁₆	—	2	12	no	—
110	C₂×C₁₆	—	17	4	no	—
111	C₂×C₁₆	—	34	12	no	—
112	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
113	C₄×C₈	—	17	12	no	—
114	C₄×C₈	—	17	4	no	—
115	C₁₇	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
117	C₂²×C₄	—	17	8	no	—
118	C₂²×C₄	—	17	6	no	—
119	C₂²×C₄	—	17	6	no	—
120	C₂²×C₄	—	34	4	no	—
121	C₂²×C₄	C₈×D₁₇	1	3	yes	—
122	C₂²×C₄	#63218	1	4	yes	—
123	C₂²×C₄	#63225	1	4	yes	—
124	C₂²×C₄	C₂×C₄×D₁₇	1	3	yes	—
125	C₂×C₈	—	17	4	no	—
126	C₂×C₈	—	34	12	no	—
127	C₂×C₈	#63225	1	12	yes	—
128	C₂×C₈	C₂×Dic₃₄	1	6	yes	—
129	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
130	C₂×C₈	C₂×C₂²⋊D₁₇	1	6	yes	—
131	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
132	C₄²	—	17	4	no	—
133	C₄²	—	17	12	no	—
134	C₁₆	—	2	4	no	—
135	C₁₆	—	34	4	no	—
136	C₂³	C₁₆×D₁₇	1	4	yes	—
137	C₂³	#10587469	1	4	yes	—
138	C₂³	#10587485	1	4	yes	—
139	C₂³	C₂×C₈×D₁₇	1	3	yes	—
140	C₂×C₄	—	17	24	no	—
141	C₂×C₄	—	17	4	no	—
142	C₂×C₄	—	17	4	no	—
143	C₂×C₄	—	34	12	no	—
144	C₂×C₄	#10587485	1	12	yes	—
145	C₂×C₄	C₂×C₈×D₁₇	1	3	yes	—
146	C₂×C₄	C₂×#63218	1	6	yes	—
147	C₂×C₄	C₂×#63225	1	6	yes	—
148	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
149	C₈	—	34	4	no	—
150	C₈	C₂×#63225	1	6	yes	—
151	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
152	C₈	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
153	C₂²	#11762627	1	16	yes	direct product factor
154	C₂²	C₂×#10587485	1	6	yes	—
155	C₂²	C₂×C₁₆×D₁₇	1	6	yes	—
156	C₂²	C₂×#10587469	1	6	yes	—
157	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
158	C₄	—	17	8	no	—
159	C₄	—	34	4	no	—
160	C₄	C₂×#10587485	1	6	yes	—
161	C₄	C₂²×#63225	1	1	yes	characteristic
162	C₄	C₂²×#63218	1	1	yes	characteristic
163	C₂	C₂×#11762627	1	12	yes	direct product factor
164	C₂	C₂²×#10587485	1	1	yes	characteristic
165	C₂	C₂²×C₁₆×D₁₇	1	1	yes	characteristic
166	C₂	C₂²×#10587469	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
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×C₄⋊C₁₆
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×#11762627	C₂	direct (N × H)
11. #11762627	C₂²	direct (N × H)
38. C₂²×Dic₁₇	C₁₆	semidirect
63. C₂×Dic₁₇	C₂×C₁₆	semidirect
74. C₂²×C₁₇	C₄⋊C₁₆	semidirect
83. Dic₁₇	C₂²×C₁₆	semidirect
94. C₂×C₁₇	C₂×C₄⋊C₁₆	semidirect
115. C₁₇	C₂²×C₄⋊C₁₆	semidirect
153. C₂²	#11762627	direct (N × H)
163. C₂	C₂×#11762627	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×#11762627	2
C₂³×C₁₆×C₁₇	2
C₂²×C₄⋊C₁₆	17

Subgroups — C22×#11762627 #17187298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762627 #17187298