Subgroups — #17957996

Order 4352.

350 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
2	C₈×#63242	C₂	1	1	yes	maximal, characteristic
3	#14543698	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587529	C₄	1	2	yes	—
6	C₄×#63242	C₄	1	1	yes	characteristic
7	C₄×#63242	C₄	1	1	yes	characteristic
8	#11763695	C₂²	1	2	yes	—
9	C₄×#63216	C₄	1	2	yes	—
10	C₈×#3624	C₄	1	2	yes	—
11	C₈×#3624	C₂²	1	2	yes	—
12	C₂×C₁₆×D₁₇	C₄	1	2	yes	—
13	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
14	C₄×#63214	C₄	1	1	yes	characteristic
15	C₁₆×Dic₁₇	C₂²	1	2	yes	—
16	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
17	C₂×#63242	C₈	1	2	yes	—
18	C₂×#63242	C₈	1	2	yes	—
19	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×#3624	C₈	1	2	yes	—
23	C₄×#3624	C₈	1	2	yes	—
24	C₄×#3624	C₂×C₄	1	2	yes	—
25	C₄×#3624	C₂×C₄	1	2	yes	—
26	C₂×#63216	C₂×C₄	1	4	yes	—
27	#10587529	—	2	4	no	—
28	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
32	C₂×#63214	C₂×C₄	1	2	yes	—
33	C₄×#3622	C₂×C₄	1	1	yes	characteristic
34	C₁₆×D₁₇	—	2	4	no	—
35	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
36	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂×#3624	C₂×C₈	1	4	yes	—
39	C₂×#3624	C₂×C₈	1	4	yes	—
40	#63242	—	2	4	no	—
41	#63242	C₁₆	1	8	yes	complemented
42	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₁₇	C₄²	1	1	yes	characteristic
45	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
46	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
47	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3622	C₄²	1	1	yes	characteristic
49	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
50	C₈×D₁₇	M₁₆	1	4	yes	—
51	C₈×D₁₇	M₁₆	1	4	yes	—
52	#63216	—	2	4	no	—
53	C₁₆×C₁₇	—	2	2	no	—
54	#63214	—	2	2	no	—
55	C₂×#1312	—	17	1	no	maximal, Sylow
56	C₂²×D₁₇	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
60	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
61	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
62	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
63	C₄×D₁₇	#97	1	2	yes	—
64	C₄×D₁₇	#97	1	2	yes	—
65	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
66	C₄×D₁₇	M₃₂	1	4	yes	—
67	#3624	—	2	4	no	—
68	#3624	C₂×C₁₆	1	8	yes	complemented
69	C₈×C₁₇	#97	1	2	yes	—
70	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
71	#3622	#97	1	2	yes	—
72	#3622	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₁₆	—	17	1	no	—
74	C₂×C₄×C₁₆	—	17	1	no	—
75	C₂×C₈²	—	17	1	no	—
76	#1312	—	17	4	no	—
77	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
78	C₂×D₁₇	#322	1	2	yes	—
79	C₂×D₁₇	C₄×C₁₆	1	2	yes	—
80	C₂×D₁₇	#346	1	2	yes	—
81	C₄×C₁₇	C₄×C₁₆	1	2	yes	—
82	C₄×C₁₇	#346	1	2	yes	—
83	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
84	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
85	Dic₁₇	#322	1	2	yes	—
86	Dic₁₇	C₂²×C₁₆	1	2	yes	—
87	Dic₁₇	C₂×M₃₂	1	2	yes	—
88	C₂²×C₁₆	—	17	2	no	—
89	C₂²×C₁₆	—	17	2	no	—
90	C₂×C₄×C₈	—	17	1	no	—
91	C₂×C₄×C₈	—	17	1	no	—
92	C₂×C₄×C₈	—	17	1	no	—
93	C₄×C₁₆	—	17	2	no	—
94	C₄×C₁₆	—	17	2	no	—
95	C₄×C₁₆	—	17	2	no	—
96	C₄×C₁₆	—	17	1	no	—
97	C₄×C₁₆	#595	1	1	yes	characteristic
98	C₈²	—	17	2	no	—
99	C₈²	—	17	2	no	—
100	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
101	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
102	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
103	D₁₇	#1312	1	4	yes	complemented
104	C₂²×C₈	—	17	2	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	2	no	—
108	C₂×C₄²	—	17	1	no	—
109	C₂×C₁₆	—	17	4	no	—
110	C₂×C₁₆	—	17	2	no	—
111	C₂×C₁₆	—	34	4	no	—
112	C₂×C₁₆	—	34	4	no	—
113	C₂×C₁₆	C₂×#595	1	2	yes	—
114	C₄×C₈	—	17	2	no	—
115	C₄×C₈	—	17	2	no	—
116	C₄×C₈	—	17	2	no	—
117	C₄×C₈	—	17	2	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	C₂×#595	1	1	yes	characteristic
122	C₁₇	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄	—	17	1	no	—
124	C₂²×C₄	—	17	1	no	—
125	C₂²×C₄	—	17	1	no	—
126	C₂×C₈	—	17	8	no	—
127	C₂×C₈	—	17	4	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	4	no	—
131	C₂×C₈	—	17	4	no	—
132	C₂×C₈	—	17	4	no	—
133	C₂×C₈	—	34	4	no	—
134	C₂×C₈	C₄×#595	1	1	yes	characteristic
135	C₂×C₈	C₂²×#595	1	1	yes	characteristic
136	C₄²	—	17	1	no	—
137	C₄²	—	17	1	no	—
138	C₄²	—	17	1	no	—
139	C₄²	C₄×#595	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	34	4	no	—
142	C₁₆	—	34	2	no	—
143	C₂³	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	1	no	—
147	C₂×C₄	—	17	2	no	—
148	C₂×C₄	—	17	2	no	—
149	C₂×C₄	—	17	4	no	—
150	C₂×C₄	—	17	4	no	—
151	C₂×C₄	C₈×#595	1	1	yes	characteristic
152	C₂×C₄	C₈×#595	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
154	C₈	—	17	8	no	—
155	C₈	—	17	2	no	—
156	C₈	—	17	2	no	—
157	C₈	—	34	4	no	—
158	C₈	#10587532	1	2	yes	—
159	C₈	D₁₇⋊M₁₆	1	2	yes	—
160	C₂²	—	17	2	no	—
161	C₂²	—	17	2	no	—
162	C₂²	—	17	2	no	—
163	C₂²	C₂×C₈×#595	1	1	yes	characteristic
164	C₄	—	17	2	no	—
165	C₄	—	17	2	no	—
166	C₄	—	17	2	no	—
167	C₄	C₁₆×#595	1	2	yes	—
168	C₄	#11763696	1	2	yes	—
169	C₄	C₂×#10587532	1	1	yes	characteristic
170	C₄	#11763768	1	1	yes	characteristic
171	C₂	—	17	4	no	—
172	C₂	#14543980	1	1	yes	characteristic
173	C₂	C₂×C₁₆×#595	1	1	yes	characteristic
174	C₂	C₂×#11763696	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#595
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₄×C₈
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	C₂×#1312
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14543698	C₂	semidirect
41. #63242	C₁₆	semidirect
68. #3624	C₂×C₁₆	semidirect
103. D₁₇	#1312	semidirect
122. C₁₇	C₂×#1312	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587529	2
C₈×#63242	2
#14543698	2
C₄×C₁₆×D₁₇	2
C₂×#1312	17