Subgroups — #17960420

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763698	C₂	1	2	yes	maximal
2	C₂×#11763697	C₂	1	2	yes	maximal
3	C₂×#11763695	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	C₂²	1	4	yes	—
6	#11763698	C₄	1	8	yes	complemented
7	#11763697	C₄	1	8	yes	complemented
8	#11763695	C₄	1	8	yes	complemented
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	C₈	1	4	yes	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂³	1	1	yes	characteristic
24	C₂×#63216	C₂×C₄	1	8	yes	—
25	C₂×#63216	C₂×C₄	1	8	yes	—
26	C₂×#63216	C₂×C₄	1	8	yes	—
27	C₄×C₈×C₁₇	C₈	1	4	yes	—
28	C₈×Dic₁₇	C₈	1	4	yes	—
29	C₈×Dic₁₇	C₈	1	4	yes	—
30	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
31	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
32	C₄×#3622	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
36	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₇	C₄²	1	2	yes	—
38	C₂×C₈×C₁₇	C₄²	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₄²×C₁₇	C₄²	1	2	yes	—
44	C₄²×C₁₇	C₂×C₈	1	2	yes	—
45	C₄×Dic₁₇	—	2	8	no	—
46	C₄×Dic₁₇	C₄²	1	2	yes	—
47	C₄×Dic₁₇	C₄²	1	2	yes	—
48	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
49	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
50	C₂×#3622	C₄²	1	2	yes	—
51	C₂×#3622	C₂×C₈	1	2	yes	—
52	C₂×#3622	C₂×C₈	1	2	yes	—
53	C₂×#3622	C₂²×C₄	1	2	yes	—
54	C₂×#3622	C₂²×C₄	1	2	yes	—
55	C₂×#3622	C₂²×C₄	1	2	yes	—
56	#63216	C₄²	1	16	yes	complemented
57	C₄×#346	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	2	4	no	—
60	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
61	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×Dic₁₇	—	2	4	no	—
68	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
69	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
70	C₂×Dic₁₇	M₃₂	1	8	yes	—
71	C₈×C₁₇	C₄×C₈	1	4	yes	—
72	C₈×C₁₇	C₄×C₈	1	4	yes	—
73	#3622	C₄×C₈	1	4	yes	—
74	#3622	C₂×C₄²	1	4	yes	—
75	C₂×#346	—	17	2	no	—
76	C₂×#346	—	17	2	no	—
77	C₂×C₄×C₁₆	—	17	2	no	—
78	C₄²×C₈	—	17	1	no	—
79	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
80	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
81	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²×C₁₇	C₂×M₃₂	1	4	yes	—
83	C₄×C₁₇	—	2	4	no	—
84	C₄×C₁₇	C₄³	1	1	yes	characteristic
85	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
88	Dic₁₇	—	2	4	no	—
89	Dic₁₇	#346	1	8	yes	complemented
90	C₂²×C₁₆	—	17	4	no	—
91	C₂×C₄×C₈	—	17	1	no	—
92	C₂×C₄×C₈	—	17	1	no	—
93	C₂×C₄×C₈	—	17	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	C₂×C₄×C₈	—	17	1	no	—
96	C₂×C₄×C₈	#595	1	1	yes	characteristic
97	C₄³	—	17	1	no	—
98	#346	—	17	8	no	—
99	#346	—	17	8	no	—
100	C₄×C₁₆	—	17	8	no	—
101	C₂×C₁₇	C₄²×C₈	1	1	yes	characteristic
102	C₂×C₁₇	C₂×#346	1	2	yes	—
103	C₂×C₁₇	C₄×M₃₂	1	2	yes	—
104	C₂×C₁₇	C₄×M₃₂	1	2	yes	—
105	C₂²×C₈	—	17	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	C₂×#595	1	1	yes	characteristic
108	C₂²×C₈	C₂×#595	1	1	yes	characteristic
109	C₂×C₄²	—	17	1	no	—
110	C₂×C₄²	—	17	1	no	—
111	C₂×C₄²	—	17	4	no	—
112	C₂×C₄²	C₂×#595	1	1	yes	characteristic
113	C₂×C₁₆	—	17	8	no	—
114	C₂×C₁₆	—	17	8	no	—
115	C₂×C₁₆	—	17	8	no	—
116	C₄×C₈	—	17	4	no	—
117	C₄×C₈	—	17	4	no	—
118	C₄×C₈	—	17	4	no	—
119	C₄×C₈	—	17	4	no	—
120	C₄×C₈	—	17	4	no	—
121	C₄×C₈	#3624	1	4	yes	—
122	C₁₇	C₄×#346	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄	—	17	2	no	—
124	C₂²×C₄	—	17	2	no	—
125	C₂²×C₄	#63242	1	2	yes	—
126	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
127	C₂×C₈	—	17	2	no	—
128	C₂×C₈	—	17	2	no	—
129	C₂×C₈	—	17	2	no	—
130	C₂×C₈	—	17	2	no	—
131	C₂×C₈	—	17	2	no	—
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	#63242	1	2	yes	—
134	C₂×C₈	#63242	1	2	yes	—
135	C₂×C₈	C₄×#595	1	2	yes	—
136	C₂×C₈	C₄×#595	1	2	yes	—
137	C₂×C₈	C₂×#3624	1	2	yes	—
138	C₂×C₈	C₂×#3624	1	2	yes	—
139	C₄²	—	17	2	no	—
140	C₄²	—	17	2	no	—
141	C₄²	—	17	2	no	—
142	C₄²	—	17	2	no	—
143	C₄²	—	34	8	no	—
144	C₄²	C₄×#595	1	2	yes	—
145	C₄²	C₂×#3624	1	2	yes	—
146	C₁₆	—	17	16	no	—
147	C₂³	C₂×#63242	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	17	4	no	—
150	C₂×C₄	—	17	8	no	—
151	C₂×C₄	—	17	4	no	—
152	C₂×C₄	—	34	4	no	—
153	C₂×C₄	C₄×#3624	1	4	yes	—
154	C₂×C₄	C₂×#63242	1	1	yes	characteristic
155	C₂×C₄	C₂×#63242	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
159	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
160	C₈	—	17	4	no	—
161	C₈	—	17	4	no	—
162	C₈	C₄×#3624	1	4	yes	—
163	C₈	C₄×#3624	1	4	yes	—
164	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
165	C₂²	C₄×#63242	1	1	yes	characteristic
166	C₂²	C₄×#63242	1	1	yes	characteristic
167	C₂²	#11763865	1	4	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	17	8	no	—
170	C₄	—	34	4	no	—
171	C₄	C₄²×#595	1	1	yes	characteristic
172	C₄	C₂×C₄×#3624	1	1	yes	characteristic
173	C₄	C₂×C₄×#3624	1	1	yes	characteristic
174	C₄	C₄×#63242	1	1	yes	characteristic
175	C₂	C₄²×#3624	1	1	yes	characteristic
176	C₂	#14544985	1	2	yes	—
177	C₂	#14544990	1	2	yes	—
178	C₂	#14544991	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#595
commutator	C₂×C₁₇	C₄²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	C₄×#346
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #11763698	C₄	semidirect
7. #11763697	C₄	semidirect
8. #11763695	C₄	semidirect
56. #63216	C₄²	semidirect
89. Dic₁₇	#346	semidirect
122. C₁₇	C₄×#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763698	2
C₂×#11763697	2
C₂×#11763695	2
C₄×C₈×Dic₁₇	2
C₄×#346	17