Subgroups — #16950756

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762635	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762634	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#14528942	C₂	1	2	yes	maximal
5	C₂×C₁₇×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	1	yes	characteristic
7	C₂×#10587489	C₂²	1	2	yes	—
8	C₂×#10587478	—	2	2	no	—
9	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
10	#11762678	—	2	2	no	—
11	#11762635	—	2	2	no	—
12	#11762634	—	2	4	no	—
13	C₁₆×Dic₁₇	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#107	C₂²	1	2	yes	—
16	C₁₇×#366	—	2	2	no	—
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₂×#63225	—	4	4	no	—
21	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
22	C₂²×#3622	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
25	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
26	C₁₇×#107	—	2	2	no	—
27	C₁₇×#107	—	2	2	no	—
28	#10587489	—	2	2	no	—
29	#10587489	—	2	2	no	—
30	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₇	Q₈	1	2	yes	—
32	C₂×#63214	C₂×C₄	1	2	yes	—
33	C₂×#63214	D₄	1	2	yes	—
34	C₂×#63214	Q₈	1	2	yes	—
35	#10587478	—	4	4	no	—
36	C₂²×C₄×C₁₇	—	4	2	no	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	—	4	2	no	—
39	C₂²×Dic₁₇	D₈	1	2	yes	—
40	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊C₄×C₁₇	—	4	2	no	—
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	#63225	—	8	8	no	—
51	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₁₇	Q₁₆	1	2	yes	—
53	C₂×#3622	C₄⋊C₄	1	2	yes	—
54	C₂×#3622	D₈	1	2	yes	—
55	C₂×#3622	Q₁₆	1	2	yes	—
56	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	complemented
57	#63214	C₄⋊C₄	1	4	yes	complemented
58	#12905	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	8	2	no	—
61	C₂×C₄×C₁₇	—	8	4	no	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₁₇	—	8	2	no	—
69	C₂×Dic₁₇	—	8	4	no	—
70	C₂×Dic₁₇	#107	1	4	yes	—
71	C₂×Dic₁₇	D₁₆	1	4	yes	—
72	C₂×Dic₁₇	Q₃₂	1	4	yes	—
73	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
77	#3622	#107	1	4	yes	—
78	C₂×#366	—	17	1	no	—
79	C₂×#366	—	17	2	no	—
80	C₂×#366	D₁₇	1	1	yes	characteristic
81	C₂×C₄×C₁₆	—	17	1	no	—
82	#1845	—	17	2	no	—
83	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
84	C₂²×C₁₇	#493	1	1	yes	characteristic
85	C₂²×C₁₇	#500	1	1	yes	characteristic
86	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
87	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
88	C₄×C₁₇	—	16	2	no	—
89	C₄×C₁₇	#384	1	1	yes	characteristic
90	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
91	C₄×C₁₇	#494	1	1	yes	characteristic
92	C₄×C₁₇	#500	1	1	yes	characteristic
93	Dic₁₇	—	16	2	no	—
94	Dic₁₇	#366	1	8	yes	complemented
95	C₂²×C₁₆	—	17	1	no	—
96	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
97	C₂×#107	—	17	2	no	—
98	C₂×#107	—	34	2	no	—
99	C₂×#107	C₂×D₁₇	1	2	yes	—
100	C₂×C₄×C₈	—	17	1	no	—
101	#384	—	34	2	no	—
102	#366	—	2	2	no	—
103	#366	—	34	4	no	—
104	#366	—	34	2	no	—
105	C₄×C₁₆	—	17	4	no	—
106	C₂×C₁₇	#1845	1	1	yes	characteristic
107	C₂×C₁₇	C₂×#366	1	2	yes	—
108	C₂×C₁₇	#2246	1	1	yes	characteristic
109	C₂×C₁₇	#2247	1	1	yes	characteristic
110	C₂×C₁₇	#2252	1	2	yes	—
111	C₂²×C₈	—	17	1	no	—
112	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	34	2	no	—
115	C₂×C₄⋊C₄	—	68	4	no	—
116	C₂×C₄²	—	17	1	no	—
117	C₂×C₁₆	—	17	2	no	—
118	C₂×C₁₆	—	17	2	no	—
119	C₂×C₁₆	—	17	2	no	—
120	C₂×C₁₆	Dic₃₄	1	2	yes	—
121	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
122	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	34	2	no	—
126	#107	—	34	2	no	—
127	#107	—	68	4	no	—
128	C₄×C₈	—	17	2	no	—
129	C₄×C₈	—	17	2	no	—
130	C₁₇	#12905	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	17	2	no	—
133	C₂²×C₄	—	68	2	no	—
134	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
135	C₂×C₈	—	17	2	no	—
136	C₂×C₈	—	17	2	no	—
137	C₂×C₈	—	17	2	no	—
138	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
140	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	68	2	no	—
147	C₄⋊C₄	—	68	2	no	—
148	C₄⋊C₄	—	136	8	no	—
149	C₄²	—	17	2	no	—
150	C₄²	—	17	2	no	—
151	C₁₆	—	17	4	no	—
152	C₁₆	#63225	1	4	yes	complemented
153	C₂³	D₈×D₁₇	1	1	yes	characteristic
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	17	4	no	—
157	C₂×C₄	—	17	4	no	—
158	C₂×C₄	—	17	4	no	—
159	C₂×C₄	—	136	4	no	—
160	C₂×C₄	—	136	2	no	—
161	C₂×C₄	#10587574	1	1	yes	characteristic
162	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
163	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
164	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
166	C₂×C₄	#10587626	1	1	yes	characteristic
167	C₈	—	17	4	no	—
168	C₈	#10587574	1	1	yes	characteristic
169	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
170	C₈	C₂×#63225	1	1	yes	characteristic
171	C₈	#10587630	1	1	yes	characteristic
172	C₂²	#107×D₁₇	1	1	yes	characteristic
173	C₂²	#11763184	1	1	yes	characteristic
174	C₂²	#11763213	1	1	yes	characteristic
175	C₂²	D₁₆×D₁₇	1	2	yes	—
176	C₂²	Q₃₂×D₁₇	1	2	yes	—
177	C₄	—	16	2	no	—
178	C₄	—	272	2	no	—
179	C₄	—	17	8	no	—
180	C₄	#11762678	1	1	yes	characteristic
181	C₄	#107×D₁₇	1	1	yes	characteristic
182	C₄	#11763187	1	1	yes	characteristic
183	C₄	#11763213	1	1	yes	characteristic
184	C₂	#14528942	1	1	yes	characteristic
185	C₂	D₁₇×#366	1	2	yes	—
186	C₂	#14531032	1	1	yes	characteristic
187	C₂	#14531035	1	1	yes	characteristic
188	C₂	#14531058	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#366	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#366	D₁₇
hypocenter	C₁₇	#12905
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₁₇	C₄	semidirect
56. C₁₆×C₁₇	C₄⋊C₄	semidirect
57. #63214	C₄⋊C₄	semidirect
94. Dic₁₇	#366	semidirect
130. C₁₇	#12905	semidirect
152. C₁₆	#63225	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762635	2
C₂×#11762634	2
C₂×C₁₆×Dic₁₇	2
#14528942	2
C₂×C₁₇×#366	2
#12905	17