Subgroups — #16950759

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762637	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762636	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×#367	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	#11762637	—	2	2	no	—
12	#11762636	—	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×#367	—	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	#12906	—	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	SD₃₂	1	8	yes	—
72	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
76	#3622	#107	1	4	yes	—
77	C₂×#367	—	17	1	no	—
78	C₂×#367	—	17	2	no	—
79	C₂×#367	D₁₇	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	17	1	no	—
81	#1845	—	17	2	no	—
82	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
83	C₂²×C₁₇	#493	1	1	yes	characteristic
84	C₂²×C₁₇	#500	1	1	yes	characteristic
85	C₂²×C₁₇	C₂×SD₃₂	1	4	yes	—
86	C₄×C₁₇	—	16	2	no	—
87	C₄×C₁₇	#384	1	1	yes	characteristic
88	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
89	C₄×C₁₇	#494	1	1	yes	characteristic
90	C₄×C₁₇	#500	1	1	yes	characteristic
91	Dic₁₇	—	16	2	no	—
92	Dic₁₇	#367	1	8	yes	complemented
93	C₂²×C₁₆	—	17	1	no	—
94	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
95	C₂×#107	—	17	2	no	—
96	C₂×#107	—	34	2	no	—
97	C₂×#107	C₂×D₁₇	1	2	yes	—
98	C₂×C₄×C₈	—	17	1	no	—
99	#384	—	34	2	no	—
100	#367	—	2	2	no	—
101	#367	—	34	4	no	—
102	#367	—	34	2	no	—
103	C₄×C₁₆	—	17	4	no	—
104	C₂×C₁₇	#1845	1	1	yes	characteristic
105	C₂×C₁₇	C₂×#367	1	2	yes	—
106	C₂×C₁₇	#2248	1	2	yes	—
107	C₂×C₁₇	#2254	1	2	yes	—
108	C₂²×C₈	—	17	1	no	—
109	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	34	2	no	—
112	C₂×C₄⋊C₄	—	68	4	no	—
113	C₂×C₄²	—	17	1	no	—
114	C₂×C₁₆	—	17	2	no	—
115	C₂×C₁₆	—	17	2	no	—
116	C₂×C₁₆	—	17	2	no	—
117	C₂×C₁₆	Dic₃₄	1	2	yes	—
118	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
119	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
120	#107	—	2	2	no	—
121	#107	—	2	2	no	—
122	#107	—	34	2	no	—
123	#107	—	34	2	no	—
124	#107	—	68	4	no	—
125	C₄×C₈	—	17	2	no	—
126	C₄×C₈	—	17	2	no	—
127	C₁₇	#12906	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	17	2	no	—
130	C₂²×C₄	—	68	2	no	—
131	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	—	17	2	no	—
134	C₂×C₈	—	17	2	no	—
135	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
137	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	68	2	no	—
144	C₄⋊C₄	—	68	2	no	—
145	C₄⋊C₄	—	136	8	no	—
146	C₄²	—	17	2	no	—
147	C₄²	—	17	2	no	—
148	C₁₆	—	17	4	no	—
149	C₁₆	#63225	1	4	yes	complemented
150	C₂³	D₈×D₁₇	1	1	yes	characteristic
151	C₂×C₄	—	8	4	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	17	8	no	—
154	C₂×C₄	—	17	4	no	—
155	C₂×C₄	—	136	4	no	—
156	C₂×C₄	—	136	2	no	—
157	C₂×C₄	#10587574	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
159	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
160	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
162	C₂×C₄	#10587626	1	1	yes	characteristic
163	C₈	—	17	4	no	—
164	C₈	#10587574	1	1	yes	characteristic
165	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
166	C₈	C₂×#63225	1	1	yes	characteristic
167	C₈	#10587630	1	1	yes	characteristic
168	C₂²	#107×D₁₇	1	1	yes	characteristic
169	C₂²	#11763184	1	1	yes	characteristic
170	C₂²	#11763213	1	1	yes	characteristic
171	C₂²	SD₃₂×D₁₇	1	4	yes	—
172	C₄	—	16	2	no	—
173	C₄	—	272	2	no	—
174	C₄	—	17	8	no	—
175	C₄	#11762678	1	1	yes	characteristic
176	C₄	#107×D₁₇	1	1	yes	characteristic
177	C₄	#11763187	1	1	yes	characteristic
178	C₄	#11763213	1	1	yes	characteristic
179	C₂	#14528942	1	1	yes	characteristic
180	C₂	D₁₇×#367	1	2	yes	—
181	C₂	#14531038	1	2	yes	—
182	C₂	#14531065	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#367	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#367	D₁₇
hypocenter	C₁₇	#12906
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

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
92. Dic₁₇	#367	semidirect
127. C₁₇	#12906	semidirect
149. 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×#11762637	2
C₂×#11762636	2
C₂×C₁₆×Dic₁₇	2
#14528942	2
C₂×C₁₇×#367	2
#12906	17