Finite Group Databasebeta

Subgroups — #16955268

Order 4352.

176 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530876	C₂	1	1	yes	maximal, characteristic
2	D₁₇×#366	C₂	1	1	yes	maximal, characteristic
3	#14526985	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526966	C₂	1	1	yes	maximal, characteristic, complemented
5	#14526964	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526921	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₃₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
10	#11763051	—	2	1	no	—
11	#107×D₁₇	—	2	1	no	—
12	C₂×#10587465	C₂²	1	1	yes	characteristic
13	#11762635	C₂²	1	1	yes	characteristic
14	#11762634	—	2	1	no	—
15	#11762612	—	2	1	no	—
16	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
17	C₁₇×#366	C₂²	1	1	yes	characteristic
18	C₁₇×#366	C₂²	1	1	yes	characteristic
19	C₂×Dic₆₈	—	2	1	no	—
20	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
21	#10587583	—	4	1	no	—
22	C₄⋊C₄×D₁₇	—	4	1	no	—
23	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
24	C₁₇×#107	—	2	1	no	—
25	C₁₇×#107	—	2	1	no	—
26	#10587489	—	2	1	no	—
27	C₂×#63214	D₄	1	1	yes	characteristic
28	#10587481	—	4	1	no	—
29	#10587478	—	4	1	no	—
30	Dic₁₃₆	—	2	1	no	—
31	Dic₁₃₆	—	2	1	no	—
32	C₁₆×D₁₇	—	2	1	no	—
33	C₁₆×D₁₇	—	2	1	no	—
34	C₁₇×C₃₂	—	2	1	no	—
35	#10587465	—	2	1	no	—
36	C₂×C₄×D₁₇	—	8	1	no	—
37	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
38	C₂×Dic₃₄	—	4	1	no	—
39	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₁₇	—	4	1	no	—
41	C₄⋊C₄×C₁₇	—	4	1	no	—
42	#63227	—	8	1	no	—
43	C₄⋊Dic₁₇	—	4	1	no	—
44	C₄⋊Dic₁₇	—	8	1	no	—
45	#63225	—	8	1	no	—
46	C₂×#3622	D₈	1	1	yes	characteristic
47	Dic₆₈	—	4	1	no	—
48	Dic₆₈	—	4	1	no	—
49	C₈×D₁₇	—	2	1	no	—
50	C₈×D₁₇	—	2	1	no	—
51	C₁₆×C₁₇	#41	1	1	yes	characteristic
52	C₁₆×C₁₇	#41	1	1	yes	characteristic
53	#63214	—	2	1	no	—
54	#13714	—	17	1	no	maximal, Sylow
55	C₂²×D₁₇	D₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	8	1	no	—
57	C₂×C₄×C₁₇	—	8	1	no	—
58	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₁₇	—	8	1	no	—
60	C₂×Dic₁₇	—	8	1	no	—
61	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
62	C₄×D₁₇	—	2	1	no	—
63	C₄×D₁₇	—	2	1	no	—
64	C₄×D₁₇	—	16	2	no	—
65	Dic₃₄	—	8	1	no	—
66	Dic₃₄	—	8	1	no	—
67	C₈×C₁₇	#123	1	1	yes	characteristic
68	C₈×C₁₇	#137	1	1	yes	characteristic
69	#3622	—	2	1	no	—
70	#2216	—	17	1	no	—
71	C₂×#366	—	17	1	no	—
72	#1424	—	17	1	no	—
73	#1424	D₁₇	1	1	yes	characteristic, complemented
74	#1416	—	17	1	no	—
75	#1416	—	17	1	no	—
76	C₂²⋊C₃₂	—	17	1	no	—
77	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
78	C₂×D₁₇	—	2	1	no	—
79	C₂×D₁₇	—	2	1	no	—
80	C₂×D₁₇	SD₆₄	1	2	yes	—
81	C₄×C₁₇	—	16	1	no	—
82	C₄×C₁₇	—	16	1	no	—
83	C₄×C₁₇	#480	1	1	yes	characteristic
84	C₄×C₁₇	#510	1	1	yes	characteristic
85	Dic₁₇	—	2	1	no	—
86	Dic₁₇	—	16	1	no	—
87	Dic₁₇	—	16	1	no	—
88	C₂×Q₃₂	—	17	1	no	—
89	C₂²×C₁₆	—	17	1	no	—
90	#467	—	34	1	no	—
91	C₂×#107	—	34	1	no	—
92	C₂×C₃₂	—	17	1	no	—
93	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
94	#366	—	17	1	no	—
95	#366	—	34	1	no	—
96	#366	C₂×D₁₇	1	1	yes	characteristic
97	#366	C₂×D₁₇	1	1	yes	characteristic
98	#358	—	34	1	no	—
99	C₂×C₁₇	#2232	1	1	yes	characteristic
100	C₂×C₁₇	C₂×SD₆₄	1	1	yes	characteristic
101	C₂×C₁₇	#2264	1	1	yes	characteristic
102	D₁₇	—	2	2	no	—
103	C₂×Q₁₆	—	34	1	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²⋊Q₈	—	68	1	no	—
106	C₂×C₄⋊C₄	—	68	1	no	—
107	Q₃₂	—	34	1	no	—
108	Q₃₂	—	34	1	no	—
109	C₂×C₁₆	—	17	1	no	—
110	C₂×C₁₆	—	34	1	no	—
111	C₂×C₁₆	—	34	1	no	—
112	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
113	#107	—	2	1	no	—
114	#107	—	2	1	no	—
115	#107	—	34	1	no	—
116	#107	—	68	1	no	—
117	#103	—	68	1	no	—
118	C₃₂	—	2	1	no	—
119	C₃₂	—	34	1	no	—
120	C₁₇	#13714	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	68	1	no	—
122	C₂²×C₄	—	17	1	no	—
123	C₂²×C₄	—	136	1	no	—
124	Q₁₆	—	68	1	no	—
125	Q₁₆	—	68	1	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	—	34	1	no	—
128	C₂×C₈	—	34	1	no	—
129	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	68	1	no	—
133	C₄⋊C₄	—	136	1	no	—
134	C₄⋊C₄	—	136	1	no	—
135	C₂²⋊C₄	—	136	1	no	—
136	C₁₆	—	34	1	no	—
137	C₁₆	#63252	1	1	yes	characteristic
138	C₁₆	#63256	1	1	yes	characteristic
139	C₂³	—	17	1	no	—
140	Q₈	—	136	1	no	—
141	Q₈	—	136	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	272	2	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	34	1	no	—
147	C₂×C₄	—	34	1	no	—
148	C₂×C₄	—	136	1	no	—
149	C₂×C₄	—	136	1	no	—
150	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
151	C₈	—	34	1	no	—
152	C₈	#10587580	1	1	yes	characteristic
153	C₈	#10587604	1	1	yes	characteristic
154	C₂²	—	17	2	no	—
155	C₂²	—	34	1	no	—
156	C₂²	—	34	1	no	—
157	C₂²	D₁₆×D₁₇	1	1	yes	characteristic
158	C₄	—	16	1	no	—
159	C₄	—	16	1	no	—
160	C₄	—	272	1	no	—
161	C₄	—	272	1	no	—
162	C₄	—	34	1	no	—
163	C₄	#11763124	1	1	yes	characteristic
164	C₄	#11763251	1	1	yes	characteristic
165	C₂	—	34	2	no	—
166	C₂	#14530966	1	1	yes	characteristic
167	C₂	D₁₇×SD₆₄	1	1	yes	characteristic
168	C₂	#14531100	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₁₇
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×#1424	C₂
socle	C₂²×C₁₇	C₂×D₁₆
prime radical	C₂²×D₁₇	D₁₆
hypercenter	#1424	D₁₇
hypocenter	C₁₇	#13714
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
3. #14526985	C₂	semidirect
4. #14526966	C₂	semidirect
5. #14526964	C₂	semidirect
7. C₁₇×#1424	C₂	semidirect
73. #1424	D₁₇	semidirect
120. C₁₇	#13714	semidirect

Maximal subgroups

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

subgroup	index
#14530876	2
D₁₇×#366	2
#14526985	2
#14526966	2
#14526964	2
#14526921	2
C₁₇×#1424	2
#13714	17