Finite Group Databasebeta

Subgroups — #14530234

Order 2176.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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	#11762836	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762805	C₂	1	1	yes	maximal, characteristic
3	#11762793	C₂	1	1	yes	maximal, characteristic
4	#11762580	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762573	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762565	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#421	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×#117	C₂²	1	1	yes	characteristic
11	#10587612	C₂²	1	1	yes	characteristic
12	C₂×#63223	C₂²	1	1	yes	characteristic
13	C₂×#63223	C₂²	1	1	yes	characteristic
14	C₂²×#3622	C₂²	1	1	yes	characteristic
15	C₁₇×#104	—	2	2	no	—
16	#10587507	—	2	1	no	—
17	#10587506	—	2	1	no	—
18	#10587504	—	2	1	no	—
19	#10587503	—	2	1	no	—
20	C₄⋊#3622	—	2	2	no	—
21	#41×C₁₇	—	2	1	no	—
22	#41×C₁₇	—	2	1	no	—
23	#41×C₁₇	—	2	1	no	—
24	#41×C₁₇	—	2	1	no	—
25	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂×D₄×C₁₇	—	2	1	no	—
27	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	—	2	1	no	—
29	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₇	—	2	1	no	—
31	M₁₆×C₁₇	—	2	1	no	—
32	C₂×C₈×C₁₇	—	2	1	no	—
33	C₄⋊C₄×C₁₇	—	2	1	no	—
34	C₂²⋊C₄×C₁₇	—	2	1	no	—
35	C₄²×C₁₇	D₄	1	2	yes	—
36	C₂²⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	—	2	1	no	—
38	C₄×Dic₁₇	—	2	1	no	—
39	#63223	—	2	1	no	—
40	#63223	—	2	1	no	—
41	#63223	—	2	1	no	—
42	#63223	—	2	1	no	—
43	C₂×#3622	—	2	1	no	—
44	C₂×#3622	—	2	1	no	—
45	C₂×#3622	—	2	1	no	—
46	C₂×#3622	—	2	1	no	—
47	C₂×#3622	—	2	1	no	—
48	C₂³×C₁₇	—	2	1	no	—
49	C₂³×C₁₇	#41	1	1	yes	characteristic
50	Q₈×C₁₇	—	2	1	no	—
51	Q₈×C₁₇	—	2	1	no	—
52	D₄×C₁₇	—	2	1	no	—
53	D₄×C₁₇	—	2	1	no	—
54	D₄×C₁₇	—	4	1	no	—
55	D₄×C₁₇	—	4	1	no	—
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	—	4	1	no	—
59	C₂×C₄×C₁₇	—	4	1	no	—
60	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
66	C₂×Dic₁₇	—	4	1	no	—
67	C₈×C₁₇	—	4	1	no	—
68	#3622	—	4	1	no	—
69	#3622	—	4	1	no	—
70	#3622	—	4	1	no	—
71	#2080	—	17	1	no	maximal, Sylow
72	C₂²×C₁₇	—	4	1	no	—
73	C₂²×C₁₇	—	4	1	no	—
74	C₂²×C₁₇	—	8	1	no	—
75	C₂²×C₁₇	#123	1	1	yes	characteristic
76	C₂²×C₁₇	#123	1	1	yes	characteristic
77	C₂²×C₁₇	#126	1	1	yes	characteristic
78	C₄×C₁₇	—	4	2	no	—
79	C₄×C₁₇	—	4	1	no	—
80	C₄×C₁₇	#121	1	1	yes	characteristic
81	C₄×C₁₇	#123	1	1	yes	characteristic
82	C₄×C₁₇	#124	1	1	yes	characteristic
83	C₄×C₁₇	#126	1	1	yes	characteristic
84	Dic₁₇	—	8	1	no	—
85	#424	—	17	1	no	—
86	#421	D₁₇	1	1	yes	characteristic
87	#416	—	17	1	no	—
88	#413	—	17	1	no	—
89	#344	—	17	1	no	—
90	#341	—	17	1	no	—
91	H(3,ℤ/4)	—	17	1	no	—
92	C₂×C₁₇	—	2	1	no	—
93	C₂×C₁₇	—	2	1	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	—	8	1	no	—
96	C₂×C₁₇	#397	1	1	yes	characteristic
97	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
98	C₂×M₁₆	—	17	1	no	—
99	C₂×M₁₆	—	17	1	no	—
100	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
101	C₂²×C₈	—	17	1	no	—
102	#117	—	17	1	no	—
103	#117	C₂×D₁₇	1	1	yes	characteristic
104	C₄⋊C₈	—	34	2	no	—
105	#104	—	2	2	no	—
106	#103	—	34	1	no	—
107	#102	—	34	1	no	—
108	#101	—	34	1	no	—
109	#100	—	34	1	no	—
110	C₁₇	#2080	1	1	yes	characteristic, complemented, Sylow
111	#41	—	2	1	no	—
112	#41	—	2	1	no	—
113	#41	—	2	1	no	—
114	#41	—	2	1	no	—
115	C₂×Q₈	D₆₈	1	1	yes	characteristic
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	D₆₈	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
120	M₁₆	—	2	1	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	—	34	1	no	—
123	M₁₆	—	34	1	no	—
124	M₁₆	—	34	1	no	—
125	M₁₆	—	34	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	34	1	no	—
128	C₂×C₈	—	34	1	no	—
129	C₂×C₈	—	34	1	no	—
130	C₂×C₈	—	34	1	no	—
131	C₂×C₈	—	34	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	34	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	34	1	no	—
136	C₄²	—	34	1	no	—
137	C₄²	C₂²⋊D₁₇	1	2	yes	—
138	C₂³	—	2	1	no	—
139	C₂³	#63254	1	1	yes	characteristic
140	Q₈	—	2	1	no	—
141	Q₈	—	2	1	no	—
142	D₄	—	2	1	no	—
143	D₄	—	2	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	68	1	no	—
151	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
152	C₂×C₄	#63252	1	1	yes	characteristic
153	C₂×C₄	#63254	1	1	yes	characteristic
154	C₂×C₄	#63254	1	1	yes	characteristic
155	C₂×C₄	#63254	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	68	1	no	—
159	C₈	—	68	1	no	—
160	C₈	—	68	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	#10587565	1	1	yes	characteristic
165	C₂²	#10587572	1	1	yes	characteristic
166	C₂²	#10587621	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	136	1	no	—
170	C₄	#10587565	1	1	yes	characteristic
171	C₄	#10587572	1	1	yes	characteristic
172	C₄	#10587616	1	1	yes	characteristic
173	C₄	#10587622	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#11762734	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10587622
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#421	C₂
socle	C₂×C₁₇	#397
prime radical	C₂×#41×C₁₇	C₂²
hypercenter	#421	D₁₇
hypocenter	C₁₇	#2080
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
1. #11762836	C₂	semidirect
4. #11762580	C₂	semidirect
5. #11762573	C₂	semidirect
6. #11762565	C₂	semidirect
110. C₁₇	#2080	semidirect

Maximal subgroups

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

subgroup	index
#11762836	2
#11762805	2
#11762793	2
#11762580	2
#11762573	2
#11762565	2
C₁₇×#421	2
#2080	17