Finite Group Databasebeta

Subgroups — #14530182

Order 2176.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11762858	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762857	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762857	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10587504	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587504	C₂	1	1	yes	maximal, characteristic
6	#11762567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#100	C₂	1	1	yes	maximal, characteristic
8	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×#63223	C₂²	1	1	yes	characteristic
12	C₂×#63223	C₂²	1	1	yes	characteristic
13	C₂×#63223	C₂²	1	1	yes	characteristic
14	C₂×#63223	C₂²	1	1	yes	characteristic
15	C₁₇×#100	—	2	2	no	—
16	#10587504	—	2	2	no	—
17	#10587504	—	2	2	no	—
18	#10587493	—	2	2	no	—
19	#10587493	—	2	2	no	—
20	#10587490	—	2	2	no	—
21	C₂⁴×C₁₇	Q₈	1	2	yes	—
22	C₂×D₄×C₁₇	—	2	1	no	—
23	C₂×D₄×C₁₇	—	2	1	no	—
24	C₂×D₄×C₁₇	—	2	1	no	—
25	C₂×D₄×C₁₇	—	2	1	no	—
26	C₂×D₄×C₁₇	—	2	1	no	—
27	C₂×D₄×C₁₇	—	2	1	no	—
28	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
29	M₁₆×C₁₇	—	2	1	no	—
30	M₁₆×C₁₇	—	2	1	no	—
31	M₁₆×C₁₇	—	2	1	no	—
32	M₁₆×C₁₇	D₄	1	2	yes	—
33	C₂×C₈×C₁₇	—	2	1	no	—
34	C₂×C₈×C₁₇	—	2	1	no	—
35	#63223	—	2	1	no	—
36	#63223	—	2	1	no	—
37	#63223	—	2	1	no	—
38	#63223	—	2	1	no	—
39	#63223	—	2	1	no	—
40	#63223	—	2	1	no	—
41	#63223	D₄	1	2	yes	—
42	#63223	D₄	1	2	yes	—
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	C₂³×C₁₇	—	2	2	no	—
48	C₂³×C₁₇	—	2	2	no	—
49	C₂³×C₁₇	—	2	2	no	—
50	C₂³×C₁₇	—	8	2	no	—
51	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	D₄×C₁₇	—	4	1	no	—
53	D₄×C₁₇	—	4	1	no	—
54	D₄×C₁₇	—	4	1	no	—
55	D₄×C₁₇	—	4	1	no	—
56	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
62	C₈×C₁₇	—	2	2	no	—
63	C₈×C₁₇	—	4	1	no	—
64	#3622	—	2	2	no	—
65	#3622	—	2	2	no	—
66	#3622	—	4	1	no	—
67	#3622	—	4	1	no	—
68	#2069	—	17	1	no	maximal, Sylow
69	C₂²×C₁₇	—	4	1	no	—
70	C₂²×C₁₇	—	4	2	no	—
71	C₂²×C₁₇	—	8	2	no	—
72	C₂²×C₁₇	—	8	2	no	—
73	C₂²×C₁₇	—	8	2	no	—
74	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₄×C₁₇	#121	1	1	yes	characteristic
78	C₄×C₁₇	#121	1	1	yes	characteristic
79	C₄×C₁₇	#121	1	1	yes	characteristic
80	C₄×C₁₇	#126	1	1	yes	characteristic
81	#430	—	17	1	no	—
82	#430	—	17	1	no	—
83	#430	—	17	1	no	—
84	C₂×#100	—	17	1	no	—
85	C₂×#100	—	17	1	no	—
86	C₂×#100	D₁₇	1	1	yes	characteristic
87	#338	—	17	1	no	—
88	C₂×C₁₇	—	2	1	no	—
89	C₂×C₁₇	—	2	1	no	—
90	C₂×C₁₇	—	2	1	no	—
91	C₂×C₁₇	—	8	2	no	—
92	C₂×C₁₇	#396	1	1	yes	characteristic
93	C₂²×D₄	C₂×D₁₇	1	1	yes	characteristic
94	C₂×M₁₆	—	17	1	no	—
95	C₂×M₁₆	—	17	1	no	—
96	C₂×M₁₆	—	17	1	no	—
97	C₂×M₁₆	—	17	1	no	—
98	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
100	#108	—	34	2	no	—
101	#108	—	34	2	no	—
102	#108	—	34	2	no	—
103	#100	—	2	2	no	—
104	#100	—	34	2	no	—
105	#100	—	34	2	no	—
106	C₁₇	#2069	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴	Dic₃₄	1	2	yes	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	17	2	no	—
119	M₁₆	—	17	2	no	—
120	M₁₆	—	34	1	no	—
121	M₁₆	—	34	1	no	—
122	M₁₆	—	34	1	no	—
123	M₁₆	—	34	1	no	—
124	M₁₆	—	34	1	no	—
125	M₁₆	—	34	1	no	—
126	M₁₆	C₂²⋊D₁₇	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	34	1	no	—
130	C₂×C₈	—	34	1	no	—
131	C₂×C₈	—	34	1	no	—
132	C₂×C₈	—	34	1	no	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	8	2	no	—
137	C₂³	C₂×Dic₃₄	1	1	yes	characteristic
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	C₂×C₄	#63252	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
144	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
145	C₂×C₄	#63254	1	1	yes	characteristic
146	C₂×C₄	#63254	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	34	2	no	—
151	C₈	—	34	2	no	—
152	C₈	—	68	1	no	—
153	C₈	—	68	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	8	2	no	—
159	C₂²	#10587564	1	1	yes	characteristic
160	C₂²	#10587564	1	1	yes	characteristic
161	C₂²	#10587610	1	1	yes	characteristic
162	C₄	#10587565	1	1	yes	characteristic
163	C₄	#10587570	1	1	yes	characteristic
164	C₄	#10587624	1	1	yes	characteristic
165	C₄	#10587625	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	8	2	no	—
170	C₂	#11762732	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762732
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#100	C₂
socle	C₂×C₁₇	#396
prime radical	C₂²×D₄×C₁₇	C₂²
hypercenter	C₂×#100	D₁₇
hypocenter	C₁₇	#2069
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

Complements

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

N	H ≅ G/N	type
1. #11762858	C₂	semidirect
2. #11762857	C₂	semidirect
3. #11762857	C₂	semidirect
6. #11762567	C₂	semidirect
106. C₁₇	#2069	semidirect

Maximal subgroups

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

subgroup	index
#11762858	2
#11762857	2
C₂×#10587504	2
#11762567	2
C₂×C₁₇×#100	2
#2069	17