Finite Group Databasebeta

Subgroups — #14529563

Order 2176.

196 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#10587493	C₂	1	1	yes	maximal, characteristic
2	#11762851	C₂	1	1	yes	maximal, characteristic
3	#11762849	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587479	C₂	1	1	yes	maximal, characteristic
5	#11762767	C₂	1	1	yes	maximal, characteristic
6	#11762577	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#343	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	#10587612	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×#63223	C₂²	1	1	yes	characteristic
14	C₂²×#3622	C₂²	1	1	yes	characteristic
15	#10587493	C₄	1	4	yes	complemented
16	#10587489	—	2	1	no	—
17	#10587488	—	2	1	no	—
18	#10587486	—	2	1	no	—
19	C₈×Dic₁₇	—	2	1	no	—
20	#10587479	—	2	1	no	—
21	#10587479	—	2	1	no	—
22	#10587479	—	2	1	no	—
23	#10587478	—	2	1	no	—
24	C₂²×C₄×C₁₇	—	2	1	no	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₇	—	2	1	no	—
27	M₁₆×C₁₇	—	2	1	no	—
28	M₁₆×C₁₇	—	2	1	no	—
29	M₁₆×C₁₇	C₂×C₄	1	2	yes	—
30	M₁₆×C₁₇	Q₈	1	2	yes	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₂×C₈×C₁₇	—	2	1	no	—
33	C₄⋊C₄×C₁₇	—	2	1	no	—
34	C₄⋊C₄×C₁₇	—	2	1	no	—
35	C₂²⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	—	2	1	no	—
38	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
39	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
40	C₄×Dic₁₇	—	2	1	no	—
41	#63223	—	2	1	no	—
42	#63223	C₂×C₄	1	2	yes	—
43	C₂×#3622	—	2	1	no	—
44	C₂×#3622	C₂×C₄	1	2	yes	—
45	C₂×#3622	D₄	1	1	yes	characteristic
46	C₂×#3622	D₄	1	1	yes	characteristic
47	C₂×#3622	Q₈	1	2	yes	—
48	C₂³×C₁₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₇	—	4	1	no	—
50	C₂×C₄×C₁₇	—	8	1	no	—
51	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
57	C₂×Dic₁₇	—	2	1	no	—
58	C₂×Dic₁₇	—	2	1	no	—
59	C₂×Dic₁₇	—	4	1	no	—
60	C₂×Dic₁₇	—	8	1	no	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	—	2	2	no	—
63	#3622	—	2	2	no	—
64	#3622	C₄⋊C₄	1	2	yes	—
65	#3622	C₄⋊C₄	1	2	yes	—
66	#1951	—	17	1	no	maximal, Sylow
67	C₂²×C₁₇	—	2	1	no	—
68	C₂²×C₁₇	—	2	1	no	—
69	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
70	C₂²×C₁₇	#121	1	1	yes	characteristic
71	C₂²×C₁₇	#125	1	1	yes	characteristic
72	C₄×C₁₇	—	8	1	no	—
73	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
75	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
77	Dic₁₇	—	4	1	no	—
78	Dic₁₇	—	4	1	no	—
79	Dic₁₇	—	8	1	no	—
80	C₂×#108	—	17	1	no	—
81	#428	—	17	1	no	—
82	#428	—	17	1	no	—
83	C₂×#106	—	17	1	no	—
84	#405	—	17	1	no	—
85	#343	—	17	1	no	—
86	#343	D₁₇	1	1	yes	characteristic
87	C₂×C₁₇	—	2	1	no	—
88	C₂×C₁₇	—	2	1	no	—
89	C₂×C₁₇	#389	1	1	yes	characteristic
90	C₂×C₁₇	#471	1	1	yes	characteristic
91	C₂×C₁₇	#471	1	1	yes	characteristic
92	C₂×M₁₆	—	17	1	no	—
93	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
94	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
95	C₂²×C₈	—	17	1	no	—
96	#117	—	17	1	no	—
97	C₂×C₄⋊C₄	—	17	1	no	—
98	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
99	#108	—	17	4	no	—
100	#107	—	34	1	no	—
101	#107	—	34	1	no	—
102	#106	—	34	1	no	—
103	#106	—	34	1	no	—
104	#106	—	34	1	no	—
105	#106	—	34	1	no	—
106	#97	—	34	1	no	—
107	C₄×C₈	—	34	1	no	—
108	C₁₇	#1951	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	34	1	no	—
111	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	17	2	no	—
115	M₁₆	—	34	1	no	—
116	M₁₆	Dic₃₄	1	2	yes	—
117	M₁₆	C₄×D₁₇	1	2	yes	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	2	no	—
124	C₂×C₈	—	34	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	17	1	no	—
128	C₄⋊C₄	—	17	1	no	—
129	C₄⋊C₄	—	34	1	no	—
130	C₄⋊C₄	—	34	1	no	—
131	C₂²⋊C₄	—	34	1	no	—
132	C₄²	—	34	1	no	—
133	C₂³	#63252	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	34	1	no	—
137	C₂×C₄	—	34	1	no	—
138	C₂×C₄	—	68	1	no	—
139	C₂×C₄	—	136	1	no	—
140	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
142	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
144	C₂×C₄	#63254	1	1	yes	characteristic
145	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	2	no	—
148	C₈	—	17	2	no	—
149	C₈	—	17	2	no	—
150	C₈	—	34	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	C₄×Dic₃₄	1	1	yes	characteristic
154	C₂²	#10587570	1	1	yes	characteristic
155	C₂²	#10587575	1	1	yes	characteristic
156	C₄	—	8	1	no	—
157	C₄	—	68	1	no	—
158	C₄	—	68	1	no	—
159	C₄	—	136	1	no	—
160	C₄	#10587564	1	1	yes	characteristic
161	C₄	#10587567	1	1	yes	characteristic
162	C₄	#10587574	1	1	yes	characteristic
163	C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#11762706	1	1	yes	characteristic
167	C₂	#11763071	1	1	yes	characteristic
168	C₂	#11763075	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587570
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#343	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#343	D₁₇
hypocenter	C₁₇	#1951
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

Complements

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

N	H ≅ G/N	type
15. #10587493	C₄	semidirect
108. C₁₇	#1951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587493	2
#11762851	2
#11762849	2
C₂×#10587479	2
#11762767	2
#11762577	2
C₁₇×#343	2
#1951	17