Finite Group Databasebeta

Subgroups — #14530513

Order 2176.

220 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 81 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×#10587590	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587490	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762617	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#63252	—	2	1	no	—
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×#63223	—	2	1	no	—
8	C₂×Dic₆₈	C₄	1	1	yes	characteristic
9	#10587590	—	2	1	no	—
10	#10587590	—	2	1	no	—
11	#10587590	C₄	1	2	yes	—
12	#10587590	C₂²	1	2	yes	—
13	C₂×D₁₃₆	C₄	1	1	yes	characteristic
14	C₂×#63219	—	2	1	no	—
15	C₁₇×M₃₂	—	2	1	no	—
16	C₁₇×M₃₂	—	2	1	no	—
17	C₂×C₁₆×C₁₇	C₂²	1	2	yes	complemented
18	#10587490	—	2	1	no	—
19	#10587490	C₂²	1	2	yes	complemented
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₇	—	4	1	no	—
22	#63252	—	2	2	no	—
23	#63252	—	2	2	no	—
24	#63252	—	4	1	no	—
25	#63252	—	4	1	no	—
26	C₂×D₆₈	—	2	1	no	—
27	C₂×C₄×D₁₇	—	4	1	no	—
28	C₂×Dic₃₄	—	2	1	no	—
29	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	#63223	—	2	2	no	—
36	#63223	—	4	1	no	—
37	C₂×#3622	—	4	1	no	—
38	Dic₆₈	—	2	1	no	—
39	Dic₆₈	C₂×C₄	1	2	yes	—
40	D₁₃₆	—	2	1	no	—
41	D₁₃₆	C₂×C₄	1	2	yes	—
42	#63219	—	2	2	no	—
43	#63219	—	2	2	no	—
44	C₁₆×C₁₇	—	2	2	no	—
45	C₂³×C₁₇	SD₁₆	1	1	yes	characteristic
46	C₂²×D₁₇	—	4	1	no	—
47	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₁₇	—	4	2	no	—
54	C₂²⋊D₁₇	—	4	2	no	—
55	C₂×Dic₁₇	—	4	1	no	—
56	D₆₈	—	2	2	no	—
57	D₆₈	—	4	1	no	—
58	C₄×D₁₇	—	4	2	no	—
59	C₄×D₁₇	—	4	2	no	—
60	Dic₃₄	—	2	2	no	—
61	Dic₃₄	—	4	1	no	—
62	C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
66	#3622	—	4	2	no	—
67	#2143	—	17	1	no	maximal, Sylow
68	C₂²×C₁₇	—	4	1	no	—
69	C₂²×C₁₇	#102	1	1	yes	characteristic
70	C₂²×C₁₇	#102	1	1	yes	characteristic
71	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₁₇	—	4	2	no	—
73	C₂×D₁₇	—	8	2	no	—
74	C₄×C₁₇	#102	1	1	yes	characteristic
75	C₄×C₁₇	#102	1	1	yes	characteristic
76	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
78	Dic₁₇	—	4	2	no	—
79	C₂×#135	—	17	1	no	—
80	C₂×M₃₂	D₁₇	1	1	yes	characteristic, complemented
81	C₂×#108	—	17	1	no	—
82	#359	—	17	4	no	—
83	C₂×C₁₇	—	2	1	no	—
84	C₂×C₁₇	—	2	1	no	—
85	C₂×C₁₇	—	2	1	no	—
86	C₂×C₁₇	C₂×#102	1	1	yes	characteristic
87	D₁₇	—	8	2	no	—
88	C₂×#41	—	34	1	no	—
89	#135	—	17	2	no	—
90	#135	—	17	2	no	—
91	#135	—	34	1	no	—
92	#135	—	34	1	no	—
93	C₂×Q₁₆	—	17	1	no	—
94	C₂×SD₁₆	—	34	1	no	—
95	C₂×D₈	—	17	1	no	—
96	C₂×M₁₆	—	34	1	no	—
97	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
98	M₃₂	—	2	1	no	—
99	M₃₂	—	2	1	no	—
100	C₂×C₁₆	C₂×D₁₇	1	2	yes	complemented
101	#108	—	17	2	no	—
102	#108	—	34	1	no	—
103	C₁₇	#2143	1	1	yes	characteristic, complemented, Sylow
104	#41	—	34	2	no	—
105	#41	—	34	2	no	—
106	#41	—	68	1	no	—
107	#41	—	68	1	no	—
108	C₂×Q₈	—	34	1	no	—
109	C₂×D₄	—	34	1	no	—
110	C₂×D₄	—	68	1	no	—
111	C₂²×C₄	—	68	1	no	—
112	C₂²×C₄	D₆₈	1	1	yes	characteristic
113	Q₁₆	—	17	2	no	—
114	Q₁₆	—	34	1	no	—
115	SD₁₆	—	34	2	no	—
116	SD₁₆	—	34	2	no	—
117	D₈	—	17	2	no	—
118	D₈	—	34	1	no	—
119	M₁₆	—	34	2	no	—
120	M₁₆	—	68	1	no	—
121	C₂×C₈	—	68	1	no	—
122	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
123	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
124	C₂×C₈	D₆₈	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₂³	—	68	1	no	—
130	C₂³	#63219	1	1	yes	characteristic
131	Q₈	—	34	2	no	—
132	Q₈	—	68	1	no	—
133	D₄	—	34	2	no	—
134	D₄	—	68	2	no	—
135	D₄	—	68	2	no	—
136	D₄	—	68	1	no	—
137	C₂×C₄	—	68	1	no	—
138	C₂×C₄	—	68	2	no	—
139	C₂×C₄	—	68	2	no	—
140	C₂×C₄	#63219	1	1	yes	characteristic
141	C₂×C₄	D₁₃₆	1	1	yes	characteristic
142	C₂×C₄	D₁₃₆	1	1	yes	characteristic
143	C₂×C₄	#63227	1	1	yes	characteristic
144	C₂×C₄	#63227	1	1	yes	characteristic
145	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
146	C₈	—	68	2	no	—
147	C₈	#63227	1	1	yes	characteristic
148	C₈	#63227	1	1	yes	characteristic
149	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
150	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
151	C₂²	—	4	1	no	—
152	C₂²	—	68	2	no	—
153	C₂²	—	136	2	no	—
154	C₂²	#10587492	1	1	yes	characteristic
155	C₂²	#10587492	1	1	yes	characteristic
156	C₂²	C₂×#63219	1	1	yes	characteristic
157	C₄	—	68	2	no	—
158	C₄	#10587492	1	1	yes	characteristic
159	C₄	#10587492	1	1	yes	characteristic
160	C₄	C₂×D₁₃₆	1	1	yes	characteristic
161	C₄	C₂×#63227	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	136	2	no	—
166	C₂	C₂×#10587492	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₃₆
commutator	C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×M₃₂	C₂
socle	C₂×C₁₇	C₂×#102
prime radical	C₂×#10587590	C₂
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	C₂×M₃₂	D₁₇
hypocenter	C₁₇	#2143
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#10587490	C₂	semidirect
3. #11762617	C₂	semidirect
4. C₂×C₁₇×M₃₂	C₂	semidirect
17. C₂×C₁₆×C₁₇	C₂²	semidirect
19. #10587490	C₂²	semidirect
80. C₂×M₃₂	D₁₇	semidirect
100. C₂×C₁₆	C₂×D₁₇	semidirect
103. C₁₇	#2143	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587590	2
C₂×#10587490	2
#11762617	2
C₂×C₁₇×M₃₂	2
#2143	17