Finite Group Databasebeta

Subgroups — #14530744

Order 2176.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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	#11763610	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763247	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10587497	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763080	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762615	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762594	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#10587691	—	2	1	no	—
9	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587618	—	2	1	no	—
11	#10587598	—	2	1	no	—
12	#10587597	—	2	2	no	—
13	D₈×D₁₇	—	2	1	no	—
14	#10587594	—	2	1	no	—
15	#10587590	C₂²	1	1	yes	characteristic
16	C₂×D₁₃₆	C₂²	1	1	yes	characteristic, complemented
17	#10587587	C₂²	1	1	yes	characteristic
18	C₁₇×M₃₂	C₂²	1	1	yes	characteristic, complemented
19	C₁₇×#108	C₂²	1	1	yes	characteristic, complemented
20	#10587497	—	2	2	no	—
21	#10587494	—	2	1	no	—
22	C₂×#63214	C₂²	1	1	yes	characteristic, complemented
23	#10587471	—	2	1	no	—
24	D₂₇₂	—	2	1	no	—
25	C₂×D₄×C₁₇	—	2	1	no	—
26	C₂×C₂²⋊D₁₇	—	4	2	no	—
27	#63254	—	2	2	no	—
28	#63254	—	4	1	no	—
29	D₄×D₁₇	—	2	2	no	—
30	D₄×D₁₇	—	4	1	no	—
31	#63252	—	2	1	no	—
32	#63252	D₄	1	1	yes	characteristic
33	C₂×D₆₈	—	2	1	no	—
34	D₈×C₁₇	—	2	1	no	—
35	D₈×C₁₇	D₄	1	2	yes	—
36	M₁₆×C₁₇	—	2	1	no	—
37	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	#63229	—	2	2	no	—
39	#63228	—	2	2	no	—
40	#63223	D₄	1	1	yes	characteristic
41	Dic₆₈	D₄	1	1	yes	characteristic
42	D₁₃₆	—	2	1	no	—
43	D₁₃₆	—	2	1	no	—
44	D₁₃₆	—	4	1	no	—
45	D₁₃₆	D₄	1	1	yes	characteristic
46	#63219	—	2	1	no	—
47	#63219	—	4	1	no	—
48	#63218	—	2	1	no	—
49	C₈×D₁₇	—	2	1	no	—
50	C₁₆×C₁₇	—	2	1	no	—
51	#63214	—	2	1	no	—
52	C₂³×C₁₇	—	4	1	no	—
53	C₂²×D₁₇	—	4	2	no	—
54	C₂²×D₁₇	—	4	1	no	—
55	D₄×C₁₇	—	2	2	no	—
56	D₄×C₁₇	—	4	1	no	—
57	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₁₇	—	2	1	no	—
59	C₂²⋊D₁₇	—	4	2	no	—
60	C₂²⋊D₁₇	—	4	2	no	—
61	C₂²⋊D₁₇	—	4	1	no	—
62	C₂×Dic₁₇	—	4	2	no	—
63	D₆₈	—	2	1	no	—
64	D₆₈	—	4	1	no	—
65	D₆₈	—	4	1	no	—
66	D₆₈	D₈	1	1	yes	characteristic
67	C₄×D₁₇	—	2	1	no	—
68	C₄×D₁₇	—	4	1	no	—
69	Dic₃₄	—	2	1	no	—
70	Dic₃₄	D₈	1	1	yes	characteristic
71	C₈×C₁₇	—	4	1	no	—
72	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
74	#3622	—	2	1	no	—
75	#2192	—	17	1	no	maximal, Sylow
76	C₂²×C₁₇	—	4	2	no	—
77	C₂²×C₁₇	—	16	1	no	—
78	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₁₇	—	2	1	no	—
80	C₂×D₁₇	—	4	1	no	—
81	C₂×D₁₇	—	8	1	no	—
82	C₂×D₁₇	—	16	2	no	—
83	C₂×D₁₇	—	16	1	no	—
84	C₄×C₁₇	#120	1	1	yes	characteristic
85	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	—	4	1	no	—
88	#576	—	17	1	no	—
89	#509	—	17	1	no	—
90	C₂×D₁₆	—	17	1	no	—
91	#472	—	17	1	no	—
92	#361	D₁₇	1	1	yes	characteristic, complemented
93	#359	—	17	1	no	—
94	#350	—	17	1	no	—
95	C₂×C₁₇	—	2	1	no	—
96	C₂×C₁₇	—	8	2	no	—
97	C₂×C₁₇	#447	1	1	yes	characteristic
98	D₁₇	—	4	1	no	—
99	D₁₇	—	8	1	no	—
100	D₁₇	—	16	1	no	—
101	H(5,𝔽₂)	—	34	1	no	—
102	AGL(1,ℤ/8)	—	34	1	no	—
103	AGL(1,ℤ/8)	—	34	1	no	—
104	AGL(1,ℤ/8)	—	34	2	no	—
105	#135	—	17	1	no	—
106	#135	—	34	1	no	—
107	C₂×D₈	—	17	1	no	—
108	C₂×D₈	—	34	1	no	—
109	C₂×D₈	C₂×D₁₇	1	1	yes	characteristic
110	#131	—	17	1	no	—
111	SD₃₂	—	34	1	no	—
112	D₁₆	—	34	1	no	—
113	D₁₆	—	34	2	no	—
114	M₃₂	C₂×D₁₇	1	1	yes	characteristic, complemented
115	C₂×C₁₆	—	17	1	no	—
116	#108	C₂×D₁₇	1	1	yes	characteristic, complemented
117	#100	—	34	1	no	—
118	C₁₇	#2192	1	1	yes	characteristic, complemented, Sylow
119	#41	—	17	1	no	—
120	#41	—	34	2	no	—
121	#41	—	34	1	no	—
122	#41	—	68	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	34	2	no	—
125	C₂×D₄	—	34	1	no	—
126	C₂×D₄	—	68	1	no	—
127	C₂×D₄	—	68	2	no	—
128	Q₁₆	—	17	1	no	—
129	SD₁₆	—	34	2	no	—
130	SD₁₆	—	34	1	no	—
131	SD₁₆	—	68	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	17	1	no	—
134	D₈	—	34	2	no	—
135	D₈	—	34	1	no	—
136	D₈	—	34	1	no	—
137	D₈	—	68	1	no	—
138	D₈	D₆₈	1	2	yes	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	17	1	no	—
141	M₁₆	—	34	1	no	—
142	C₂×C₈	—	34	1	no	—
143	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
144	C₁₆	—	2	1	no	—
145	C₁₆	—	34	1	no	—
146	C₂³	—	4	1	no	—
147	C₂³	—	68	2	no	—
148	C₂³	—	68	1	no	—
149	Q₈	—	17	1	no	—
150	Q₈	—	34	1	no	—
151	D₄	—	2	2	no	—
152	D₄	—	4	1	no	—
153	D₄	—	17	1	no	—
154	D₄	—	34	1	no	—
155	D₄	—	34	1	no	—
156	D₄	—	68	2	no	—
157	D₄	—	68	2	no	—
158	D₄	—	68	1	no	—
159	D₄	—	68	1	no	—
160	D₄	—	68	1	no	—
161	C₂×C₄	—	34	1	no	—
162	C₂×C₄	—	68	2	no	—
163	C₂×C₄	—	68	1	no	—
164	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
165	C₈	—	4	1	no	—
166	C₈	—	34	1	no	—
167	C₈	C₂×D₆₈	1	1	yes	characteristic
168	C₈	D₄×D₁₇	1	1	yes	characteristic
169	C₂²	—	4	2	no	—
170	C₂²	—	16	1	no	—
171	C₂²	—	272	1	no	—
172	C₂²	—	272	2	no	—
173	C₂²	—	34	1	no	—
174	C₂²	—	68	1	no	—
175	C₂²	—	136	1	no	—
176	C₂²	#10587600	1	1	yes	characteristic
177	C₄	—	34	1	no	—
178	C₄	—	68	1	no	—
179	C₄	#10587568	1	1	yes	characteristic
180	C₄	D₈×D₁₇	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	8	2	no	—
183	C₂	—	272	1	no	—
184	C₂	—	68	1	no	—
185	C₂	—	136	1	no	—
186	C₂	#11762944	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762944
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#361	C₂
socle	C₂×C₁₇	#447
hypercenter	#361	D₁₇
hypocenter	C₁₇	#2192
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

Complements

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

N	H ≅ G/N	type
1. #11763610	C₂	semidirect
2. #11763247	C₂	semidirect
3. C₂×#10587497	C₂	semidirect
4. #11763080	C₂	semidirect
5. #11762615	C₂	semidirect
6. #11762594	C₂	semidirect
7. C₁₇×#361	C₂	semidirect
16. C₂×D₁₃₆	C₂²	semidirect
18. C₁₇×M₃₂	C₂²	semidirect
19. C₁₇×#108	C₂²	semidirect
22. C₂×#63214	C₂²	semidirect
92. #361	D₁₇	semidirect
114. M₃₂	C₂×D₁₇	semidirect
116. #108	C₂×D₁₇	semidirect
118. C₁₇	#2192	semidirect

Maximal subgroups

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

subgroup	index
#11763610	2
#11763247	2
C₂×#10587497	2
#11763080	2
#11762615	2
#11762594	2
C₁₇×#361	2
#2192	17