Finite Group Databasebeta

Subgroups — #14530749

Order 2176.

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

normal subgroups only

This group has 210 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	#11763623	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763244	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10587471	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763080	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762624	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762592	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#10587697	—	2	1	no	—
9	C₁₇×#135	C₂²	1	1	yes	characteristic
10	#10587637	—	2	1	no	—
11	Q₁₆×D₁₇	—	2	1	no	—
12	#10587601	—	2	2	no	—
13	#10587598	—	2	1	no	—
14	#10587594	—	2	1	no	—
15	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
16	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
17	#10587587	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#108	C₂²	1	1	yes	characteristic
20	#10587499	—	2	1	no	—
21	#10587497	—	2	1	no	—
22	#10587494	—	2	1	no	—
23	#10587483	C₂²	1	1	yes	characteristic
24	#10587471	—	2	2	no	—
25	#41×C₁₇	—	2	1	no	—
26	Q₈×D₁₇	—	2	2	no	—
27	#63254	—	2	2	no	—
28	#63254	—	4	2	no	—
29	#63252	—	4	1	no	—
30	#63252	D₄	1	1	yes	characteristic
31	C₂×D₆₈	—	2	1	no	—
32	C₂×Dic₃₄	—	2	1	no	—
33	C₂×Dic₃₄	—	4	1	no	—
34	Q₁₆×C₁₇	D₄	1	1	yes	characteristic
35	SD₁₆×C₁₇	—	2	1	no	—
36	D₈×C₁₇	D₄	1	1	yes	characteristic
37	M₁₆×C₁₇	—	2	1	no	—
38	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	#63231	—	2	2	no	—
40	#63229	—	2	2	no	—
41	#63223	D₄	1	1	yes	characteristic
42	Dic₆₈	—	2	1	no	—
43	Dic₆₈	D₄	1	2	yes	—
44	D₁₃₆	—	2	1	no	—
45	D₁₃₆	—	2	1	no	—
46	D₁₃₆	—	4	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	C₂²×D₁₇	—	4	1	no	—
53	Q₈×C₁₇	—	2	1	no	—
54	D₄×C₁₇	—	2	1	no	—
55	D₄×C₁₇	—	4	1	no	—
56	C₂×C₄×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	C₂×Dic₁₇	—	4	2	no	—
61	C₂×Dic₁₇	—	4	1	no	—
62	D₆₈	—	4	1	no	—
63	D₆₈	—	4	1	no	—
64	D₆₈	D₈	1	1	yes	characteristic
65	C₄×D₁₇	—	2	1	no	—
66	C₄×D₁₇	—	4	2	no	—
67	Dic₃₄	—	2	2	no	—
68	Dic₃₄	—	4	2	no	—
69	Dic₃₄	—	4	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	#2193	—	17	1	no	maximal, Sylow
76	C₂²×C₁₇	—	4	1	no	—
77	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×D₁₇	—	2	1	no	—
79	C₂×D₁₇	—	8	1	no	—
80	C₂×D₁₇	—	16	1	no	—
81	C₄×C₁₇	—	4	1	no	—
82	C₄×C₁₇	#120	1	1	yes	characteristic
83	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
84	Dic₁₇	—	2	1	no	—
85	Dic₁₇	—	4	2	no	—
86	#578	—	17	1	no	—
87	#509	—	17	1	no	—
88	C₂×SD₃₂	—	17	1	no	—
89	#472	—	17	1	no	—
90	#362	—	17	1	no	—
91	#359	D₁₇	1	1	yes	characteristic, complemented
92	#350	—	17	1	no	—
93	C₂×C₁₇	—	2	1	no	—
94	C₂×C₁₇	—	8	1	no	—
95	C₂×C₁₇	#447	1	1	yes	characteristic
96	D₁₇	—	4	1	no	—
97	D₁₇	—	16	1	no	—
98	2₋¹⁺⁴	—	34	1	no	—
99	#137	—	34	1	no	—
100	#137	—	34	2	no	—
101	AGL(1,ℤ/8)	—	34	1	no	—
102	#135	—	34	1	no	—
103	#135	C₂×D₁₇	1	1	yes	characteristic
104	C₂×Q₁₆	—	17	1	no	—
105	C₂×Q₁₆	—	34	1	no	—
106	C₂×D₈	—	17	1	no	—
107	#131	—	17	1	no	—
108	SD₃₂	—	34	2	no	—
109	SD₃₂	—	34	1	no	—
110	D₁₆	—	34	1	no	—
111	M₃₂	—	17	1	no	—
112	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
113	#108	C₂×D₁₇	1	1	yes	characteristic
114	#100	—	34	1	no	—
115	C₁₇	#2193	1	1	yes	characteristic, complemented, Sylow
116	#41	—	2	1	no	—
117	#41	—	17	1	no	—
118	#41	—	34	2	no	—
119	#41	—	68	1	no	—
120	#41	—	68	2	no	—
121	C₂×Q₈	—	34	2	no	—
122	C₂×Q₈	—	34	1	no	—
123	C₂×Q₈	—	68	1	no	—
124	C₂×D₄	—	34	1	no	—
125	Q₁₆	—	17	2	no	—
126	Q₁₆	—	34	2	no	—
127	Q₁₆	—	34	1	no	—
128	Q₁₆	D₆₈	1	1	yes	characteristic
129	SD₁₆	—	2	1	no	—
130	SD₁₆	—	34	2	no	—
131	SD₁₆	—	68	1	no	—
132	D₈	—	34	1	no	—
133	D₈	—	34	1	no	—
134	D₈	—	68	1	no	—
135	D₈	D₆₈	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	17	1	no	—
138	M₁₆	—	34	1	no	—
139	C₂×C₈	—	34	1	no	—
140	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	34	1	no	—
143	C₂³	—	68	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	17	1	no	—
146	Q₈	—	34	2	no	—
147	Q₈	—	68	2	no	—
148	Q₈	—	68	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	17	1	no	—
152	D₄	—	34	1	no	—
153	D₄	—	68	2	no	—
154	D₄	—	68	1	no	—
155	D₄	—	68	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	34	1	no	—
158	C₂×C₄	—	68	2	no	—
159	C₂×C₄	—	68	2	no	—
160	C₂×C₄	—	68	1	no	—
161	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
162	C₈	—	4	1	no	—
163	C₈	—	34	1	no	—
164	C₈	C₂×D₆₈	1	1	yes	characteristic
165	C₈	D₄×D₁₇	1	1	yes	characteristic
166	C₂²	—	4	1	no	—
167	C₂²	—	272	1	no	—
168	C₂²	—	34	1	no	—
169	C₂²	—	136	1	no	—
170	C₂²	#10587600	1	1	yes	characteristic
171	C₄	—	4	1	no	—
172	C₄	—	34	1	no	—
173	C₄	—	68	2	no	—
174	C₄	#10587568	1	1	yes	characteristic
175	C₄	D₈×D₁₇	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	—	272	1	no	—
179	C₂	—	68	1	no	—
180	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×#359	C₂
socle	C₂×C₁₇	#447
hypercenter	#359	D₁₇
hypocenter	C₁₇	#2193
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #11763623	C₂	semidirect
2. #11763244	C₂	semidirect
3. C₂×#10587471	C₂	semidirect
4. #11763080	C₂	semidirect
5. #11762624	C₂	semidirect
6. #11762592	C₂	semidirect
7. C₁₇×#359	C₂	semidirect
91. #359	D₁₇	semidirect
115. C₁₇	#2193	semidirect

Maximal subgroups

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

subgroup	index
#11763623	2
#11763244	2
C₂×#10587471	2
#11763080	2
#11762624	2
#11762592	2
C₁₇×#359	2
#2193	17