Finite Group Databasebeta

Subgroups — #14539320

Order 2176.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#11763440	C₂	1	2	yes	maximal
2	#11763217	C₂	1	4	yes	maximal, complemented
3	#11763213	C₂	1	2	yes	maximal, complemented
4	#11763210	C₂	1	2	yes	maximal, complemented
5	#11762849	C₂	1	2	yes	maximal
6	#11762847	C₂	1	1	yes	maximal, characteristic
7	#11762764	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₇×#107	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
11	#10587612	C₂²	1	2	yes	—
12	#10587612	C₂²	1	1	yes	characteristic
13	#10587610	—	2	4	no	—
14	C₂×#63223	C₂²	1	1	yes	characteristic
15	#10587576	—	2	2	no	—
16	#10587576	C₂²	1	4	yes	—
17	#10587574	—	2	2	no	—
18	#10587574	C₂²	1	4	yes	—
19	C₁₇×#107	C₂²	1	2	yes	—
20	C₁₇×#107	C₂²	1	2	yes	—
21	#10587489	C₂²	1	1	yes	characteristic
22	#10587489	C₂²	1	1	yes	characteristic
23	#10587488	C₂²	1	2	yes	—
24	#10587486	C₂²	1	2	yes	—
25	C₈×Dic₁₇	C₂²	1	2	yes	—
26	#10587479	C₂²	1	4	yes	—
27	#10587478	C₂²	1	4	yes	—
28	C₂²×C₄×C₁₇	—	2	2	no	—
29	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₃₄	—	2	4	no	—
31	C₂×C₈×C₁₇	Q₈	1	4	yes	complemented
32	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₇	—	2	2	no	—
35	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
36	C₂²⋊Dic₁₇	—	2	2	no	—
37	C₂²⋊Dic₁₇	D₄	1	2	yes	—
38	C₄⋊Dic₁₇	—	2	4	no	—
39	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
40	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
41	C₄⋊Dic₁₇	C₂³	1	2	yes	—
42	C₄⋊Dic₁₇	C₂³	1	2	yes	—
43	#63225	—	2	4	no	—
44	#63225	—	2	4	no	—
45	C₄×Dic₁₇	—	2	2	no	—
46	C₄×Dic₁₇	C₂³	1	2	yes	—
47	#63223	Q₈	1	4	yes	complemented
48	C₂×#3622	C₂³	1	2	yes	—
49	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	—	2	4	no	—
51	C₂×C₄×C₁₇	—	8	2	no	—
52	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₇	—	2	2	no	—
57	C₂×Dic₁₇	—	2	2	no	—
58	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
59	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
60	Dic₃₄	—	8	4	no	—
61	C₈×C₁₇	C₂×Q₈	1	2	yes	—
62	C₈×C₁₇	C₂×Q₈	1	2	yes	—
63	#3622	C₂×Q₈	1	4	yes	—
64	#3164	—	17	1	no	maximal, Sylow
65	C₂²×C₁₇	—	2	1	no	—
66	C₂²×C₁₇	—	2	1	no	—
67	C₂²×C₁₇	C₄⋊Q₈	1	2	yes	—
68	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₁₇	—	4	4	no	—
70	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
71	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
73	Dic₁₇	—	2	2	no	—
74	Dic₁₇	—	2	2	no	—
75	Dic₁₇	—	4	2	no	—
76	Dic₁₇	—	4	2	no	—
77	#544	—	17	2	no	—
78	#501	—	17	4	no	—
79	#500	—	17	2	no	—
80	#499	—	17	2	no	—
81	#428	—	17	2	no	—
82	#427	—	17	1	no	—
83	C₂×#107	D₁₇	1	1	yes	characteristic
84	#405	—	17	1	no	—
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₁₇	#576	1	1	yes	characteristic
88	C₂×C₁₇	#578	1	1	yes	characteristic
89	C₂×M₁₆	—	17	1	no	—
90	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
91	C₄⋊Q₈	—	17	4	no	—
92	C₄⋊Q₈	—	34	2	no	—
93	#125	—	17	4	no	—
94	#125	—	34	2	no	—
95	C₂²⋊Q₈	—	34	4	no	—
96	#117	—	17	1	no	—
97	#117	—	17	2	no	—
98	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
99	#107	—	17	1	no	—
100	#107	—	17	1	no	—
101	#107	—	17	4	no	—
102	#107	C₂×D₁₇	1	2	yes	—
103	#107	C₂×D₁₇	1	2	yes	—
104	#106	—	17	2	no	—
105	#106	—	17	4	no	—
106	#97	—	17	2	no	—
107	C₄×C₈	—	17	2	no	—
108	C₁₇	#3164	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	34	4	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
112	M₁₆	—	17	4	no	—
113	C₂×C₈	—	17	2	no	—
114	C₂×C₈	Dic₃₄	1	4	yes	complemented
115	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	17	1	no	—
119	C₄⋊C₄	—	17	1	no	—
120	C₄⋊C₄	—	17	2	no	—
121	C₄⋊C₄	—	17	2	no	—
122	C₄⋊C₄	—	34	4	no	—
123	C₄⋊C₄	—	34	4	no	—
124	C₄⋊C₄	—	34	4	no	—
125	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
126	C₂²⋊C₄	—	17	2	no	—
127	C₂²⋊C₄	—	34	2	no	—
128	C₄²	—	17	2	no	—
129	C₄²	—	34	2	no	—
130	C₂³	D₄×D₁₇	1	1	yes	characteristic
131	Q₈	—	136	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	17	2	no	—
135	C₂×C₄	—	17	2	no	—
136	C₂×C₄	—	34	2	no	—
137	C₂×C₄	—	34	2	no	—
138	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
139	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
140	C₂×C₄	Q₈×D₁₇	1	2	yes	—
141	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
142	C₈	—	17	4	no	—
143	C₈	C₂×Dic₃₄	1	2	yes	—
144	C₈	C₂×Dic₃₄	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#10587574	1	2	yes	—
148	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
149	C₄	—	4	4	no	—
150	C₄	—	34	2	no	—
151	C₄	—	34	2	no	—
152	C₄	—	68	2	no	—
153	C₄	—	68	2	no	—
154	C₄	#10587574	1	2	yes	—
155	C₄	C₂²×Dic₃₄	1	1	yes	characteristic
156	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#10587574	1	1	yes	characteristic
159	C₂	#11763610	1	1	yes	characteristic
160	C₂	#11763622	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×#107	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂×#107	D₁₇
hypocenter	C₁₇	#3164
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. #11763217	C₂	semidirect
3. #11763213	C₂	semidirect
4. #11763210	C₂	semidirect
31. C₂×C₈×C₁₇	Q₈	semidirect
47. #63223	Q₈	semidirect
108. C₁₇	#3164	semidirect
114. C₂×C₈	Dic₃₄	semidirect

Maximal subgroups

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

subgroup	index
#11763440	2
#11763217	2
#11763213	2
#11763210	2
#11762849	2
#11762847	2
#11762764	2
C₂×C₁₇×#107	2
#3164	17