Finite Group Databasebeta

Subgroups — #14545692

Order 2176.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 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	#11763860	C₂	1	4	yes	maximal, complemented
2	#11763859	C₂	1	2	yes	maximal, complemented
3	#11763858	C₂	1	2	yes	maximal, complemented
4	#11763834	C₂	1	4	yes	maximal
5	#11763831	C₂	1	1	yes	maximal, characteristic
6	#11763827	C₂	1	1	yes	maximal, characteristic
7	#11763364	C₂	1	1	yes	maximal, characteristic
8	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	#10587632	C₄	1	2	yes	—
10	Q₈×Dic₁₇	C₂²	1	2	yes	—
11	#10587583	C₄	1	2	yes	—
12	#10587578	C₂²	1	2	yes	—
13	#10587576	C₄	1	2	yes	—
14	#10587573	C₂²	1	2	yes	—
15	C₄²×D₁₇	C₂²	1	1	yes	characteristic
16	#10587555	C₄	1	1	yes	characteristic
17	#10587544	C₂²	1	2	yes	—
18	#10587544	C₂²	1	4	yes	—
19	#10587543	C₂²	1	2	yes	—
20	#10587543	C₂²	1	2	yes	—
21	#10587542	C₂²	1	2	yes	—
22	#10587542	C₂²	1	4	yes	—
23	C₄⋊#3624	C₂²	1	2	yes	—
24	C₄⋊#3624	C₂²	1	4	yes	—
25	C₄×#3624	C₂²	1	1	yes	characteristic
26	C₄×#3624	C₂²	1	1	yes	characteristic
27	C₄×#3624	C₂²	1	4	yes	—
28	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
33	C₂×#3624	C₂³	1	2	yes	—
34	C₂×#3624	C₂³	1	2	yes	—
35	C₂×#3624	C₂³	1	4	yes	—
36	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
38	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
39	#63227	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
43	#63225	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
46	C₄×Dic₁₇	C₂³	1	4	yes	—
47	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
48	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₇	—	4	2	no	—
50	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
55	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
58	C₄×D₁₇	—	2	2	no	—
59	C₄×D₁₇	—	2	2	no	—
60	C₄×D₁₇	—	4	1	no	—
61	Dic₃₄	—	2	4	no	—
62	#3624	—	2	2	no	—
63	#3624	—	2	2	no	—
64	#3624	—	2	4	no	—
65	#2985	—	17	1	no	maximal, Sylow
66	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₁₇	—	2	1	no	—
68	C₂×D₁₇	—	2	1	no	—
69	C₂×D₁₇	—	2	1	no	—
70	C₄×C₁₇	—	2	1	no	—
71	C₄×C₁₇	—	2	1	no	—
72	C₄×C₁₇	—	2	4	no	—
73	C₄×C₁₇	#131	1	2	yes	—
74	Dic₁₇	—	2	4	no	—
75	Dic₁₇	—	2	1	no	—
76	Dic₁₇	—	2	1	no	—
77	Dic₁₇	2₋¹⁺⁴	1	2	yes	—
78	#533	—	17	1	no	—
79	#446	—	17	2	no	—
80	#446	—	17	4	no	—
81	C₈×Q₈	—	17	2	no	—
82	#433	—	17	4	no	—
83	#432	—	17	1	no	—
84	#431	—	17	1	no	—
85	C₂×C₁₇	#519	1	1	yes	characteristic
86	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
87	C₂×C₁₇	#568	1	1	yes	characteristic
88	D₁₇	—	4	1	no	—
89	C₄⋊Q₈	—	17	1	no	—
90	C₄⋊Q₈	#595	1	1	yes	characteristic
91	#125	—	17	2	no	—
92	C₂²⋊Q₈	—	17	2	no	—
93	C₂²⋊Q₈	—	17	2	no	—
94	C₄×Q₈	—	17	2	no	—
95	C₄×Q₈	—	17	2	no	—
96	#117	—	17	2	no	—
97	C₂×C₄²	—	17	1	no	—
98	C₄⋊C₈	—	17	2	no	—
99	C₄⋊C₈	—	17	4	no	—
100	C₄⋊C₈	—	17	2	no	—
101	C₄⋊C₈	—	17	4	no	—
102	C₂²⋊C₈	—	17	2	no	—
103	C₂²⋊C₈	—	17	2	no	—
104	#97	—	17	2	no	—
105	#97	—	17	4	no	—
106	C₄×C₈	—	17	1	no	—
107	C₄×C₈	—	17	1	no	—
108	C₄×C₈	—	17	4	no	—
109	C₁₇	#2985	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	—	17	2	no	—
111	C₂×Q₈	C₂×#595	1	2	yes	—
112	C₂²×C₄	—	17	1	no	—
113	C₂²×C₄	—	17	1	no	—
114	C₂²×C₄	—	17	1	no	—
115	C₂×C₈	—	17	2	no	—
116	C₂×C₈	—	17	2	no	—
117	C₂×C₈	—	17	4	no	—
118	C₄⋊C₄	—	17	2	no	—
119	C₄⋊C₄	—	17	4	no	—
120	C₄⋊C₄	—	17	4	no	—
121	C₄⋊C₄	—	17	2	no	—
122	C₄⋊C₄	C₂×#595	1	2	yes	—
123	C₄⋊C₄	C₂×#595	1	2	yes	—
124	C₂²⋊C₄	—	17	4	no	—
125	C₄²	—	17	1	no	—
126	C₄²	—	17	4	no	—
127	C₄²	—	17	1	no	—
128	C₄²	—	17	1	no	—
129	C₄²	C₂×#595	1	1	yes	characteristic
130	C₂³	—	17	1	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	34	4	no	—
133	C₂×C₄	—	17	1	no	—
134	C₂×C₄	—	17	1	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	4	no	—
137	C₂×C₄	—	34	2	no	—
138	C₂×C₄	—	34	2	no	—
139	C₂×C₄	—	68	1	no	—
140	C₂×C₄	C₂²×#595	1	1	yes	characteristic
141	C₂×C₄	C₂²×#595	1	1	yes	characteristic
142	C₂×C₄	C₂²×#595	1	4	yes	—
143	C₂×C₄	C₂²×#595	1	1	yes	characteristic
144	C₈	—	34	2	no	—
145	C₈	—	34	2	no	—
146	C₈	—	34	4	no	—
147	C₂²	—	34	1	no	—
148	C₂²	—	34	1	no	—
149	C₂²	—	34	1	no	—
150	C₂²	C₂³×#595	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	17	2	no	—
155	C₄	—	34	1	no	—
156	C₄	—	34	4	no	—
157	C₄	—	34	1	no	—
158	C₄	#10587680	1	2	yes	—
159	C₂	—	68	1	no	—
160	C₂	#11763877	1	1	yes	characteristic
161	C₂	C₂×#10587680	1	1	yes	characteristic
162	C₂	#11763887	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×C₄⋊Q₈	C₄
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂×Dic₁₇	C₂⁴
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#2985
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #11763860	C₂	semidirect
2. #11763859	C₂	semidirect
3. #11763858	C₂	semidirect
109. C₁₇	#2985	semidirect

Maximal subgroups

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

subgroup	index
#11763860	2
#11763859	2
#11763858	2
#11763834	2
#11763831	2
#11763827	2
#11763364	2
#2985	17