Finite Group Databasebeta

Subgroups — #14528844

Order 2176.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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×#10587587	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587490	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762808	C₂	1	2	yes	maximal
4	#11762578	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63252	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	1	yes	characteristic
11	#10587587	—	2	2	no	—
12	#10587587	C₄	1	4	yes	complemented
13	C₂×#63218	—	2	1	no	—
14	C₂×C₈×D₁₇	—	2	1	no	—
15	C₁₇×#106	—	2	1	no	—
16	C₁₇×#106	—	2	1	no	—
17	#10587490	—	2	2	no	—
18	#10587481	—	2	2	no	—
19	#10587480	—	2	2	no	—
20	C₂²×C₄×C₁₇	—	2	2	no	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₇	—	2	1	no	—
23	#63252	—	2	1	no	—
24	#63252	—	2	1	no	—
25	#63252	C₂×C₄	1	2	yes	—
26	#63252	Q₈	1	2	yes	—
27	C₂×D₆₈	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₇	—	2	1	no	—
29	C₂×Dic₃₄	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₇	D₄	1	2	yes	—
32	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₁₇	—	2	2	no	—
35	C₄⋊C₄×C₁₇	—	2	2	no	—
36	#63223	—	2	1	no	—
37	#63223	—	2	2	no	—
38	#63223	—	2	2	no	—
39	#63223	C₂×C₄	1	2	yes	—
40	C₂×#3622	—	2	1	no	—
41	C₂×#3622	—	2	2	no	—
42	#63218	—	2	4	no	—
43	C₈×D₁₇	—	2	4	no	—
44	C₂³×C₁₇	#41	1	1	yes	characteristic
45	C₂²×D₁₇	—	2	1	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	—	8	2	no	—
48	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
54	C₂²⋊D₁₇	—	2	2	no	—
55	C₂²⋊D₁₇	—	2	2	no	—
56	C₂×Dic₁₇	—	2	1	no	—
57	D₆₈	—	2	1	no	—
58	D₆₈	C₄⋊C₄	1	2	yes	—
59	C₄×D₁₇	—	2	2	no	—
60	C₄×D₁₇	—	2	2	no	—
61	Dic₃₄	—	2	1	no	—
62	Dic₃₄	C₄⋊C₄	1	2	yes	—
63	C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
64	C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
65	#3622	—	2	2	no	—
66	#3622	—	4	2	no	—
67	#1815	—	17	1	no	maximal, Sylow
68	C₂²×C₁₇	—	2	2	no	—
69	C₂²×C₁₇	#117	1	1	yes	characteristic
70	C₂²×C₁₇	#121	1	1	yes	characteristic
71	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×D₁₇	—	2	2	no	—
73	C₂×D₁₇	—	4	2	no	—
74	C₄×C₁₇	—	8	2	no	—
75	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₁₇	#121	1	1	yes	characteristic
78	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
79	Dic₁₇	—	2	2	no	—
80	C₂×#131	—	17	1	no	—
81	C₂×#108	—	17	1	no	—
82	C₂×#106	D₁₇	1	1	yes	characteristic, complemented
83	#417	—	17	2	no	—
84	#343	—	17	2	no	—
85	C₂×C₁₇	—	2	1	no	—
86	C₂×C₁₇	—	2	1	no	—
87	C₂×C₁₇	#380	1	1	yes	characteristic
88	C₂×C₁₇	#471	1	2	yes	—
89	D₁₇	—	4	2	no	—
90	C₂×#41	—	17	1	no	—
91	#131	—	17	4	no	—
92	#131	—	34	2	no	—
93	C₂×M₁₆	—	17	1	no	—
94	C₂×M₁₆	—	17	2	no	—
95	C₂×M₁₆	—	34	1	no	—
96	C₂²×C₈	—	34	1	no	—
97	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
99	#108	—	34	2	no	—
100	#106	—	2	1	no	—
101	#106	—	2	1	no	—
102	#103	—	34	2	no	—
103	#102	—	34	2	no	—
104	C₁₇	#1815	1	1	yes	characteristic, complemented, Sylow
105	#41	—	17	2	no	—
106	#41	—	17	2	no	—
107	#41	—	34	1	no	—
108	#41	—	34	1	no	—
109	C₂×Q₈	—	17	1	no	—
110	C₂×D₄	—	17	1	no	—
111	C₂×D₄	—	34	1	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	34	1	no	—
114	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
115	M₁₆	—	17	2	no	—
116	M₁₆	—	34	1	no	—
117	M₁₆	—	34	4	no	—
118	M₁₆	—	34	2	no	—
119	M₁₆	—	34	2	no	—
120	C₂×C₈	—	34	1	no	—
121	C₂×C₈	—	34	4	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	C₄×D₁₇	1	2	yes	—
124	C₂×C₈	D₆₈	1	2	yes	—
125	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₂³	—	34	1	no	—
130	C₂³	#63254	1	1	yes	characteristic
131	Q₈	—	17	2	no	—
132	Q₈	—	34	1	no	—
133	D₄	—	17	2	no	—
134	D₄	—	34	2	no	—
135	D₄	—	34	2	no	—
136	D₄	—	34	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	34	1	no	—
140	C₂×C₄	—	34	2	no	—
141	C₂×C₄	—	34	2	no	—
142	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
143	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
144	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
145	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
146	C₂×C₄	#63256	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
148	C₈	—	34	2	no	—
149	C₈	—	68	2	no	—
150	C₈	#63227	1	2	yes	—
151	C₈	#63227	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	34	2	no	—
154	C₂²	—	68	2	no	—
155	C₂²	#10587578	1	1	yes	characteristic
156	C₂²	#10587583	1	1	yes	characteristic
157	C₂²	#10587624	1	1	yes	characteristic
158	C₄	—	8	2	no	—
159	C₄	—	34	2	no	—
160	C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
161	C₄	#10587581	1	1	yes	characteristic
162	C₄	C₂×#63227	1	1	yes	characteristic
163	C₄	#10587632	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	68	2	no	—
167	C₂	#11762666	1	1	yes	characteristic
168	C₂	#11763069	1	2	yes	—

	subgroup	G/H
center	C₂²	#10587624
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#106	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂×#63252	C₂²
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#106	D₁₇
hypocenter	C₁₇	#1815
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#10587490	C₂	semidirect
4. #11762578	C₂	semidirect
5. C₂×C₁₇×#106	C₂	semidirect
12. #10587587	C₄	semidirect
82. C₂×#106	D₁₇	semidirect
104. C₁₇	#1815	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587587	2
C₂×#10587490	2
#11762808	2
#11762578	2
C₂×C₁₇×#106	2
#1815	17