Finite Group Databasebeta

Subgroups — #14530752

Order 2176.

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

normal subgroups only

This group has 216 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	#11763611	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587499	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763081	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762623	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762616	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762594	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#510	C₂	1	1	yes	maximal, characteristic, complemented
8	#10587696	—	2	1	no	—
9	C₁₇×#135	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic, complemented
11	#10587635	—	2	1	no	—
12	#10587628	—	2	1	no	—
13	#10587605	—	2	1	no	—
14	#10587600	—	2	2	no	—
15	D₈×D₁₇	—	2	1	no	—
16	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
17	#10587587	C₂²	1	1	yes	characteristic
18	C₁₇×Q₃₂	—	2	1	no	—
19	C₁₇×SD₃₂	—	2	1	no	—
20	C₁₇×M₃₂	C₂²	1	1	yes	characteristic, complemented
21	#10587507	—	2	1	no	—
22	#10587499	—	2	2	no	—
23	#10587490	C₂²	1	1	yes	characteristic, complemented
24	C₂×#63214	C₂²	1	1	yes	characteristic, complemented
25	#41×C₁₇	—	2	1	no	—
26	C₂×Q₈×C₁₇	—	2	1	no	—
27	#63256	—	2	2	no	—
28	D₄×D₁₇	—	2	2	no	—
29	D₄×D₁₇	—	4	2	no	—
30	#63252	—	4	1	no	—
31	#63252	D₄	1	1	yes	characteristic
32	C₂×D₆₈	—	2	1	no	—
33	C₂×D₆₈	—	4	1	no	—
34	Q₁₆×C₁₇	—	2	1	no	—
35	Q₁₆×C₁₇	—	2	1	no	—
36	Q₁₆×C₁₇	D₄	1	1	yes	characteristic
37	SD₁₆×C₁₇	—	2	1	no	—
38	D₈×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
40	#63231	—	4	1	no	—
41	#63230	—	2	2	no	—
42	#63230	—	4	1	no	—
43	#63228	—	2	2	no	—
44	#63223	—	2	1	no	—
45	#63223	D₄	1	1	yes	characteristic
46	D₁₃₆	—	2	1	no	—
47	D₁₃₆	D₄	1	2	yes	—
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	2	no	—
53	C₂²×D₁₇	—	4	1	no	—
54	Q₈×C₁₇	—	2	1	no	—
55	Q₈×C₁₇	—	4	1	no	—
56	Q₈×C₁₇	—	4	1	no	—
57	D₄×C₁₇	—	2	1	no	—
58	D₄×C₁₇	—	4	1	no	—
59	C₂×C₄×C₁₇	—	4	1	no	—
60	C₂×C₄×C₁₇	—	4	1	no	—
61	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₁₇	—	2	1	no	—
63	C₂²⋊D₁₇	—	4	2	no	—
64	D₆₈	—	2	2	no	—
65	D₆₈	—	4	2	no	—
66	D₆₈	—	4	1	no	—
67	D₆₈	D₈	1	1	yes	characteristic
68	C₄×D₁₇	—	2	1	no	—
69	C₄×D₁₇	—	4	2	no	—
70	Dic₃₄	D₈	1	1	yes	characteristic
71	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
73	#3622	—	2	1	no	—
74	#3622	—	4	1	no	—
75	#2194	—	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	—	4	2	no	—
80	C₂×D₁₇	—	16	2	no	—
81	C₂×D₁₇	—	16	1	no	—
82	C₄×C₁₇	—	4	1	no	—
83	C₄×C₁₇	—	8	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	#576	—	17	1	no	—
88	#510	D₁₇	1	1	yes	characteristic, complemented
89	C₂×SD₃₂	—	17	1	no	—
90	#473	—	17	1	no	—
91	#361	—	17	1	no	—
92	#359	—	17	1	no	—
93	#350	—	17	1	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	—	8	1	no	—
96	C₂×C₁₇	#447	1	1	yes	characteristic
97	D₁₇	—	4	1	no	—
98	D₁₇	—	8	2	no	—
99	H(5,𝔽₂)	—	34	1	no	—
100	#137	—	34	1	no	—
101	AGL(1,ℤ/8)	—	34	1	no	—
102	AGL(1,ℤ/8)	—	34	2	no	—
103	#135	—	34	1	no	—
104	#135	C₂×D₁₇	1	1	yes	characteristic
105	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic, complemented
106	C₂×D₈	—	17	1	no	—
107	C₂×D₈	—	34	1	no	—
108	#131	—	17	1	no	—
109	Q₃₂	—	2	1	no	—
110	SD₃₂	—	2	1	no	—
111	SD₃₂	—	34	2	no	—
112	M₃₂	C₂×D₁₇	1	1	yes	characteristic, complemented
113	C₂×C₁₆	—	17	1	no	—
114	#108	—	17	1	no	—
115	#101	—	34	1	no	—
116	C₁₇	#2194	1	1	yes	characteristic, complemented, Sylow
117	#41	—	2	1	no	—
118	#41	—	17	1	no	—
119	#41	—	34	2	no	—
120	#41	—	68	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×D₄	—	34	2	no	—
123	C₂×D₄	—	34	1	no	—
124	C₂×D₄	—	68	1	no	—
125	C₂×D₄	—	68	2	no	—
126	Q₁₆	—	2	1	no	—
127	Q₁₆	—	2	1	no	—
128	Q₁₆	—	68	1	no	—
129	Q₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
130	SD₁₆	—	2	1	no	—
131	SD₁₆	—	34	2	no	—
132	SD₁₆	—	68	1	no	—
133	D₈	—	17	2	no	—
134	D₈	—	34	2	no	—
135	D₈	—	34	1	no	—
136	D₈	C₂²⋊D₁₇	1	1	yes	characteristic
137	M₁₆	—	17	1	no	—
138	M₁₆	—	34	1	no	—
139	M₁₆	—	34	1	no	—
140	C₂×C₈	—	34	1	no	—
141	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	—	34	1	no	—
144	C₂³	—	68	2	no	—
145	C₂³	—	68	1	no	—
146	Q₈	—	2	1	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	17	1	no	—
150	D₄	—	2	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	17	1	no	—
153	D₄	—	34	1	no	—
154	D₄	—	34	2	no	—
155	D₄	—	68	2	no	—
156	D₄	—	68	2	no	—
157	D₄	—	68	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	34	1	no	—
161	C₂×C₄	—	68	2	no	—
162	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
163	C₈	—	34	1	no	—
164	C₈	—	68	1	no	—
165	C₈	D₄×D₁₇	1	1	yes	characteristic
166	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
167	C₂²	—	4	1	no	—
168	C₂²	—	272	1	no	—
169	C₂²	—	272	2	no	—
170	C₂²	—	34	1	no	—
171	C₂²	—	68	2	no	—
172	C₂²	#10587597	1	1	yes	characteristic
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	34	1	no	—
176	C₄	D₈×D₁₇	1	1	yes	characteristic
177	C₄	#10587623	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	8	1	no	—
180	C₂	—	68	1	no	—
181	C₂	—	136	2	no	—
182	C₂	#11762941	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762941
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#510	C₂
socle	C₂×C₁₇	#447
prime radical	#11763611	C₂
hypercenter	#510	D₁₇
hypocenter	C₁₇	#2194
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

Complements

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

N	H ≅ G/N	type
2. C₂×#10587499	C₂	semidirect
3. #11763081	C₂	semidirect
4. #11762623	C₂	semidirect
5. #11762616	C₂	semidirect
6. #11762594	C₂	semidirect
7. C₁₇×#510	C₂	semidirect
10. C₂×Q₁₆×C₁₇	C₂²	semidirect
20. C₁₇×M₃₂	C₂²	semidirect
23. #10587490	C₂²	semidirect
24. C₂×#63214	C₂²	semidirect
88. #510	D₁₇	semidirect
105. C₂×Q₁₆	C₂×D₁₇	semidirect
112. M₃₂	C₂×D₁₇	semidirect
116. C₁₇	#2194	semidirect

Maximal subgroups

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

subgroup	index
#11763611	2
C₂×#10587499	2
#11763081	2
#11762623	2
#11762616	2
#11762594	2
C₁₇×#510	2
#2194	17