Finite Group Databasebeta

Subgroups — #14544943

Order 2176.

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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	#11763819	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763808	C₂	1	2	yes	maximal, complemented
3	#11763790	C₂	1	2	yes	maximal, complemented
4	#11763772	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10587605	C₂	1	1	yes	maximal, characteristic
6	C₂×#63256	C₂²	1	2	yes	—
7	#10587677	C₂²	1	2	yes	—
8	#10587677	C₂²	1	1	yes	characteristic
9	D₁₇⋊M₁₆	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#63230	C₄	1	2	yes	complemented
12	#10587605	—	2	1	no	—
13	#10587605	—	2	1	no	—
14	#10587605	—	2	1	no	—
15	#10587605	—	2	1	no	—
16	C₂×D₁₃₆	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	#10587550	—	2	4	no	—
19	#10587539	—	2	4	no	—
20	#10587534	—	2	1	no	—
21	#10587533	—	2	1	no	—
22	#10587532	—	2	1	no	—
23	C₈×#595	—	2	1	no	—
24	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
25	#63256	—	2	2	no	—
26	#63256	—	2	2	no	—
27	#63256	—	2	2	no	—
28	#63256	—	2	2	no	—
29	C₂×D₆₈	—	2	2	no	—
30	C₂×D₆₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₇	—	2	2	no	—
32	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
33	C₂²⋊#595	—	2	2	no	—
34	C₂²⋊#595	D₄	1	2	yes	—
35	#63247	—	2	1	no	—
36	C₄⋊#595	—	2	2	no	—
37	C₄⋊#595	—	2	1	no	—
38	C₄×#595	—	2	1	no	—
39	C₄×#595	—	2	2	no	—
40	#63243	D₄	1	2	yes	—
41	#63242	—	2	1	no	—
42	Q₁₆×C₁₇	—	2	1	no	—
43	Q₁₆×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	#63230	—	2	2	no	—
46	#63230	—	2	2	no	—
47	C₂×#3622	C₂×C₄	1	1	yes	characteristic
48	D₁₃₆	—	2	1	no	—
49	D₁₃₆	—	2	1	no	—
50	C₈×D₁₇	D₄	1	1	yes	characteristic
51	C₈×D₁₇	D₄	1	1	yes	characteristic
52	C₈×D₁₇	D₄	1	1	yes	characteristic
53	C₈×D₁₇	D₄	1	1	yes	characteristic
54	C₂²×D₁₇	—	2	2	no	—
55	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×#595	—	2	2	no	—
57	C₂×#595	—	4	2	no	—
58	Q₈×C₁₇	—	2	2	no	—
59	Q₈×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₁₇	#41	1	1	yes	characteristic
63	D₆₈	—	2	2	no	—
64	D₆₈	—	2	2	no	—
65	D₆₈	—	4	2	no	—
66	D₆₈	—	4	2	no	—
67	C₄×D₁₇	—	4	2	no	—
68	C₄×D₁₇	—	4	2	no	—
69	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
70	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
71	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
72	C₄×D₁₇	#41	1	1	yes	characteristic
73	#3624	—	2	2	no	—
74	C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	#3622	C₂²⋊C₄	1	1	yes	characteristic
77	#3622	C₂²⋊C₄	1	1	yes	characteristic
78	#1976	—	17	1	no	maximal, Sylow
79	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂×D₁₇	—	4	1	no	—
81	C₂×D₁₇	—	4	2	no	—
82	C₂×D₁₇	—	8	2	no	—
83	C₂×D₁₇	#121	1	1	yes	characteristic
84	C₂×D₁₇	#127	1	1	yes	characteristic
85	#595	—	4	2	no	—
86	#595	—	8	2	no	—
87	C₄×C₁₇	—	4	2	no	—
88	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
90	Dic₁₇	#121	1	1	yes	characteristic
91	Dic₁₇	#124	1	1	yes	characteristic
92	C₂×#135	—	17	1	no	—
93	#427	—	17	1	no	—
94	#421	—	17	2	no	—
95	#410	—	17	2	no	—
96	#405	—	17	1	no	—
97	C₂×C₁₇	—	2	1	no	—
98	C₂×C₁₇	#390	1	1	yes	characteristic
99	D₁₇	—	2	1	no	—
100	D₁₇	—	2	1	no	—
101	D₁₇	—	8	2	no	—
102	C₂×#41	—	17	2	no	—
103	#135	—	34	1	no	—
104	#135	—	34	1	no	—
105	#135	—	34	1	no	—
106	#135	—	34	1	no	—
107	C₂×Q₁₆	#595	1	1	yes	characteristic, complemented
108	C₂×SD₁₆	—	17	2	no	—
109	C₂×D₈	—	17	1	no	—
110	C₂×M₁₆	—	17	1	no	—
111	C₂²×C₈	—	17	1	no	—
112	#117	—	17	1	no	—
113	#117	—	17	2	no	—
114	#107	—	34	1	no	—
115	#106	—	34	1	no	—
116	#104	—	34	4	no	—
117	#99	—	34	4	no	—
118	#97	—	34	1	no	—
119	C₄×C₈	—	34	1	no	—
120	C₁₇	#1976	1	1	yes	characteristic, complemented, Sylow
121	#41	—	34	2	no	—
122	#41	—	34	2	no	—
123	#41	—	34	2	no	—
124	#41	—	34	2	no	—
125	C₂×Q₈	C₂×#595	1	2	yes	—
126	C₂×D₄	—	17	2	no	—
127	C₂×D₄	—	34	2	no	—
128	C₂²×C₄	—	17	1	no	—
129	C₂²×C₄	—	34	2	no	—
130	Q₁₆	—	2	1	no	—
131	Q₁₆	—	2	1	no	—
132	SD₁₆	—	34	2	no	—
133	SD₁₆	—	34	2	no	—
134	D₈	—	34	1	no	—
135	D₈	—	34	1	no	—
136	M₁₆	—	17	2	no	—
137	M₁₆	—	34	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	1	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	34	1	no	—
144	C₂×C₈	C₂×#595	1	1	yes	characteristic
145	C₄⋊C₄	—	34	1	no	—
146	C₄⋊C₄	—	34	2	no	—
147	C₂²⋊C₄	—	17	2	no	—
148	C₂²⋊C₄	—	34	2	no	—
149	C₄²	—	34	1	no	—
150	C₄²	—	34	2	no	—
151	C₂³	—	17	1	no	—
152	C₂³	—	34	2	no	—
153	Q₈	—	2	2	no	—
154	Q₈	—	2	2	no	—
155	D₄	—	34	2	no	—
156	D₄	—	34	2	no	—
157	D₄	—	68	2	no	—
158	D₄	—	68	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	17	1	no	—
161	C₂×C₄	—	17	1	no	—
162	C₂×C₄	—	17	1	no	—
163	C₂×C₄	—	17	1	no	—
164	C₂×C₄	—	17	1	no	—
165	C₂×C₄	—	34	2	no	—
166	C₂×C₄	—	68	2	no	—
167	C₂×C₄	—	68	2	no	—
168	C₂×C₄	—	68	2	no	—
169	C₂×C₄	C₂²×#595	1	1	yes	characteristic
170	C₈	—	17	1	no	—
171	C₈	—	17	1	no	—
172	C₈	—	34	2	no	—
173	C₈	C₂²⋊#595	1	1	yes	characteristic
174	C₈	C₂²⋊#595	1	1	yes	characteristic
175	C₂²	—	17	1	no	—
176	C₂²	—	17	1	no	—
177	C₂²	—	68	1	no	—
178	C₂²	—	68	2	no	—
179	C₂²	—	136	2	no	—
180	C₂²	D₄×#595	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	17	1	no	—
183	C₄	—	17	1	no	—
184	C₄	—	68	2	no	—
185	C₄	—	136	2	no	—
186	C₄	D₄×#595	1	1	yes	characteristic
187	C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	34	1	no	—
190	C₂	—	34	1	no	—
191	C₂	—	136	2	no	—
192	C₂	#11763761	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11763761
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₂×Q₁₆×C₁₇	C₄
socle	C₂×C₁₇	#390
prime radical	C₂×#10587605	C₂
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₂×Q₁₆	#595
hypocenter	C₁₇	#1976
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #11763819	C₂	semidirect
2. #11763808	C₂	semidirect
3. #11763790	C₂	semidirect
4. #11763772	C₂	semidirect
10. C₂×Q₁₆×C₁₇	C₄	semidirect
11. C₂×#63230	C₄	semidirect
16. C₂×D₁₃₆	C₄	semidirect
107. C₂×Q₁₆	#595	semidirect
120. C₁₇	#1976	semidirect

Maximal subgroups

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

subgroup	index
#11763819	2
#11763808	2
#11763790	2
#11763772	2
C₂×#10587605	2
#1976	17