Finite Group Databasebeta

Subgroups — #14544871

Order 2176.

216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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	#11763824	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763808	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587550	C₂	1	1	yes	maximal, characteristic
4	M₁₆×#595	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763692	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763681	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762792	C₂	1	1	yes	maximal, characteristic
8	C₂×#63256	C₂²	1	1	yes	characteristic
9	#10587677	C₂²	1	1	yes	characteristic
10	C₂×C₄×#595	C₂²	1	1	yes	characteristic
11	D₁₇⋊M₁₆	C₂²	1	1	yes	characteristic
12	C₂×#63243	C₂²	1	1	yes	characteristic
13	M₁₆×D₁₇	C₂²	1	1	yes	characteristic
14	M₁₆×D₁₇	C₂²	1	1	yes	characteristic
15	#10587550	—	2	1	no	—
16	#10587550	—	2	1	no	—
17	#10587550	—	2	1	no	—
18	#10587550	—	2	1	no	—
19	#10587536	—	2	1	no	—
20	#10587535	—	2	1	no	—
21	#10587532	—	2	1	no	—
22	C₈×#595	—	2	1	no	—
23	C₁₇×#101	C₄	1	1	yes	characteristic, complemented
24	#10587507	C₄	1	1	yes	characteristic, complemented
25	#10587494	C₄	1	1	yes	characteristic, complemented
26	#10587494	C₄	1	1	yes	characteristic, complemented
27	C₂²×#595	—	2	1	no	—
28	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	#63256	—	2	1	no	—
30	#63256	—	2	1	no	—
31	#63256	—	2	1	no	—
32	#63256	—	2	1	no	—
33	C₂×D₆₈	—	2	1	no	—
34	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₇	—	2	1	no	—
36	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
37	C₂²⋊#595	—	2	1	no	—
38	#63247	—	2	1	no	—
39	C₂×#3624	—	2	1	no	—
40	C₄⋊#595	—	2	1	no	—
41	C₄×#595	—	2	1	no	—
42	C₄×#595	—	2	1	no	—
43	C₄×#595	D₄	1	1	yes	characteristic
44	C₄×#595	D₄	1	1	yes	characteristic
45	#63243	—	2	1	no	—
46	#63243	—	2	1	no	—
47	#63243	D₄	1	1	yes	characteristic
48	#63243	D₄	1	1	yes	characteristic
49	#63242	—	2	1	no	—
50	M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
51	M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
52	#63223	C₂×C₄	1	1	yes	characteristic
53	#63223	C₂×C₄	1	1	yes	characteristic
54	#63218	—	2	1	no	—
55	#63218	—	2	1	no	—
56	C₈×D₁₇	—	2	1	no	—
57	C₈×D₁₇	—	2	1	no	—
58	C₂²×D₁₇	—	2	1	no	—
59	C₂²×D₁₇	#41	1	1	yes	characteristic
60	C₂×#595	—	2	1	no	—
61	C₂×#595	—	2	1	no	—
62	C₂×#595	—	4	2	no	—
63	C₂×#595	—	4	1	no	—
64	Q₈×C₁₇	—	2	1	no	—
65	Q₈×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₁₇	#41	1	1	yes	characteristic
69	D₆₈	—	2	1	no	—
70	D₆₈	—	2	1	no	—
71	D₆₈	—	4	1	no	—
72	D₆₈	—	4	1	no	—
73	C₄×D₁₇	—	4	1	no	—
74	C₄×D₁₇	—	4	1	no	—
75	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
76	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
77	C₄×D₁₇	#41	1	1	yes	characteristic
78	C₄×D₁₇	#41	1	1	yes	characteristic
79	#3624	—	2	1	no	—
80	#3624	—	2	1	no	—
81	#3624	—	4	1	no	—
82	C₈×C₁₇	—	2	1	no	—
83	C₈×C₁₇	—	2	1	no	—
84	#3622	—	2	1	no	—
85	#3622	—	2	1	no	—
86	#1932	—	17	1	no	maximal, Sylow
87	C₂²×C₁₇	#117	1	1	yes	characteristic
88	C₂×D₁₇	—	4	1	no	—
89	C₂×D₁₇	—	4	1	no	—
90	C₂×D₁₇	—	8	1	no	—
91	C₂×D₁₇	#123	1	1	yes	characteristic
92	C₂×D₁₇	#124	1	1	yes	characteristic
93	#595	—	2	2	no	—
94	#595	—	4	1	no	—
95	#595	—	8	1	no	—
96	C₄×C₁₇	—	4	1	no	—
97	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
98	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
99	Dic₁₇	#121	1	1	yes	characteristic
100	Dic₁₇	#126	1	1	yes	characteristic
101	#430	—	17	1	no	—
102	#421	—	17	1	no	—
103	C₂×#104	—	17	1	no	—
104	#413	—	17	1	no	—
105	C₄×M₁₆	—	17	1	no	—
106	#344	—	17	1	no	—
107	#339	—	17	1	no	—
108	C₂×C₁₇	—	2	1	no	—
109	C₂×C₁₇	#388	1	1	yes	characteristic
110	D₁₇	—	2	1	no	—
111	D₁₇	—	2	1	no	—
112	D₁₇	—	8	1	no	—
113	C₂×#41	—	17	1	no	—
114	C₂×M₁₆	—	17	1	no	—
115	C₂×M₁₆	—	17	1	no	—
116	C₂×M₁₆	—	17	1	no	—
117	C₂×M₁₆	—	17	1	no	—
118	#117	—	17	1	no	—
119	C₂×C₄²	—	17	1	no	—
120	#108	—	34	1	no	—
121	#108	—	34	1	no	—
122	#104	—	34	1	no	—
123	#104	—	34	1	no	—
124	#104	—	34	1	no	—
125	#104	—	34	1	no	—
126	#101	—	17	1	no	—
127	#101	#595	1	1	yes	characteristic, complemented
128	#100	—	17	1	no	—
129	#100	—	17	1	no	—
130	#97	—	34	1	no	—
131	C₄×C₈	—	34	1	no	—
132	C₁₇	#1932	1	1	yes	characteristic, complemented, Sylow
133	#41	—	34	1	no	—
134	#41	—	34	1	no	—
135	#41	—	34	1	no	—
136	#41	—	34	1	no	—
137	C₂×Q₈	C₂×#595	1	1	yes	characteristic
138	C₂×D₄	—	17	1	no	—
139	C₂×D₄	—	34	1	no	—
140	C₂²×C₄	—	17	1	no	—
141	C₂²×C₄	—	34	1	no	—
142	C₂²×C₄	—	34	1	no	—
143	M₁₆	—	17	1	no	—
144	M₁₆	—	17	1	no	—
145	M₁₆	—	17	1	no	—
146	M₁₆	—	17	1	no	—
147	M₁₆	—	34	1	no	—
148	M₁₆	—	34	1	no	—
149	M₁₆	—	34	1	no	—
150	M₁₆	—	34	1	no	—
151	M₁₆	—	34	1	no	—
152	M₁₆	C₂×#595	1	1	yes	characteristic
153	M₁₆	C₂×#595	1	1	yes	characteristic
154	C₂×C₈	—	34	1	no	—
155	C₂×C₈	—	34	1	no	—
156	C₂×C₈	—	34	1	no	—
157	C₂×C₈	—	34	1	no	—
158	C₄⋊C₄	—	34	1	no	—
159	C₂²⋊C₄	—	34	1	no	—
160	C₄²	—	17	1	no	—
161	C₄²	—	17	1	no	—
162	C₄²	—	34	1	no	—
163	C₄²	—	34	1	no	—
164	C₂³	—	17	1	no	—
165	C₂³	—	34	1	no	—
166	Q₈	—	2	1	no	—
167	Q₈	—	2	1	no	—
168	D₄	—	34	1	no	—
169	D₄	—	34	1	no	—
170	D₄	—	68	1	no	—
171	D₄	—	68	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	17	1	no	—
174	C₂×C₄	—	17	1	no	—
175	C₂×C₄	—	17	1	no	—
176	C₂×C₄	—	17	1	no	—
177	C₂×C₄	—	17	1	no	—
178	C₂×C₄	—	34	1	no	—
179	C₂×C₄	—	34	1	no	—
180	C₂×C₄	—	68	1	no	—
181	C₂×C₄	—	68	1	no	—
182	C₂×C₄	—	68	2	no	—
183	C₂×C₄	—	68	1	no	—
184	C₂×C₄	C₂²×#595	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	34	1	no	—
188	C₈	—	34	1	no	—
189	C₈	—	34	1	no	—
190	C₈	—	34	1	no	—
191	C₈	—	68	1	no	—
192	C₂²	—	17	1	no	—
193	C₂²	—	17	1	no	—
194	C₂²	—	68	1	no	—
195	C₂²	—	68	1	no	—
196	C₂²	—	136	1	no	—
197	C₂²	#10587677	1	1	yes	characteristic
198	C₄	—	4	1	no	—
199	C₄	—	17	1	no	—
200	C₄	—	17	1	no	—
201	C₄	—	34	2	no	—
202	C₄	—	68	1	no	—
203	C₄	—	136	1	no	—
204	C₄	D₄×#595	1	1	yes	characteristic
205	C₄	D₄×#595	1	1	yes	characteristic
206	C₂	—	2	1	no	—
207	C₂	—	34	1	no	—
208	C₂	—	34	1	no	—
209	C₂	—	136	1	no	—
210	C₂	#11763759	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11763759
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#101	C₄
socle	C₂×C₁₇	#388
prime radical	C₂×#63256	C₂²
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#101	#595
hypocenter	C₁₇	#1932
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
1. #11763824	C₂	semidirect
4. M₁₆×#595	C₂	semidirect
5. #11763692	C₂	semidirect
6. #11763681	C₂	semidirect
23. C₁₇×#101	C₄	semidirect
24. #10587507	C₄	semidirect
25. #10587494	C₄	semidirect
26. #10587494	C₄	semidirect
127. #101	#595	semidirect
132. C₁₇	#1932	semidirect

Maximal subgroups

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

subgroup	index
#11763824	2
#11763808	2
C₂×#10587550	2
M₁₆×#595	2
#11763692	2
#11763681	2
#11762792	2
#1932	17