Finite Group Databasebeta

Subgroups — #16782398

Order 3328.

216 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 58 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	#13129703	C₂	1	1	yes	maximal, characteristic
2	#13129703	C₂	1	1	yes	maximal, characteristic
3	#13129703	C₂	1	1	yes	maximal, characteristic
4	#13129696	C₂	1	1	yes	maximal, characteristic
5	#13129696	C₂	1	1	yes	maximal, characteristic
6	#13129696	C₂	1	1	yes	maximal, characteristic
7	#13113140	C₂	1	1	yes	maximal, characteristic
8	#11719061	C₂²	1	1	yes	characteristic
9	#11719055	C₂²	1	1	yes	characteristic
10	#11719055	C₂²	1	1	yes	characteristic
11	#11719055	C₂²	1	1	yes	characteristic
12	C₄×#6094	C₂²	1	1	yes	characteristic
13	C₄×#6094	C₂²	1	1	yes	characteristic
14	C₄×#6094	C₂²	1	1	yes	characteristic
15	C₁₃×#322	C₄	1	1	yes	characteristic, complemented
16	#11718046	C₄	1	1	yes	characteristic, complemented
17	#11718046	C₄	1	1	yes	characteristic, complemented
18	C₈×#1055	C₄	1	1	yes	characteristic, complemented
19	C₂×#6094	—	2	1	no	—
20	C₂×#6094	—	2	1	no	—
21	C₂×#6094	—	2	1	no	—
22	C₄²×D₁₃	C₂³	1	1	yes	characteristic
23	(C₂×#261)⋊C₄	—	2	1	no	—
24	(C₂×#261)⋊C₄	—	2	1	no	—
25	(C₂×#261)⋊C₄	—	2	1	no	—
26	#88112	—	2	1	no	—
27	#88112	—	2	1	no	—
28	#88112	—	2	1	no	—
29	#88111	—	2	1	no	—
30	#88111	—	2	1	no	—
31	#88111	—	2	1	no	—
32	#88110	—	2	1	no	—
33	#88110	—	2	1	no	—
34	#88110	—	2	1	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88055	—	2	1	no	—
39	#88055	—	2	1	no	—
40	#88055	—	2	1	no	—
41	C₄×#1055	C₂×C₄	1	1	yes	characteristic
42	C₄×#1055	C₂×C₄	1	1	yes	characteristic
43	C₄×#1055	C₂×C₄	1	1	yes	characteristic
44	#8803	—	13	1	no	maximal, Sylow
45	C₂²×#261	—	4	1	no	—
46	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
47	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
48	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
49	C₂×#1057	—	4	1	no	—
50	C₂×#1057	—	4	1	no	—
51	C₂×#1057	—	4	1	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₁₃	#41	1	1	yes	characteristic
57	C₄×Dic₁₃	#41	1	1	yes	characteristic
58	C₄×Dic₁₃	#41	1	1	yes	characteristic
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	2	1	no	—
62	#6094	—	4	2	no	—
63	#6094	—	4	2	no	—
64	#6094	—	4	2	no	—
65	#1455	—	13	1	no	—
66	#1291	—	13	1	no	—
67	#1291	—	13	1	no	—
68	#1291	—	13	1	no	—
69	#1288	—	13	1	no	—
70	#1288	—	13	1	no	—
71	#1288	—	13	1	no	—
72	C₂²×D₁₃	#126	1	1	yes	characteristic
73	C₂×#261	—	8	1	no	—
74	C₂×#261	—	8	1	no	—
75	C₂×#261	—	8	1	no	—
76	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
79	C₂×Dic₁₃	#126	1	1	yes	characteristic
80	C₂×Dic₁₃	#126	1	1	yes	characteristic
81	C₂×Dic₁₃	#126	1	1	yes	characteristic
82	C₄×D₁₃	—	4	1	no	—
83	C₄×D₁₃	—	4	1	no	—
84	C₄×D₁₃	—	4	1	no	—
85	#1057	—	8	1	no	—
86	#1057	—	8	1	no	—
87	#1057	—	8	1	no	—
88	C₈×C₁₃	—	2	2	no	—
89	C₈×C₁₃	—	4	1	no	—
90	C₈×C₁₃	—	4	1	no	—
91	#1055	—	2	2	no	—
92	#1055	—	2	2	no	—
93	#1055	—	4	1	no	—
94	#432	—	13	1	no	—
95	#432	—	13	1	no	—
96	#432	—	13	1	no	—
97	C₄×M₁₆	—	13	1	no	—
98	C₄×M₁₆	—	13	1	no	—
99	C₄×M₁₆	—	13	1	no	—
100	#383	—	13	1	no	—
101	#322	—	13	1	no	—
102	#322	—	13	1	no	—
103	#322	#261	1	1	yes	characteristic, complemented
104	C₈²	—	13	1	no	—
105	C₂²×C₁₃	#383	1	1	yes	characteristic
106	C₂×D₁₃	—	2	1	no	—
107	C₂×D₁₃	—	2	1	no	—
108	C₂×D₁₃	—	2	1	no	—
109	#261	—	16	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	1	no	—
113	Dic₁₃	—	2	1	no	—
114	Dic₁₃	—	2	1	no	—
115	Dic₁₃	—	2	1	no	—
116	C₂×M₁₆	—	26	1	no	—
117	C₂×M₁₆	—	26	1	no	—
118	C₂×M₁₆	—	26	1	no	—
119	C₂×C₄²	—	13	1	no	—
120	C₄⋊C₈	—	26	1	no	—
121	C₄⋊C₈	—	26	1	no	—
122	C₄⋊C₈	—	26	1	no	—
123	C₂²⋊C₈	—	26	1	no	—
124	C₂²⋊C₈	—	26	1	no	—
125	C₂²⋊C₈	—	26	1	no	—
126	#97	—	26	1	no	—
127	#97	—	26	1	no	—
128	#97	—	26	1	no	—
129	#97	—	26	1	no	—
130	#97	—	26	1	no	—
131	#97	—	26	1	no	—
132	C₄×C₈	—	13	1	no	—
133	C₄×C₈	—	13	1	no	—
134	C₄×C₈	—	13	1	no	—
135	C₄×C₈	C₂×#261	1	1	yes	characteristic
136	C₄×C₈	C₂×#261	1	1	yes	characteristic
137	C₄×C₈	C₂×#261	1	1	yes	characteristic
138	#95	—	26	1	no	—
139	#95	—	26	1	no	—
140	#95	—	26	1	no	—
141	C₂×C₁₃	#1841	1	1	yes	characteristic
142	C₂×C₁₃	#1841	1	1	yes	characteristic
143	C₂×C₁₃	#1841	1	1	yes	characteristic
144	D₁₃	—	4	1	no	—
145	C₂²×C₄	—	13	1	no	—
146	C₂²×C₄	—	13	1	no	—
147	C₂²×C₄	—	13	1	no	—
148	C₂²×C₄	—	52	1	no	—
149	M₁₆	—	52	2	no	—
150	M₁₆	—	52	2	no	—
151	M₁₆	—	52	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	26	1	no	—
156	C₂×C₈	—	26	1	no	—
157	C₂×C₈	—	26	1	no	—
158	C₂×C₈	—	52	1	no	—
159	C₂×C₈	—	52	1	no	—
160	C₂×C₈	—	52	1	no	—
161	C₄²	—	13	1	no	—
162	C₄²	—	13	1	no	—
163	C₄²	—	13	1	no	—
164	C₄²	C₂²×#261	1	1	yes	characteristic
165	C₁₃	#8803	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	13	1	no	—
167	C₂×C₄	—	13	1	no	—
168	C₂×C₄	—	13	1	no	—
169	C₂×C₄	—	13	1	no	—
170	C₂×C₄	—	52	1	no	—
171	C₂×C₄	—	52	1	no	—
172	C₂×C₄	—	52	1	no	—
173	C₂×C₄	—	104	1	no	—
174	C₂×C₄	—	104	1	no	—
175	C₂×C₄	—	104	1	no	—
176	C₂×C₄	#88237	1	1	yes	characteristic
177	C₂×C₄	#88237	1	1	yes	characteristic
178	C₂×C₄	#88237	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	4	1	no	—
181	C₈	—	4	1	no	—
182	C₈	—	26	2	no	—
183	C₈	—	26	2	no	—
184	C₈	—	52	1	no	—
185	C₈	—	104	1	no	—
186	C₈	—	104	1	no	—
187	C₈	—	104	1	no	—
188	C₂²	—	26	1	no	—
189	C₂²	—	26	1	no	—
190	C₂²	—	26	1	no	—
191	C₂²	#11719061	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	26	1	no	—
196	C₄	—	26	1	no	—
197	C₄	—	26	1	no	—
198	C₄	—	208	1	no	—
199	C₂	—	52	1	no	—
200	C₂	#13130766	1	1	yes	characteristic
201	C₂	#13130767	1	1	yes	characteristic
202	C₂	#13130767	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11719061
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#322	C₄
socle	C₂²×C₁₃	#383
prime radical	C₂²×D₁₃	#126
cosocle	C₄²×D₁₃	C₂³
hypercenter	#322	#261
hypocenter	C₁₃	#8803
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
15. C₁₃×#322	C₄	semidirect
16. #11718046	C₄	semidirect
17. #11718046	C₄	semidirect
18. C₈×#1055	C₄	semidirect
103. #322	#261	semidirect
165. C₁₃	#8803	semidirect

Maximal subgroups

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

subgroup	index
#13129703	2
#13129696	2
#13113140	2
#8803	13