Finite Group Databasebeta

Subgroups — #16788812

Order 3328.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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	#13131225	C₂	1	1	yes	maximal, characteristic, complemented
2	#13131223	C₂	1	1	yes	maximal, characteristic, complemented
3	#13129872	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129799	C₂	1	1	yes	maximal, characteristic, complemented
5	#13129798	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#1057	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116714	C₂	1	1	yes	maximal, characteristic
8	#11719162	—	2	1	no	—
9	#11719096	—	2	1	no	—
10	#11718857	C₂²	1	1	yes	characteristic
11	#11718855	C₄	1	1	yes	characteristic
12	#11718556	C₂²	1	1	yes	characteristic
13	#11718304	—	2	1	no	—
14	#11718270	—	2	1	no	—
15	#11718261	C₂²	1	1	yes	characteristic
16	#11718261	C₂²	1	1	yes	characteristic
17	#11718257	C₂²	1	1	yes	characteristic
18	C₈×#1057	C₂²	1	1	yes	characteristic
19	C₁₃×#359	C₄	1	1	yes	characteristic
20	#11718102	C₄	1	1	yes	characteristic
21	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
22	#11718085	C₄	1	1	yes	characteristic
23	C₁₃×#135	C₂×C₄	1	1	yes	characteristic
24	#88202	—	2	1	no	—
25	#88161	—	2	1	no	—
26	#88158	C₂×C₄	1	1	yes	characteristic
27	C₄×#1057	—	2	1	no	—
28	#88106	—	2	1	no	—
29	#88106	—	2	1	no	—
30	#88106	—	4	1	no	—
31	C₂×#6092	—	4	1	no	—
32	C₂×#6092	D₄	1	1	yes	characteristic
33	C₂×#6092	D₄	1	1	yes	characteristic
34	#88098	—	4	1	no	—
35	D₁₃⋊C₁₆	—	4	1	no	—
36	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#108	C₂×C₄	1	1	yes	characteristic
38	#88077	—	2	1	no	—
39	#88069	C₈	1	1	yes	characteristic, complemented
40	#88068	C₈	1	1	yes	characteristic, complemented
41	#88067	C₈	1	1	yes	characteristic, complemented
42	#88066	C₈	1	1	yes	characteristic, complemented
43	#88065	—	2	1	no	—
44	#88059	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂×#6090	C₂×C₄	1	1	yes	characteristic
47	#12046	—	13	1	no	maximal, Sylow
48	C₁₃×#41	—	2	1	no	—
49	#6127	—	2	1	no	—
50	C₂×#1057	—	2	2	no	—
51	C₁₃×Q₁₆	C₂×C₈	1	1	yes	characteristic
52	C₁₃×SD₁₆	—	2	1	no	—
53	C₁₃×D₈	C₂×C₈	1	1	yes	characteristic
54	C₁₃×M₁₆	—	2	1	no	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₁₃	#41	1	1	yes	characteristic
57	#6099	—	2	1	no	—
58	#6099	—	4	1	no	—
59	C₂×#1055	—	4	1	no	—
60	C₂×#1055	C₂×D₄	1	1	yes	characteristic
61	Dic₅₂	C₂×C₈	1	1	yes	characteristic
62	D₁₀₄	C₂×C₈	1	1	yes	characteristic
63	#6095	—	2	1	no	—
64	#6094	—	4	1	no	—
65	C₈×D₁₃	—	4	1	no	—
66	#6092	—	2	1	no	—
67	#6092	—	4	1	no	—
68	#6092	—	4	1	no	—
69	#6092	D₈	1	1	yes	characteristic
70	#6092	D₈	1	1	yes	characteristic
71	C₁₃×C₁₆	—	2	1	no	—
72	#6090	C₂×C₈	1	1	yes	characteristic
73	#6090	C₂×C₈	1	1	yes	characteristic
74	#2178	—	13	1	no	—
75	#2170	—	13	1	no	—
76	#2170	—	13	1	no	—
77	#1373	—	13	1	no	—
78	#1335	—	13	1	no	—
79	#1335	—	13	1	no	—
80	C₈×C₁₆	—	13	1	no	—
81	Q₈×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	4	1	no	—
84	C₂×C₄×C₁₃	—	4	1	no	—
85	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₂²⋊D₁₃	—	4	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	D₅₂	—	2	1	no	—
89	C₄×D₁₃	—	4	1	no	—
90	Dic₂₆	—	2	1	no	—
91	#1057	—	2	4	no	—
92	C₈×C₁₃	—	4	1	no	—
93	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
94	C₈×C₁₃	#131	1	1	yes	characteristic
95	#1055	—	4	1	no	—
96	#1055	C₂×D₈	1	1	yes	characteristic
97	#1055	#135	1	1	yes	characteristic
98	#508	—	13	1	no	—
99	#504	—	26	1	no	—
100	#504	—	26	1	no	—
101	#443	—	13	1	no	—
102	#443	—	13	1	no	—
103	#368	—	13	1	no	—
104	#364	—	13	1	no	—
105	#364	—	13	1	no	—
106	#359	—	13	1	no	—
107	#359	#261	1	1	yes	characteristic
108	#350	—	26	1	no	—
109	#350	—	26	1	no	—
110	C₄×C₁₆	—	13	1	no	—
111	C₄×C₁₆	—	13	1	no	—
112	C₈²	—	13	1	no	—
113	C₂²×C₁₃	—	4	1	no	—
114	C₂²×C₁₃	#443	1	1	yes	characteristic
115	C₂×D₁₃	—	4	1	no	—
116	C₄×C₁₃	—	4	1	no	—
117	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
118	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	—	4	1	no	—
121	#135	—	13	1	no	—
122	#135	C₂×#261	1	1	yes	characteristic
123	#131	—	26	1	no	—
124	#131	—	26	1	no	—
125	Q₃₂	—	13	1	no	—
126	SD₃₂	—	13	1	no	—
127	SD₃₂	—	13	1	no	—
128	D₁₆	—	13	1	no	—
129	M₃₂	—	26	1	no	—
130	M₃₂	—	26	1	no	—
131	M₃₂	—	52	1	no	—
132	M₃₂	—	52	1	no	—
133	C₂×C₁₆	—	13	1	no	—
134	C₂×C₁₆	—	13	1	no	—
135	C₂×C₁₆	—	13	1	no	—
136	C₂×C₁₆	—	52	1	no	—
137	C₂×C₁₆	—	52	1	no	—
138	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
139	#108	—	13	1	no	—
140	#108	C₂×#261	1	1	yes	characteristic
141	#104	—	26	1	no	—
142	#104	—	26	1	no	—
143	C₄×C₈	—	13	1	no	—
144	C₄×C₈	—	26	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	—	8	1	no	—
147	C₂×C₁₃	C₈×D₈	1	1	yes	characteristic
148	D₁₃	—	8	1	no	—
149	#41	—	2	1	no	—
150	#41	—	26	1	no	—
151	Q₁₆	—	13	1	no	—
152	Q₁₆	D₁₃⋊C₈	1	1	yes	characteristic
153	SD₁₆	—	2	1	no	—
154	SD₁₆	—	26	1	no	—
155	D₈	—	13	1	no	—
156	D₈	D₁₃⋊C₈	1	1	yes	characteristic
157	M₁₆	—	2	1	no	—
158	M₁₆	—	26	1	no	—
159	M₁₆	—	52	1	no	—
160	M₁₆	—	52	1	no	—
161	C₂×C₈	—	13	1	no	—
162	C₂×C₈	—	26	2	no	—
163	C₂×C₈	—	52	1	no	—
164	C₂×C₈	—	52	1	no	—
165	C₂×C₈	C₂²×#261	1	1	yes	characteristic
166	C₄²	—	13	1	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	13	1	no	—
169	C₁₆	—	13	1	no	—
170	C₁₆	—	13	1	no	—
171	C₁₆	—	13	1	no	—
172	C₁₆	—	26	1	no	—
173	C₁₆	—	52	1	no	—
174	C₁₆	—	52	1	no	—
175	C₁₃	#12046	1	1	yes	characteristic, complemented, Sylow
176	Q₈	—	2	1	no	—
177	Q₈	—	26	1	no	—
178	D₄	—	2	1	no	—
179	D₄	—	4	1	no	—
180	D₄	—	26	1	no	—
181	D₄	—	52	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	26	1	no	—
184	C₂×C₄	—	52	1	no	—
185	C₂×C₄	D₄×#261	1	1	yes	characteristic
186	C₈	—	4	1	no	—
187	C₈	—	13	1	no	—
188	C₈	—	13	1	no	—
189	C₈	—	26	4	no	—
190	C₈	—	52	1	no	—
191	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
192	C₈	#88238	1	1	yes	characteristic
193	C₂²	—	4	1	no	—
194	C₂²	—	52	1	no	—
195	C₂²	#11719107	1	1	yes	characteristic
196	C₄	—	4	1	no	—
197	C₄	—	26	2	no	—
198	C₄	—	52	1	no	—
199	C₄	#11719063	1	1	yes	characteristic
200	C₄	D₈×#261	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	8	1	no	—
203	C₂	—	104	1	no	—
204	C₂	#13130416	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×#261
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#359	C₄
socle	C₂×C₁₃	C₈×D₈
prime radical	#11718855	C₄
cosocle	C₈×Dic₁₃	C₂³
hypercenter	#359	#261
hypocenter	C₁₃	#12046
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #13131225	C₂	semidirect
2. #13131223	C₂	semidirect
3. #13129872	C₂	semidirect
4. #13129799	C₂	semidirect
5. #13129798	C₂	semidirect
6. C₁₆×#1057	C₂	semidirect
39. #88069	C₈	semidirect
40. #88068	C₈	semidirect
41. #88067	C₈	semidirect
42. #88066	C₈	semidirect
175. C₁₃	#12046	semidirect

Maximal subgroups

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

subgroup	index
#13131225	2
#13131223	2
#13129872	2
#13129799	2
#13129798	2
C₁₆×#1057	2
#13116714	2
#12046	13