Subgroups — #16790799

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6092	C₂	1	2	yes	maximal
3	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic
4	#13129874	C₂	1	2	yes	maximal
5	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6092	C₂²	1	2	yes	—
7	C₂²×#6092	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₄⋊#6092	—	2	2	no	—
14	C₄⋊#6092	—	2	4	no	—
15	C₄×#6092	—	2	2	no	—
16	#11718116	C₄	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1055	C₂×C₄	1	2	yes	—
22	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	C₂×#6092	—	2	1	no	—
25	C₂×#6092	—	2	1	no	—
26	C₂×#6092	—	2	1	no	—
27	C₂×#6092	—	2	1	no	—
28	C₂×#6092	—	2	2	no	—
29	C₂×#6092	—	2	2	no	—
30	C₂×#6092	—	2	2	no	—
31	C₂×#6092	D₄	1	2	yes	—
32	C₂×#6092	Q₈	1	2	yes	—
33	C₁₃×#95	C₈	1	2	yes	—
34	C₄⋊#1055	—	2	2	no	—
35	C₄×#1055	—	2	2	no	—
36	C₄×#1055	—	2	2	no	—
37	C₄×#1055	D₄	1	2	yes	—
38	C₄×#1055	Q₈	1	2	yes	—
39	#13159	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₁₃×C₄⋊C₄	C₁₆	1	4	yes	complemented
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₄²×C₁₃	M₁₆	1	2	yes	—
51	C₂×#1055	—	2	1	no	—
52	C₂×#1055	—	2	1	no	—
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	—	2	2	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	C₄⋊C₄	1	2	yes	—
57	C₂×#1055	C₂×D₄	1	1	yes	characteristic
58	C₂×#1055	C₂×D₄	1	1	yes	characteristic
59	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
60	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
61	C₂×#1055	#41	1	1	yes	characteristic
62	C₂×#1055	#41	1	1	yes	characteristic
63	#6092	—	2	2	no	—
64	#6092	—	2	2	no	—
65	#6092	—	4	2	no	—
66	C₂×C₄⋊C₁₆	—	13	1	no	—
67	C₂×C₄⋊C₁₆	—	13	2	no	—
68	C₂×C₄×C₁₆	—	13	1	no	—
69	C₈×C₄⋊C₄	—	13	1	no	—
70	#1374	—	13	2	no	—
71	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
82	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
90	#1055	—	2	2	no	—
91	#1055	—	4	2	no	—
92	#1055	#121	1	1	yes	characteristic
93	#1055	C₂²⋊Q₈	1	1	yes	characteristic
94	#1055	#125	1	1	yes	characteristic
95	#1055	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₁₆	—	13	1	no	—
97	C₂²×C₁₆	—	13	1	no	—
98	C₂²×C₁₆	—	13	2	no	—
99	C₂×C₄⋊C₈	—	13	1	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	2	no	—
102	C₄×C₄⋊C₄	#261	1	1	yes	characteristic
103	C₄⋊C₁₆	—	26	2	no	—
104	C₄⋊C₁₆	—	26	4	no	—
105	C₄×C₁₆	—	26	2	no	—
106	#336	—	13	2	no	—
107	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
109	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
110	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
112	C₂²×C₁₃	#504	1	1	yes	characteristic
113	C₂²×C₁₃	#504	1	1	yes	characteristic
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	C₄⋊C₁₆	1	4	yes	complemented
118	C₄×C₁₃	#389	1	1	yes	characteristic
119	C₄×C₁₃	#424	1	1	yes	characteristic
120	C₄×C₁₃	#436	1	1	yes	characteristic
121	C₄×C₁₃	#446	1	1	yes	characteristic
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	—	13	2	no	—
125	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
126	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
127	C₂×C₄²	C₂×#261	1	2	yes	—
128	C₂×C₄²	C₂×#261	1	1	yes	characteristic
129	C₂×C₁₆	—	13	2	no	—
130	C₂×C₁₆	—	13	2	no	—
131	C₂×C₁₆	—	26	1	no	—
132	C₂×C₁₆	—	26	1	no	—
133	C₂×C₁₆	—	26	1	no	—
134	C₂×C₁₆	—	26	1	no	—
135	C₂×C₁₆	—	26	2	no	—
136	C₂×C₁₆	—	26	2	no	—
137	C₂×C₁₆	—	26	2	no	—
138	C₄⋊C₈	—	26	2	no	—
139	C₄×C₈	—	13	2	no	—
140	C₄×C₈	—	13	2	no	—
141	C₄×C₈	—	26	2	no	—
142	C₄×C₈	—	26	2	no	—
143	#95	#1057	1	2	yes	—
144	C₂×C₁₃	#1939	1	1	yes	characteristic
145	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
146	C₂×C₁₃	#2151	1	1	yes	characteristic
147	C₂×C₁₃	D₄×C₁₆	1	1	yes	characteristic
148	C₂×C₁₃	#2169	1	1	yes	characteristic
149	C₂×C₁₃	Q₈×C₁₆	1	1	yes	characteristic
150	C₂×C₁₃	#2183	1	1	yes	characteristic
151	C₂²×C₄	C₂×#1057	1	2	yes	—
152	C₂²×C₄	C₂×#1057	1	2	yes	—
153	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
154	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
155	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
156	C₂×C₈	—	13	1	no	—
157	C₂×C₈	—	13	1	no	—
158	C₂×C₈	—	13	1	no	—
159	C₂×C₈	—	13	1	no	—
160	C₂×C₈	—	13	1	no	—
161	C₂×C₈	—	13	1	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	26	1	no	—
164	C₂×C₈	—	26	1	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	26	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	#6092	1	4	yes	complemented
170	C₄²	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	#6118	1	2	yes	—
173	C₄²	C₄⋊#261	1	2	yes	—
174	C₁₆	—	26	2	no	—
175	C₁₆	—	26	2	no	—
176	C₁₆	—	52	2	no	—
177	C₁₃	#13159	1	1	yes	characteristic, complemented, Sylow
178	C₂³	C₂²×#1057	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₂×#6092	1	4	yes	—
187	C₂×C₄	C₂×#6092	1	2	yes	—
188	C₂×C₄	#88106	1	2	yes	—
189	C₂×C₄	C₄⋊#1057	1	2	yes	—
190	C₂×C₄	C₄⋊#1057	1	2	yes	—
191	C₂×C₄	C₂×#6118	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
193	C₂×C₄	#88238	1	1	yes	characteristic
194	C₂×C₄	D₄×#261	1	1	yes	characteristic
195	C₂×C₄	#88240	1	1	yes	characteristic
196	C₂×C₄	Q₈×#261	1	1	yes	characteristic
197	C₈	—	13	1	no	—
198	C₈	—	13	1	no	—
199	C₈	—	13	1	no	—
200	C₈	—	13	1	no	—
201	C₈	—	26	2	no	—
202	C₈	—	52	2	no	—
203	C₂²	C₂²×#6092	1	1	yes	characteristic
204	C₂²	C₂×#88106	1	1	yes	characteristic
205	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
206	C₂²	D₄×#1057	1	1	yes	characteristic
207	C₂²	Q₈×#1057	1	1	yes	characteristic
208	C₂²	#11719162	1	1	yes	characteristic
209	C₂²	#11719162	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	2	no	—
213	C₄	C₄⋊#6092	1	4	yes	complemented
214	C₄	#11719076	1	1	yes	characteristic
215	C₄	#11719079	1	1	yes	characteristic
216	C₄	#11719080	1	1	yes	characteristic
217	C₄	#11719083	1	1	yes	characteristic
218	C₂	#13130933	1	1	yes	characteristic
219	C₂	C₂×C₄⋊#6092	1	1	yes	characteristic
220	C₂	#13131171	1	1	yes	characteristic
221	C₂	D₄×#6092	1	1	yes	characteristic
222	C₂	#13131220	1	1	yes	characteristic
223	C₂	Q₈×#6092	1	1	yes	characteristic
224	C₂	#13131251	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₄×C₁₃×C₄⋊C₄	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₄×C₄⋊C₄	#261
hypocenter	C₁₃	#13159
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
46. C₁₃×C₄⋊C₄	C₁₆	semidirect
117. C₄×C₁₃	C₄⋊C₁₆	semidirect
169. C₄⋊C₄	#6092	semidirect
177. C₁₃	#13159	semidirect
213. C₄	C₄⋊#6092	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×C₄×#6092	2
#13129874	2
C₄⋊C₄×#1055	2
#13159	13