Subgroups — #16789613

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718260	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718258	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718231	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718257	C₂	1	1	yes	maximal, characteristic
5	#13129875	C₂	1	1	yes	maximal, characteristic
6	#13129874	C₂	1	1	yes	maximal, characteristic
7	#13114697	C₂	1	1	yes	maximal, characteristic
8	C₂²×#6092	C₂²	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	1	yes	characteristic
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	C₂×#88054	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₄×#6101	C₄	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718260	C₄	1	4	yes	complemented
18	#11718258	C₄	1	4	yes	complemented
19	#11718257	—	2	2	no	—
20	#11718231	—	2	2	no	—
21	C₁₃×#336	C₄	1	1	yes	characteristic
22	#11718142	C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#6101	C₈	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂×#6092	—	2	1	no	—
32	C₂×#6092	—	2	1	no	—
33	C₂×#6092	—	2	1	no	—
34	C₂×#6092	—	2	1	no	—
35	C₂×#6092	—	2	1	no	—
36	C₂×#6092	—	2	1	no	—
37	C₂×#6092	C₂×C₄	1	2	yes	—
38	C₂×#6092	C₂×C₄	1	2	yes	—
39	C₂×#6092	C₂×C₄	1	2	yes	—
40	C₂×#6092	C₂×C₄	1	2	yes	—
41	C₂×#6092	D₄	1	2	yes	—
42	C₂×#6092	Q₈	1	2	yes	—
43	#88071	C₈	1	2	yes	—
44	#88054	C₈	1	4	yes	—
45	C₈×Dic₁₃	—	2	1	no	—
46	C₈×Dic₁₃	—	2	1	no	—
47	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₄×#1055	—	2	1	no	—
50	C₄×#1055	—	2	1	no	—
51	#12663	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	C₄²	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	#6101	—	2	4	no	—
66	C₄×Dic₁₃	—	2	1	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	C₄²	1	2	yes	—
69	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	C₂×C₈	1	2	yes	—
73	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
74	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
75	C₂×#1055	C₂×D₄	1	1	yes	characteristic
76	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
77	C₂×#1055	#41	1	1	yes	characteristic
78	C₂×#1055	#41	1	1	yes	characteristic
79	#6092	—	2	2	no	—
80	#6092	—	2	2	no	—
81	#6092	—	2	2	no	—
82	#6092	C₄⋊C₄	1	4	yes	complemented
83	C₂×C₄⋊C₁₆	—	13	1	no	—
84	C₂×#346	—	13	1	no	—
85	C₂×#346	—	13	1	no	—
86	C₂×C₄×C₁₆	—	13	1	no	—
87	C₈×C₄⋊C₄	—	13	1	no	—
88	#1374	—	13	1	no	—
89	#1374	—	13	1	no	—
90	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
105	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
106	C₂×Dic₁₃	M₃₂	1	4	yes	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	#1055	—	2	2	no	—
110	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
111	#1055	#117	1	1	yes	characteristic
112	#1055	C₄×D₄	1	1	yes	characteristic
113	#1055	C₄×Q₈	1	1	yes	characteristic
114	C₂²×C₁₆	—	13	1	no	—
115	C₂²×C₁₆	—	13	1	no	—
116	C₂²×C₁₆	—	13	1	no	—
117	C₂²×C₁₆	—	13	1	no	—
118	C₂×C₄⋊C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₂×C₄×C₈	—	13	1	no	—
122	C₄×C₄⋊C₄	—	13	1	no	—
123	C₄⋊C₁₆	—	13	4	no	—
124	#346	—	13	4	no	—
125	#346	—	26	2	no	—
126	C₄×C₁₆	—	26	2	no	—
127	#336	—	13	1	no	—
128	#336	#261	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
133	C₂²×C₁₃	#504	1	2	yes	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₁₃	#405	1	1	yes	characteristic
137	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
138	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	2	2	no	—
141	Dic₁₃	—	2	4	no	—
142	C₂²×C₈	—	13	1	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂²×C₈	C₂×#261	1	1	yes	characteristic
145	C₂²×C₈	C₂×#261	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	13	2	no	—
147	C₂×C₄²	—	13	1	no	—
148	C₂×C₄²	—	13	1	no	—
149	C₂×C₄²	C₂×#261	1	1	yes	characteristic
150	C₂×C₁₆	—	13	2	no	—
151	C₂×C₁₆	—	13	2	no	—
152	C₂×C₁₆	—	13	2	no	—
153	C₂×C₁₆	—	13	2	no	—
154	C₂×C₁₆	—	13	2	no	—
155	C₂×C₁₆	—	13	2	no	—
156	C₂×C₁₆	—	26	1	no	—
157	C₂×C₁₆	—	26	1	no	—
158	C₂×C₁₆	—	26	1	no	—
159	C₂×C₁₆	—	26	1	no	—
160	C₂×C₁₆	—	26	1	no	—
161	C₂×C₁₆	—	26	1	no	—
162	C₄⋊C₈	—	13	4	no	—
163	C₄×C₈	—	13	2	no	—
164	C₄×C₈	—	13	2	no	—
165	C₄×C₈	—	26	1	no	—
166	C₄×C₈	—	26	1	no	—
167	C₄×C₈	—	26	1	no	—
168	C₄×C₈	—	26	1	no	—
169	#95	—	13	2	no	—
170	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
172	C₂×C₁₃	#2108	1	1	yes	characteristic
173	C₂×C₁₃	#2168	1	2	yes	—
174	C₂×C₁₃	#2183	1	2	yes	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	2	no	—
177	C₂²×C₄	—	13	1	no	—
178	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
179	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	13	1	no	—
185	C₂×C₈	—	13	1	no	—
186	C₂×C₈	—	13	1	no	—
187	C₂×C₈	—	13	1	no	—
188	C₂×C₈	—	13	1	no	—
189	C₂×C₈	—	13	1	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	26	1	no	—
192	C₂×C₈	—	26	1	no	—
193	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
194	C₂×C₈	C₄×#261	1	2	yes	—
195	C₄⋊C₄	—	26	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	13	2	no	—
199	C₄²	—	13	2	no	—
200	C₄²	—	26	1	no	—
201	C₄²	—	26	1	no	—
202	C₁₆	—	13	4	no	—
203	C₁₆	—	26	2	no	—
204	C₁₆	—	26	2	no	—
205	C₁₆	—	26	2	no	—
206	C₁₃	#12663	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	13	2	no	—
212	C₂×C₄	—	13	2	no	—
213	C₂×C₄	—	13	4	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	2	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
220	C₂×C₄	#88238	1	1	yes	characteristic
221	C₂×C₄	#88238	1	1	yes	characteristic
222	C₂×C₄	D₄×#261	1	1	yes	characteristic
223	C₂×C₄	Q₈×#261	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	13	1	no	—
227	C₈	—	13	1	no	—
228	C₈	—	13	1	no	—
229	C₈	—	13	1	no	—
230	C₈	—	26	2	no	—
231	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
232	C₂²	#11719063	1	1	yes	characteristic
233	C₂²	#11719074	1	1	yes	characteristic
234	C₂²	D₁₃⋊M₃₂	1	2	yes	—
235	C₂²	#11719096	1	2	yes	—
236	C₄	—	4	2	no	—
237	C₄	—	26	2	no	—
238	C₄	—	26	2	no	—
239	C₄	—	26	4	no	—
240	C₄	#11719062	1	1	yes	characteristic
241	C₄	#11719063	1	1	yes	characteristic
242	C₄	#11719074	1	1	yes	characteristic
243	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
244	C₂	#13130628	1	1	yes	characteristic
245	C₂	#13131043	1	1	yes	characteristic
246	C₂	#13131046	1	1	yes	characteristic
247	C₂	#13131217	1	2	yes	—
248	C₂	#13131253	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#336	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	#336	#261
hypocenter	C₁₃	#12663
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
17. #11718260	C₄	semidirect
18. #11718258	C₄	semidirect
82. #6092	C₄⋊C₄	semidirect
206. C₁₃	#12663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718260	2
C₂×#11718258	2
C₂×#11718231	2
C₂×#11718257	2
#13129875	2
#13129874	2
#13114697	2
#12663	13