Subgroups — #15861981

Order 3328.

914 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13114706	C₂	1	3	yes	maximal
4	#13114703	C₂	1	3	yes	maximal
5	#13114691	C₂	1	3	yes	maximal
6	#13114581	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	C₄	1	2	yes	—
10	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	C₂×#88055	C₂²	1	6	yes	—
14	Q₈×#1055	C₄	1	8	yes	complemented
15	C₄⋊C₄×Dic₁₃	C₄	1	6	yes	—
16	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
17	C₄²×Dic₁₃	C₂²	1	3	yes	—
18	C₂×C₄⋊#1055	C₂²	1	3	yes	—
19	C₂×C₄×#1055	C₂²	1	3	yes	—
20	#11718142	C₂²	1	3	yes	—
21	#11718142	C₂²	1	3	yes	—
22	#11718142	C₂²	1	3	yes	—
23	#11718142	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
27	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
31	Q₈×Dic₁₃	—	2	8	no	—
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	6	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
36	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
37	C₂²×#1055	C₂³	1	3	yes	—
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
41	#88071	C₂×C₄	1	6	yes	—
42	#88055	—	4	6	no	—
43	C₄⋊#1055	C₂×C₄	1	12	yes	—
44	C₄×#1055	C₂×C₄	1	12	yes	—
45	#20572	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
47	C₂×Q₈×C₁₃	M₁₆	1	8	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
52	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₁₃	—	2	3	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	3	no	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
59	C₁₃×C₄⋊C₄	C₄²	1	12	yes	—
60	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
61	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
62	C₄⋊Dic₁₃	—	2	12	no	—
63	C₄×Dic₁₃	—	2	12	no	—
64	C₄×Dic₁₃	—	2	3	no	—
65	C₄×Dic₁₃	—	2	1	no	—
66	C₄×Dic₁₃	—	2	3	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	—	2	12	no	—
69	C₂×#1055	—	2	3	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	2	3	no	—
73	C₂×#1055	C₂²×C₄	1	2	yes	—
74	C₂×#1055	C₂²×C₄	1	6	yes	—
75	C₂×C₈×Q₈	—	13	1	no	—
76	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
77	C₄²×Q₈	—	13	1	no	—
78	#1770	—	13	3	no	—
79	#1770	—	13	3	no	—
80	#1767	—	13	3	no	—
81	#1744	—	13	3	no	—
82	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	Q₈×C₁₃	#97	1	16	yes	complemented
84	C₂×C₄×C₁₃	—	2	6	no	—
85	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
86	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
87	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
93	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	—	2	6	no	—
97	C₂×Dic₁₃	—	2	6	no	—
98	C₂×Dic₁₃	—	2	6	no	—
99	C₂×Dic₁₃	#131	1	4	yes	—
100	C₈×C₁₃	—	2	2	no	—
101	C₈×C₁₃	—	2	6	no	—
102	#1055	—	2	2	no	—
103	#1055	—	2	6	no	—
104	C₂×C₄×Q₈	—	13	2	no	—
105	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
106	C₈×Q₈	—	13	8	no	—
107	C₈×Q₈	Dic₁₃	1	8	yes	complemented
108	C₂×C₄⋊C₈	—	13	3	no	—
109	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
110	C₂×#97	—	13	6	no	—
111	C₂×C₄×C₈	—	13	3	no	—
112	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
113	C₄×C₄⋊C₄	—	13	6	no	—
114	C₄×C₄⋊C₄	—	13	3	no	—
115	C₄³	—	13	3	no	—
116	#336	—	13	1	no	—
117	#336	—	13	3	no	—
118	#336	—	13	3	no	—
119	#336	—	13	3	no	—
120	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
121	C₂²×C₁₃	#519	1	1	yes	characteristic
122	C₂²×C₁₃	#520	1	1	yes	characteristic
123	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
124	C₂²×C₁₃	C₂×#131	1	2	yes	—
125	C₄×C₁₃	—	2	6	no	—
126	C₄×C₁₃	C₂×#97	1	12	yes	—
127	C₄×C₁₃	#519	1	1	yes	characteristic
128	C₄×C₁₃	#519	1	1	yes	characteristic
129	C₄×C₁₃	#520	1	1	yes	characteristic
130	C₄×C₁₃	#520	1	1	yes	characteristic
131	Dic₁₃	—	2	4	no	—
132	Dic₁₃	—	4	6	no	—
133	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	13	3	no	—
136	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
138	C₄×Q₈	—	26	8	no	—
139	C₄×Q₈	C₄×D₁₃	1	4	yes	—
140	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
141	C₂×C₄⋊C₄	—	13	6	no	—
142	C₂×C₄⋊C₄	C₄×D₁₃	1	3	yes	—
143	C₂×C₄²	—	13	6	no	—
144	C₂×C₄²	—	13	6	no	—
145	C₂×C₄²	—	13	1	no	—
146	C₂×C₄²	—	13	3	no	—
147	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
148	C₄⋊C₈	—	13	12	no	—
149	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
150	#97	—	52	6	no	—
151	C₄×C₈	—	13	12	no	—
152	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
153	#95	—	13	6	no	—
154	C₂×C₁₃	#2276	1	1	yes	characteristic
155	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
156	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
157	C₂×C₁₃	#2984	1	2	yes	—
158	C₂×C₁₃	#2991	1	2	yes	—
159	C₂×Q₈	#6094	1	8	yes	—
160	C₂×Q₈	C₄×Dic₁₃	1	4	yes	—
161	C₂²×C₄	—	13	2	no	—
162	C₂²×C₄	—	13	6	no	—
163	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
164	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
165	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	3	no	—
169	C₂×C₈	—	2	3	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	13	6	no	—
172	C₂×C₈	—	26	1	no	—
173	C₂×C₈	—	26	3	no	—
174	C₂×C₈	—	26	3	no	—
175	C₂×C₈	—	26	1	no	—
176	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
177	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
178	C₄⋊C₄	—	26	12	no	—
179	C₄⋊C₄	C₄×Dic₁₃	1	12	yes	—
180	C₄²	—	26	12	no	—
181	C₄²	—	26	12	no	—
182	C₄²	—	26	1	no	—
183	C₄²	—	26	3	no	—
184	C₄²	—	26	3	no	—
185	C₄²	—	26	1	no	—
186	C₄²	C₂×C₄×D₁₃	1	6	yes	—
187	C₄²	C₂²×Dic₁₃	1	6	yes	—
188	C₁₃	#20572	1	1	yes	characteristic, complemented, Sylow
189	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
190	Q₈	#88055	1	16	yes	complemented
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	13	4	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	26	2	no	—
195	C₂×C₄	—	26	6	no	—
196	C₂×C₄	—	26	6	no	—
197	C₂×C₄	—	26	6	no	—
198	C₂×C₄	C₂×#6094	1	12	yes	—
199	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
200	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
201	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
202	C₂×C₄	#88254	1	1	yes	characteristic
203	C₂×C₄	#88254	1	1	yes	characteristic
204	C₂×C₄	#88256	1	1	yes	characteristic
205	C₂×C₄	#88257	1	1	yes	characteristic
206	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	6	no	—
209	C₈	—	26	2	no	—
210	C₈	—	26	6	no	—
211	C₂²	#11719277	1	1	yes	characteristic
212	C₂²	#11719279	1	1	yes	characteristic
213	C₂²	C₂²×#6094	1	2	yes	—
214	C₂²	D₁₃×#131	1	2	yes	—
215	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
216	C₄	—	2	6	no	—
217	C₄	—	26	4	no	—
218	C₄	—	52	6	no	—
219	C₄	C₂×#88055	1	12	yes	—
220	C₄	#11719277	1	1	yes	characteristic
221	C₄	#11719279	1	1	yes	characteristic
222	C₄	#11719517	1	1	yes	characteristic
223	C₄	#11719532	1	1	yes	characteristic
224	C₂	#13117201	1	1	yes	characteristic
225	C₂	C₂²×#88055	1	1	yes	characteristic
226	C₂	#131×Dic₁₃	1	1	yes	characteristic
227	C₂	#13123924	1	2	yes	—
228	C₂	#13123967	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#20572
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₁₃	C₄	semidirect
14. Q₈×#1055	C₄	semidirect
83. Q₈×C₁₃	#97	semidirect
107. C₈×Q₈	Dic₁₃	semidirect
188. C₁₃	#20572	semidirect
190. Q₈	#88055	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
C₄×Q₈×Dic₁₃	2
#13114706	2
#13114703	2
#13114691	2
#13114581	2
C₂×C₈×Q₈×C₁₃	2
#20572	13