Subgroups — #15778243

Order 3328.

358 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	M₃₂×Dic₁₃	C₂	1	2	yes	maximal
3	#13114522	C₂	1	1	yes	maximal, characteristic
4	#13112917	C₂	1	2	yes	maximal
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₄	1	1	yes	characteristic
9	C₂×#88055	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	2	yes	—
13	#11718593	C₄	1	1	yes	characteristic
14	C₂×#88043	C₄	1	1	yes	characteristic
15	C₁₃×#364	C₄	1	4	yes	complemented
16	#11718094	—	2	2	no	—
17	C₁₆×Dic₁₃	—	2	2	no	—
18	#11718055	C₄	1	4	yes	complemented
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₁₃×M₃₂	—	2	2	no	—
27	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	#88071	—	2	2	no	—
32	#88055	C₈	1	4	yes	—
33	#88055	C₈	1	4	yes	—
34	C₈×Dic₁₃	—	2	1	no	—
35	C₈×Dic₁₃	—	2	1	no	—
36	C₈×Dic₁₃	D₄	1	2	yes	—
37	C₈×Dic₁₃	Q₈	1	2	yes	—
38	#88052	—	2	2	no	—
39	#88052	C₂×C₄	1	4	yes	—
40	C₂×#6090	—	2	2	no	—
41	#88043	—	2	2	no	—
42	#12656	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	2	1	no	—
46	C₂²×Dic₁₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₃	C₄²	1	2	yes	—
48	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	C₄²	1	2	yes	—
58	C₄×Dic₁₃	—	2	1	no	—
59	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
60	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
61	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	C₄⋊C₄	1	2	yes	—
65	C₂×#1055	C₂×C₈	1	2	yes	—
66	C₂×#1055	C₂×C₈	1	2	yes	—
67	C₂×#1055	M₁₆	1	2	yes	—
68	C₁₃×C₁₆	—	2	4	no	—
69	#6090	—	2	4	no	—
70	C₂×#364	—	13	1	no	—
71	C₂×#364	D₁₃	1	1	yes	characteristic
72	C₄×M₃₂	—	13	2	no	—
73	#1726	—	13	1	no	—
74	#1376	—	13	2	no	—
75	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	4	2	no	—
86	C₂×Dic₁₃	—	4	2	no	—
87	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
88	C₈×C₁₃	C₄×C₈	1	4	yes	—
89	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
90	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
91	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
92	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
93	#1055	—	2	2	no	—
94	#1055	C₄⋊C₈	1	4	yes	—
95	C₂×M₃₂	—	13	2	no	—
96	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
97	C₂×#97	—	13	1	no	—
98	C₂×#97	—	13	1	no	—
99	C₂×#97	—	13	1	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
103	#376	—	13	1	no	—
104	#364	—	13	4	no	—
105	#364	Dic₁₃	1	4	yes	complemented
106	#346	—	26	2	no	—
107	C₄×C₁₆	—	26	2	no	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃	#431	1	1	yes	characteristic
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
115	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₄×C₁₃	#431	1	1	yes	characteristic
117	Dic₁₃	—	2	4	no	—
118	Dic₁₃	—	4	2	no	—
119	C₂²×C₈	—	13	1	no	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₄²	—	13	1	no	—
124	C₂×C₄²	—	13	1	no	—
125	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
126	M₃₂	—	2	2	no	—
127	M₃₂	—	13	4	no	—
128	M₃₂	—	26	2	no	—
129	M₃₂	C₂×Dic₁₃	1	4	yes	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	26	2	no	—
132	#97	—	13	4	no	—
133	#97	—	13	4	no	—
134	#97	—	26	2	no	—
135	C₄×C₈	—	13	2	no	—
136	C₄×C₈	—	13	2	no	—
137	C₄×C₈	—	26	1	no	—
138	C₄×C₈	—	26	1	no	—
139	C₄×C₈	C₄×D₁₃	1	2	yes	—
140	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
141	#95	—	26	2	no	—
142	C₂×C₁₃	—	2	1	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₁₃	#2153	1	2	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	13	2	no	—
148	C₂²×C₄	—	26	1	no	—
149	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	13	2	no	—
153	C₂×C₈	—	13	2	no	—
154	C₂×C₈	—	26	1	no	—
155	C₂×C₈	—	26	1	no	—
156	C₂×C₈	C₈×D₁₃	1	2	yes	—
157	C₂×C₈	C₈×D₁₃	1	2	yes	—
158	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
161	C₂×C₈	#6129	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
163	C₂×C₈	#6131	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
165	C₄²	—	2	1	no	—
166	C₄²	—	13	2	no	—
167	C₄²	—	13	2	no	—
168	C₄²	—	13	2	no	—
169	C₄²	—	26	1	no	—
170	C₄²	C₄×Dic₁₃	1	2	yes	—
171	C₁₆	—	2	4	no	—
172	C₁₆	—	26	4	no	—
173	C₁₃	#12656	1	1	yes	characteristic, complemented, Sylow
174	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	13	4	no	—
178	C₂×C₄	—	26	2	no	—
179	C₂×C₄	—	52	2	no	—
180	C₂×C₄	—	52	2	no	—
181	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
182	C₂×C₄	#88146	1	1	yes	characteristic
183	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
187	C₈	—	13	4	no	—
188	C₈	—	26	2	no	—
189	C₈	C₈×Dic₁₃	1	4	yes	—
190	C₈	#88141	1	1	yes	characteristic
191	C₈	#88147	1	1	yes	characteristic
192	C₈	D₄×Dic₁₃	1	1	yes	characteristic
193	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
196	C₂²	#11718494	1	1	yes	characteristic
197	C₂²	M₁₆×Dic₁₃	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	26	4	no	—
200	C₄	—	52	2	no	—
201	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
202	C₄	#11718494	1	1	yes	characteristic
203	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
204	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄⋊C₈×Dic₁₃	1	1	yes	characteristic
208	C₂	#13116600	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	#12656
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₁₃×#364	C₄	semidirect
18. #11718055	C₄	semidirect
105. #364	Dic₁₃	semidirect
173. C₁₃	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718055	2
M₃₂×Dic₁₃	2
#13114522	2
#13112917	2
C₂×C₁₃×#364	2
#12656	13