Subgroups — #15778242

Order 3328.

358 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718103	C₂	1	2	yes	maximal
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic
3	#13114522	C₂	1	1	yes	maximal, characteristic
4	#13112917	C₂	1	2	yes	maximal
5	C₄×C₁₃×M₃₂	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×#346	—	2	1	no	—
16	C₄×C₁₃×C₁₆	—	2	1	no	—
17	#11718103	C₄	1	8	yes	complemented
18	#11718053	—	2	1	no	—
19	C₄×#6090	—	2	1	no	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₁₃×M₃₂	—	2	2	no	—
28	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₁₆	—	2	2	no	—
30	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	#88071	—	2	2	no	—
35	#88055	—	2	2	no	—
36	#88055	C₈	1	4	yes	—
37	C₈×Dic₁₃	—	2	2	no	—
38	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
39	#88052	—	2	2	no	—
40	#88052	C₂×C₄	1	4	yes	—
41	C₂×#6090	—	2	2	no	—
42	#88043	C₈	1	4	yes	—
43	#12656	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	1	no	—
47	C₂²×Dic₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	2	yes	—
56	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₁₃	—	2	1	no	—
60	C₄×Dic₁₃	—	2	1	no	—
61	C₄×Dic₁₃	C₄²	1	2	yes	—
62	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	C₄²	1	2	yes	—
66	C₂×#1055	C₂×C₈	1	2	yes	—
67	C₂×#1055	C₂×C₈	1	2	yes	—
68	C₂×#1055	C₂×C₈	1	2	yes	—
69	C₁₃×C₁₆	—	2	4	no	—
70	#6090	—	2	4	no	—
71	C₂×#364	—	13	2	no	—
72	C₄×M₃₂	—	13	1	no	—
73	C₄×M₃₂	D₁₃	1	1	yes	characteristic
74	#1726	—	13	1	no	—
75	#1376	—	13	2	no	—
76	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
91	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
92	C₈×C₁₃	C₄×D₄	1	2	yes	—
93	C₈×C₁₃	C₄×Q₈	1	2	yes	—
94	#1055	—	2	2	no	—
95	#1055	C₄×C₈	1	4	yes	—
96	C₂×M₃₂	—	13	2	no	—
97	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
98	C₂×#97	—	13	1	no	—
99	C₂×#97	—	13	1	no	—
100	C₂×#97	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
104	#376	—	13	1	no	—
105	#364	—	13	8	no	—
106	#346	—	2	1	no	—
107	#346	—	26	1	no	—
108	C₄×C₁₆	—	2	1	no	—
109	C₄×C₁₆	—	26	1	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂²×C₁₃	#431	1	1	yes	characteristic
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₄×C₁₃	#431	1	1	yes	characteristic
120	Dic₁₃	—	4	2	no	—
121	Dic₁₃	—	4	2	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	13	4	no	—
131	M₃₂	—	26	2	no	—
132	M₃₂	C₄×D₁₃	1	4	yes	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	26	2	no	—
135	#97	—	13	4	no	—
136	#97	—	13	4	no	—
137	#97	—	26	2	no	—
138	C₄×C₈	—	13	4	no	—
139	C₄×C₈	—	26	2	no	—
140	C₄×C₈	Dic₂₆	1	1	yes	characteristic
141	C₄×C₈	Dic₂₆	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
144	#95	—	26	2	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	—	2	1	no	—
147	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₁₃	#2153	1	2	yes	—
149	C₂²×C₄	—	26	1	no	—
150	C₂²×C₄	—	26	1	no	—
151	C₂²×C₄	#6094	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
153	C₂×C₈	—	13	2	no	—
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	26	1	no	—
158	C₂×C₈	—	26	1	no	—
159	C₂×C₈	C₈×D₁₃	1	2	yes	—
160	C₂×C₈	#6094	1	2	yes	—
161	C₂×C₈	#6101	1	1	yes	characteristic
162	C₂×C₈	#6101	1	1	yes	characteristic
163	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
165	C₂×C₈	#6127	1	2	yes	—
166	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₄²	—	13	2	no	—
168	C₄²	—	13	2	no	—
169	C₄²	—	26	1	no	—
170	C₄²	—	26	1	no	—
171	C₄²	C₈×D₁₃	1	2	yes	—
172	C₄²	#6101	1	1	yes	characteristic
173	C₄²	#6101	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₁₆	—	26	4	no	—
176	C₁₃	#12656	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×#6094	1	1	yes	characteristic
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	26	2	no	—
180	C₂×C₄	—	26	2	no	—
181	C₂×C₄	—	52	2	no	—
182	C₂×C₄	—	52	2	no	—
183	C₂×C₄	#88054	1	2	yes	—
184	C₂×C₄	#88054	1	2	yes	—
185	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
186	C₂×C₄	#88126	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂×#6094	1	1	yes	characteristic
190	C₂×C₄	C₂×#6101	1	1	yes	characteristic
191	C₈	—	13	4	no	—
192	C₈	—	26	2	no	—
193	C₈	#88054	1	2	yes	—
194	C₈	#88054	1	2	yes	—
195	C₈	C₄×Dic₂₆	1	2	yes	—
196	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	#11718346	1	1	yes	characteristic
199	C₂²	C₄×#6094	1	1	yes	characteristic
200	C₂²	C₂×#88054	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	52	2	no	—
204	C₄	—	52	2	no	—
205	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
206	C₄	#11718346	1	1	yes	characteristic
207	C₄	C₄×#6101	1	1	yes	characteristic
208	C₄	C₂×#88054	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄×#88054	1	1	yes	characteristic
212	C₂	#13116601	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12656
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
17. #11718103	C₄	semidirect
176. 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×#11718103	2
C₄×#88052	2
#13114522	2
#13112917	2
C₄×C₁₃×M₃₂	2
#12656	13