Subgroups — #15777759

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	2	yes	maximal
2	#13112914	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	—	2	4	no	—
8	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×#88052	—	2	4	no	—
11	C₂×#88052	—	2	4	no	—
12	C₂×#88052	C₄	1	8	yes	complemented
13	C₂²×#6090	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₃	C₈	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₈	1	4	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	4	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₈	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	—	2	4	no	—
31	C₄×C₈×C₁₃	—	2	4	no	—
32	#88052	—	2	16	no	—
33	C₂×#6090	—	2	4	no	—
34	C₂×#6090	—	2	4	no	—
35	C₂×#6090	C₂×C₄	1	8	yes	complemented
36	#12544	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	—	2	8	no	—
40	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
47	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	—	2	8	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	C₄²	1	2	yes	—
53	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₁₃	—	2	4	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	#6090	—	2	8	no	—
66	C₂²×M₃₂	—	13	2	no	—
67	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
68	#1374	—	13	4	no	—
69	C₂³×C₁₃	—	2	4	no	—
70	C₂³×C₁₃	C₄×C₈	1	2	yes	—
71	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
72	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
73	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	16	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	#95	1	4	yes	—
78	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₁₃	#97	1	4	yes	—
80	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
82	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
84	C₂×C₄×C₁₃	M₃₂	1	8	yes	complemented
85	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	#95	1	4	yes	—
93	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
94	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
96	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
97	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
98	C₂×M₃₂	—	13	8	no	—
99	C₂×M₃₂	—	26	4	no	—
100	C₂×M₃₂	—	26	4	no	—
101	C₂²×C₁₆	—	13	4	no	—
102	C₂×C₄×C₈	—	2	4	no	—
103	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	4	no	—
107	C₂²×C₁₃	—	2	4	no	—
108	C₂²×C₁₃	#336	1	4	yes	—
109	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
113	C₄×C₁₃	—	2	8	no	—
114	C₄×C₁₃	#336	1	4	yes	—
115	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
116	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
117	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂³×C₄	#1055	1	2	yes	—
120	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	#1055	1	4	yes	—
124	C₂²×C₈	Dic₂₆	1	2	yes	—
125	C₂²×C₈	C₄×D₁₃	1	4	yes	—
126	C₂²×C₈	D₅₂	1	2	yes	—
127	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
128	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	#1055	1	2	yes	—
134	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
135	M₃₂	—	26	16	no	—
136	C₂×C₁₆	—	13	8	no	—
137	C₂×C₁₆	—	26	4	no	—
138	C₂×C₁₆	—	26	4	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	2	4	no	—
141	C₂×C₁₃	—	2	4	no	—
142	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
143	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
144	C₂×C₁₃	#2112	1	2	yes	—
145	C₂×C₁₃	#2150	1	2	yes	—
146	C₂⁴	C₂×#1055	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	C₂×#1055	1	4	yes	—
150	C₂²×C₄	C₂×#1055	1	2	yes	—
151	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
152	C₂²×C₄	#6099	1	2	yes	—
153	C₂²×C₄	#6099	1	2	yes	—
154	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
155	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂×#1055	1	4	yes	—
162	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
163	C₂×C₈	#6101	1	8	yes	—
164	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
165	C₂×C₈	#6103	1	8	yes	—
166	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
167	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
169	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
171	C₄²	—	2	4	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₁₆	—	26	8	no	—
175	C₁₃	#12544	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	C₄×#1055	1	2	yes	—
178	C₂³	C₄⋊#1055	1	2	yes	—
179	C₂³	C₂²⋊#1055	1	2	yes	—
180	C₂³	C₂²×#1055	1	1	yes	characteristic
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	C₄×#1055	1	2	yes	—
185	C₂×C₄	#88043	1	4	yes	—
186	C₂×C₄	C₄⋊#1055	1	4	yes	—
187	C₂×C₄	C₄⋊#1055	1	2	yes	—
188	C₂×C₄	#88052	1	8	yes	complemented
189	C₂×C₄	C₂²⋊#1055	1	4	yes	—
190	C₂×C₄	C₂²⋊#1055	1	2	yes	—
191	C₂×C₄	#88071	1	4	yes	—
192	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
193	C₂×C₄	C₂×#6099	1	1	yes	characteristic
194	C₂×C₄	C₂×#6099	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	#88071	1	4	yes	—
200	C₈	C₂×#6101	1	2	yes	—
201	C₈	C₂×#6103	1	2	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#11718116	1	4	yes	—
206	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
207	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
208	C₂²	C₂×#88052	1	4	yes	—
209	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	#11718116	1	4	yes	—
212	C₄	C₂×#88043	1	1	yes	characteristic
213	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
215	C₄	C₂×#88071	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₂×#11718116	1	1	yes	characteristic
218	C₂	C₄×#88052	1	2	yes	—
219	C₂	#13116447	1	2	yes	—
220	C₂	#13116592	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#12544
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13112914	C₂	semidirect
12. C₂×#88052	C₄	semidirect
35. C₂×#6090	C₂×C₄	semidirect
84. C₂×C₄×C₁₃	M₃₂	semidirect
175. C₁₃	#12544	semidirect
188. C₂×C₄	#88052	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
#13112914	2
C₂²×C₄×C₈×C₁₃	2
#12544	13