Subgroups — #15779386

Order 3328.

438 conjugacy classes of subgroups.

The subgroup lattice has 234 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	1	yes	maximal, characteristic
2	C₂×C₄⋊#6090	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
4	#13112914	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
9	C₄×C₁₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
10	C₂×#88052	—	2	4	no	—
11	C₂×#88052	—	2	2	no	—
12	C₂²×#6090	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	1	yes	characteristic
14	C₂²×#6090	C₂²	1	2	yes	—
15	C₁₃×#336	C₄	1	2	yes	—
16	C₂²⋊#6090	—	2	2	no	—
17	C₄⋊#6090	—	2	4	no	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
28	C₁₃×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₁₃×#95	C₈	1	1	yes	characteristic
32	C₁₃×#95	C₈	1	1	yes	characteristic
33	#88052	—	2	8	no	—
34	C₂×#6090	—	2	2	no	—
35	C₂×#6090	—	2	2	no	—
36	C₂×#6090	—	2	2	no	—
37	C₂×#6090	—	2	1	no	—
38	C₂×#6090	—	2	1	no	—
39	C₂×#6090	—	2	2	no	—
40	C₂×#6090	D₄	1	4	yes	—
41	#12913	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₃	#41	1	2	yes	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	#6090	—	2	4	no	—
67	#6090	—	4	1	no	—
68	#6090	—	4	1	no	—
69	C₂²×M₃₂	—	13	1	no	—
70	C₂×C₄⋊C₁₆	—	13	2	no	—
71	C₂×C₂²⋊C₁₆	—	13	1	no	—
72	C₈×C₂²⋊C₄	D₁₃	1	1	yes	characteristic
73	#1374	—	13	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#131	1	2	yes	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	4	2	no	—
97	C₈×C₁₃	#120	1	1	yes	characteristic
98	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
99	C₈×C₁₃	#123	1	1	yes	characteristic
100	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
101	C₂×M₃₂	—	26	4	no	—
102	C₂×M₃₂	—	26	2	no	—
103	C₂²×C₁₆	—	13	2	no	—
104	C₂²×C₁₆	—	13	1	no	—
105	C₂²×C₁₆	—	13	1	no	—
106	C₂×C₂²⋊C₈	Dic₁₃	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
108	C₄×C₂²⋊C₄	Dic₁₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	26	4	no	—
110	C₂²⋊C₁₆	—	26	2	no	—
111	#336	Dic₁₃	1	2	yes	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₈×D₄	1	2	yes	—
119	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
120	C₂²×C₁₃	#504	1	1	yes	characteristic
121	C₂²×C₁₃	#504	1	1	yes	characteristic
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	4	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	#385	1	1	yes	characteristic
126	C₄×C₁₃	#424	1	1	yes	characteristic
127	C₄×C₁₃	#435	1	2	yes	—
128	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	Dic₂₆	1	2	yes	—
132	C₂²×C₈	D₅₂	1	2	yes	—
133	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
134	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	#1055	1	2	yes	—
137	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
138	M₃₂	—	26	8	no	—
139	C₂×C₁₆	—	13	4	no	—
140	C₂×C₁₆	—	26	2	no	—
141	C₂×C₁₆	—	26	2	no	—
142	C₂×C₁₆	—	26	1	no	—
143	C₂×C₁₆	—	26	1	no	—
144	C₂×C₁₆	—	26	2	no	—
145	C₂×C₁₆	—	26	2	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	#95	#1055	1	1	yes	characteristic
150	#95	#1055	1	1	yes	characteristic
151	C₂×C₁₃	—	4	2	no	—
152	C₂×C₁₃	#1850	1	1	yes	characteristic
153	C₂×C₁₃	#2150	1	1	yes	characteristic
154	C₂×C₁₃	#2151	1	1	yes	characteristic
155	C₂×C₁₃	#2168	1	2	yes	—
156	C₂×C₁₃	#2169	1	2	yes	—
157	C₂⁴	C₂×#1055	1	1	yes	characteristic
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	C₂×#1055	1	2	yes	—
161	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
162	C₂²×C₄	C₂×#1055	1	2	yes	—
163	C₂²×C₄	#6099	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
174	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
175	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
176	C₂×C₈	D₄×D₁₃	1	2	yes	—
177	C₂×C₈	#6129	1	2	yes	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₁₆	—	26	4	no	—
182	C₁₆	—	52	1	no	—
183	C₁₆	—	52	1	no	—
184	C₁₃	#12913	1	1	yes	characteristic, complemented, Sylow
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	C₄⋊#1055	1	2	yes	—
190	C₂³	C₂²×#1055	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₄⋊#1055	1	2	yes	—
201	C₂×C₄	#88052	1	4	yes	complemented
202	C₂×C₄	C₂×#6099	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
205	C₂×C₄	#88202	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	4	2	no	—
208	C₈	#88132	1	2	yes	—
209	C₈	#88136	1	1	yes	characteristic
210	C₈	#88140	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
217	C₂²	D₄×#1055	1	2	yes	—
218	C₂²	C₂×#88052	1	2	yes	—
219	C₂²	#11718801	1	1	yes	characteristic
220	C₂²	#11718801	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	#11718611	1	1	yes	characteristic
225	C₄	#11718660	1	1	yes	characteristic
226	C₄	#11718672	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#13115016	1	1	yes	characteristic
229	C₂	#13116592	1	1	yes	characteristic
230	C₂	#13116595	1	1	yes	characteristic
231	C₂	#13116662	1	2	yes	—
232	C₂	#13116668	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator, socle	C₂³×C₁₃	C₂²×C₈
Fitting	C₈×C₁₃×C₂²⋊C₄	C₂
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₈×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	#12913
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#6090	C₂	semidirect
4. #13112914	C₂	semidirect
90. C₂×C₄×C₁₃	M₃₂	semidirect
184. C₁₃	#12913	semidirect
201. 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
C₂×C₄⋊#6090	2
C₂×C₂²⋊#6090	2
#13112914	2
C₈×C₁₃×C₂²⋊C₄	2
#12913	13