Subgroups — #15772697

Order 3328.

386 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic
3	#13112805	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₆	—	2	1	no	—
7	C₁₃×#431	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₄	1	1	yes	characteristic
10	C₂×#88056	—	2	1	no	—
11	C₂×#88052	—	2	1	no	—
12	C₄×Dic₅₂	—	2	1	no	—
13	C₄×Dic₅₂	—	2	1	no	—
14	C₄×Dic₅₂	C₄	1	2	yes	—
15	C₄×Dic₅₂	C₂²	1	2	yes	—
16	C₁₃×#335	—	2	1	no	—
17	C₁₃×#334	C₂²	1	2	yes	—
18	#11718053	C₂²	1	2	yes	—
19	C₄×#6090	—	2	1	no	—
20	C₂²×Dic₂₆	—	2	1	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	1	no	—
25	C₂×#6101	—	4	1	no	—
26	C₂×C₄×Dic₁₃	—	4	1	no	—
27	C₂×Dic₅₂	—	2	1	no	—
28	C₂×Dic₅₂	—	2	2	no	—
29	C₂×Dic₅₂	C₈	1	4	yes	complemented
30	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
31	C₄×Dic₂₆	—	2	2	no	—
32	C₄×Dic₂₆	—	2	2	no	—
33	C₄×Dic₂₆	—	4	1	no	—
34	C₄×Dic₂₆	—	4	1	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₂²⋊C₈	—	4	1	no	—
37	C₄×C₈×C₁₃	—	4	1	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	#88058	—	2	1	no	—
43	#88058	C₂×C₄	1	2	yes	—
44	#88056	—	2	2	no	—
45	#88056	—	2	2	no	—
46	#88052	—	2	4	no	—
47	C₂×#6090	—	2	2	no	—
48	#11600	—	13	1	no	maximal, Sylow
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	4	1	no	—
53	C₂²×Dic₁₃	—	4	1	no	—
54	C₂×Dic₂₆	—	2	4	no	—
55	C₂×Dic₂₆	—	2	2	no	—
56	C₂×Dic₂₆	—	4	1	no	—
57	C₂×Dic₂₆	—	4	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	D₈	1	1	yes	characteristic
68	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄⋊Dic₁₃	—	2	2	no	—
71	C₄⋊Dic₁₃	—	4	1	no	—
72	#6101	—	4	2	no	—
73	#6101	—	4	2	no	—
74	C₄×Dic₁₃	—	4	2	no	—
75	C₄×Dic₁₃	—	4	2	no	—
76	Dic₅₂	—	2	4	no	—
77	Dic₅₂	—	2	2	no	—
78	Dic₅₂	C₂×C₈	1	4	yes	—
79	#6090	—	2	4	no	—
80	C₂×C₄×Q₁₆	—	13	1	no	—
81	C₄×M₃₂	—	13	1	no	—
82	#1564	D₁₃	1	1	yes	characteristic
83	#1334	—	13	4	no	—
84	C₂³×C₁₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	4	1	no	—
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
88	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#104	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×Dic₁₃	—	4	4	no	—
96	C₂×Dic₁₃	—	4	2	no	—
97	C₂×Dic₁₃	—	4	2	no	—
98	C₂×Dic₁₃	—	8	2	no	—
99	Dic₂₆	—	2	4	no	—
100	Dic₂₆	—	4	4	no	—
101	Dic₂₆	—	4	2	no	—
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	—	8	2	no	—
105	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
106	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂²×Q₁₆	—	13	1	no	—
109	C₂×C₄×Q₈	—	26	1	no	—
110	C₂×M₃₂	—	26	1	no	—
111	C₄×Q₁₆	—	13	2	no	—
112	C₄×Q₁₆	—	13	2	no	—
113	C₄×Q₁₆	—	26	1	no	—
114	C₄×Q₁₆	—	26	1	no	—
115	#431	—	2	1	no	—
116	C₂×#107	—	13	1	no	—
117	C₂×#103	—	26	1	no	—
118	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
119	#346	—	13	2	no	—
120	C₄×C₁₆	—	26	1	no	—
121	#335	—	2	1	no	—
122	#334	C₂×D₁₃	1	2	yes	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	D₄⋊C₈	1	2	yes	—
126	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
131	C₄×C₁₃	#362	1	2	yes	—
132	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
134	Dic₁₃	—	4	4	no	—
135	Dic₁₃	—	8	2	no	—
136	C₂²×Q₈	—	26	1	no	—
137	C₂×Q₁₆	—	13	4	no	—
138	C₂×Q₁₆	—	13	2	no	—
139	C₂×Q₁₆	—	26	2	no	—
140	C₂×Q₁₆	—	26	1	no	—
141	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
142	C₂²×C₈	D₅₂	1	1	yes	characteristic
143	C₄×Q₈	—	26	2	no	—
144	C₄×Q₈	—	26	2	no	—
145	C₄×Q₈	—	52	1	no	—
146	C₄×Q₈	—	52	1	no	—
147	C₂×C₄⋊C₄	—	26	1	no	—
148	C₂×C₄⋊C₄	—	52	1	no	—
149	C₂×C₄²	—	52	1	no	—
150	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
151	M₃₂	—	26	4	no	—
152	C₂×C₁₆	—	26	2	no	—
153	#107	—	13	2	no	—
154	#107	—	26	1	no	—
155	C₄⋊C₈	—	2	2	no	—
156	#103	—	26	2	no	—
157	#103	—	26	2	no	—
158	C₂²⋊C₈	—	4	1	no	—
159	C₄×C₈	—	4	1	no	—
160	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
161	C₄×C₈	D₅₂	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
166	C₂×C₁₃	#2145	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
168	C₂×Q₈	—	26	2	no	—
169	C₂×Q₈	—	26	4	no	—
170	C₂×Q₈	—	52	2	no	—
171	C₂×Q₈	—	52	1	no	—
172	C₂²×C₄	—	52	1	no	—
173	C₂²×C₄	—	52	1	no	—
174	C₂²×C₄	#6103	1	1	yes	characteristic
175	C₂²×C₄	#6104	1	1	yes	characteristic
176	C₂²×C₄	#6106	1	1	yes	characteristic
177	Q₁₆	—	13	4	no	—
178	Q₁₆	—	26	2	no	—
179	Q₁₆	—	26	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	C₈×D₁₃	1	2	yes	—
183	C₂×C₈	#6094	1	2	yes	—
184	C₂×C₈	#6103	1	1	yes	characteristic
185	C₂×C₈	#6103	1	1	yes	characteristic
186	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
188	C₄⋊C₄	—	26	2	no	—
189	C₄⋊C₄	—	52	2	no	—
190	C₄⋊C₄	—	52	1	no	—
191	C₄⋊C₄	—	52	2	no	—
192	C₄²	—	52	2	no	—
193	C₄²	—	52	2	no	—
194	C₄²	#6103	1	1	yes	characteristic
195	C₄²	#6104	1	1	yes	characteristic
196	C₄²	#6106	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	C₁₆	—	26	4	no	—
199	C₁₃	#11600	1	1	yes	characteristic, complemented, Sylow
200	C₂³	#88049	1	1	yes	characteristic
201	Q₈	—	26	4	no	—
202	Q₈	—	52	2	no	—
203	Q₈	—	52	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	52	4	no	—
207	C₂×C₄	—	52	2	no	—
208	C₂×C₄	—	52	2	no	—
209	C₂×C₄	—	104	2	no	—
210	C₂×C₄	#88049	1	1	yes	characteristic
211	C₂×C₄	#88049	1	1	yes	characteristic
212	C₂×C₄	#88049	1	1	yes	characteristic
213	C₂×C₄	#88060	1	2	yes	—
214	C₂×C₄	#88065	1	2	yes	—
215	C₂×C₄	C₂×#6103	1	1	yes	characteristic
216	C₂×C₄	C₂×#6104	1	1	yes	characteristic
217	C₂×C₄	C₂×#6106	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	8	2	no	—
221	C₈	#88060	1	2	yes	—
222	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
223	C₈	C₂×#6094	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#11718075	1	2	yes	—
227	C₂²	C₂×#88049	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	52	4	no	—
232	C₄	—	104	2	no	—
233	C₄	#11718075	1	2	yes	—
234	C₄	#11718088	1	2	yes	—
235	C₄	C₂×#88060	1	1	yes	characteristic
236	C₄	C₂×#88065	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂×#11718075	1	1	yes	characteristic
239	C₂	#13116571	1	1	yes	characteristic
240	C₂	C₂×#11718088	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88049
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	#102
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#11600
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #13112805	C₂	semidirect
29. C₂×Dic₅₂	C₈	semidirect
199. C₁₃	#11600	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
C₄×#88052	2
#13112805	2
C₁₃×#1564	2
#11600	13