Subgroups — #15863532

Order 3328.

758 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13115010	C₂	1	2	yes	maximal, complemented
4	#13114976	C₂	1	2	yes	maximal, complemented
5	#13114876	C₂	1	4	yes	maximal
6	C₄×#88062	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₁₃×#432	—	2	2	no	—
11	C₁₃×#431	—	2	2	no	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₁₃×#407	C₂²	1	2	yes	—
14	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
15	#11718868	C₂²	1	2	yes	—
16	C₂×#88062	C₂²	1	8	yes	—
17	#11718587	—	2	2	no	—
18	#11718586	—	2	2	no	—
19	#11718585	C₂²	1	2	yes	—
20	C₄×#6099	C₄	1	8	yes	—
21	C₄×#6099	C₂²	1	2	yes	—
22	C₂×#88043	C₂²	1	1	yes	characteristic
23	C₂×#88043	C₂²	1	1	yes	characteristic
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718150	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×#6099	—	2	8	no	—
36	C₂×#6099	C₂×C₄	1	8	yes	—
37	C₂×#6099	C₂³	1	4	yes	—
38	C₂²×#1055	C₂³	1	2	yes	—
39	C₂²×#1055	C₂³	1	2	yes	—
40	C₁₃×C₄⋊C₈	—	2	4	no	—
41	C₁₃×C₂²⋊C₈	—	2	4	no	—
42	C₁₃×#97	—	2	2	no	—
43	C₄×C₈×C₁₃	—	2	2	no	—
44	C₂²⋊#1055	—	2	4	no	—
45	#88062	—	2	16	no	—
46	C₄⋊#1055	—	2	4	no	—
47	#88043	—	2	2	no	—
48	#88043	C₂×C₄	1	4	yes	—
49	#88043	C₂×C₄	1	4	yes	—
50	C₄×#1055	—	2	2	no	—
51	C₄×#1055	C₂×C₄	1	8	yes	—
52	#20827	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	4	4	no	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×M₁₆	—	2	8	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	2	yes	—
72	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
73	#6099	—	2	16	no	—
74	#6099	—	2	8	no	—
75	C₂×#1055	—	2	4	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	C₂²×C₄	1	4	yes	—
78	C₂×#1055	C₂²×C₄	1	4	yes	—
79	C₂×#1055	#41	1	8	yes	—
80	#2920	—	13	1	no	—
81	#2920	D₁₃	1	1	yes	characteristic
82	C₂×C₄×M₁₆	—	13	1	no	—
83	#1849	—	13	2	no	—
84	#1838	—	13	2	no	—
85	#1811	—	13	4	no	—
86	C₄×#108	—	13	4	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	4	2	no	—
89	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	8	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
105	C₈×C₁₃	—	4	4	no	—
106	#1055	—	2	4	no	—
107	#1055	—	4	4	no	—
108	#1055	#117	1	8	yes	—
109	C₂²×M₁₆	—	13	2	no	—
110	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
111	#432	—	2	2	no	—
112	#432	—	26	2	no	—
113	#431	—	2	2	no	—
114	#431	—	26	2	no	—
115	C₂×#108	—	13	8	no	—
116	#423	—	13	2	no	—
117	#423	C₂×D₁₃	1	2	yes	—
118	#407	—	13	2	no	—
119	#407	C₂×D₁₃	1	2	yes	—
120	C₄×M₁₆	—	13	8	no	—
121	C₄×M₁₆	—	13	2	no	—
122	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
123	C₂×#97	—	13	1	no	—
124	C₂×#97	—	13	1	no	—
125	C₂×C₄×C₈	—	13	2	no	—
126	#341	—	13	8	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	4	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	4	no	—
131	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₃	#533	1	2	yes	—
133	C₄×C₁₃	—	2	4	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	#430	1	4	yes	—
137	C₄×C₁₃	C₂×#117	1	2	yes	—
138	C₄×C₁₃	#532	1	2	yes	—
139	C₂³×C₄	Dic₂₆	1	2	yes	—
140	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
141	C₂×M₁₆	—	13	8	no	—
142	C₂×M₁₆	—	13	4	no	—
143	C₂×M₁₆	—	26	8	no	—
144	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
145	C₂²×C₈	—	13	2	no	—
146	C₂²×C₈	—	13	2	no	—
147	C₂×C₄²	Dic₂₆	1	2	yes	—
148	C₂×C₄²	C₄×D₁₃	1	4	yes	—
149	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
150	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
151	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
153	#108	—	26	16	no	—
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	26	4	no	—
156	C₂²⋊C₈	—	2	4	no	—
157	C₂²⋊C₈	—	26	4	no	—
158	#97	—	2	2	no	—
159	#97	—	13	4	no	—
160	#97	—	13	4	no	—
161	#97	—	26	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	13	8	no	—
164	C₄×C₈	—	26	2	no	—
165	C₂×C₁₃	—	2	2	no	—
166	C₂×C₁₃	—	4	2	no	—
167	C₂×C₁₃	#2285	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#430	1	2	yes	—
169	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	#6101	1	4	yes	—
173	C₂²×C₄	#6101	1	4	yes	—
174	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
176	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
179	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
180	M₁₆	—	2	8	no	—
181	M₁₆	—	26	16	no	—
182	M₁₆	—	26	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	13	4	no	—
185	C₂×C₈	—	13	8	no	—
186	C₂×C₈	—	13	4	no	—
187	C₂×C₈	—	26	4	no	—
188	C₂×C₈	—	26	4	no	—
189	C₄²	#6101	1	4	yes	—
190	C₄²	#6101	1	4	yes	—
191	C₄²	C₂×C₄×D₁₃	1	2	yes	—
192	C₄²	C₂×C₄×D₁₃	1	2	yes	—
193	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
195	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
196	C₁₃	#20827	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×#6101	1	2	yes	—
200	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂×#6101	1	2	yes	—
209	C₂×C₄	C₂×#6101	1	2	yes	—
210	C₂×C₄	C₂×#6101	1	4	yes	—
211	C₂×C₄	C₂×#6101	1	2	yes	—
212	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄	#41×D₁₃	1	4	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₈	—	4	4	no	—
216	C₈	—	13	8	no	—
217	C₈	—	26	4	no	—
218	C₈	—	52	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	4	4	no	—
223	C₂²	#11719222	1	2	yes	—
224	C₂²	C₂²×#6101	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	2	no	—
228	C₄	#11718785	1	4	yes	—
229	C₄	D₁₃×#117	1	2	yes	—
230	C₄	#11719537	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	#13117239	1	1	yes	characteristic
234	C₂	C₂×#11718785	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2920	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2920	D₁₃
hypocenter	C₁₃	#20827
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #13115010	C₂	semidirect
4. #13114976	C₂	semidirect
6. C₄×#88062	C₂	semidirect
196. C₁₃	#20827	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
C₂×C₄×#6099	2
#13115010	2
#13114976	2
#13114876	2
C₄×#88062	2
C₁₃×#2920	2
#20827	13