Subgroups — #13123551

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719537	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88161	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718779	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718771	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88055	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	4	yes	maximal, complemented
9	#11718389	C₂	1	4	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂×#6127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#88202	—	2	4	no	—
15	#88201	C₄	1	1	yes	characteristic
16	#88198	C₄	1	1	yes	characteristic
17	#88194	C₄	1	1	yes	characteristic
18	#88193	C₄	1	4	yes	—
19	#88190	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	2	yes	—
24	C₂²×#1055	C₂²	1	2	yes	—
25	C₂²×#1055	C₂²	1	1	yes	characteristic
26	#88161	—	2	4	no	—
27	C₂×#6094	C₂²	1	2	yes	—
28	C₂×C₈×D₁₃	C₂²	1	2	yes	—
29	C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
30	#88060	C₂²	1	4	yes	—
31	#88055	C₂²	1	2	yes	—
32	#88055	C₂²	1	2	yes	—
33	#88054	C₂²	1	4	yes	—
34	C₁₃×#41	—	2	2	no	—
35	C₁₃×#41	—	2	2	no	—
36	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
43	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
44	#6127	—	2	2	no	—
45	#6127	—	2	2	no	—
46	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂³	1	2	yes	—
48	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
49	C₁₃×M₁₆	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂×C₈×C₁₃	C₂³	1	2	yes	—
53	C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
54	#6103	C₂×C₄	1	4	yes	—
55	#6101	C₂×C₄	1	4	yes	—
56	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₃	C₂³	1	2	yes	—
59	#6099	—	2	2	no	—
60	#6099	—	2	4	no	—
61	C₂×#1055	—	2	4	no	—
62	C₂×#1055	D₄	1	4	yes	complemented
63	C₂×#1055	C₂³	1	2	yes	—
64	C₂×#1055	C₂³	1	1	yes	characteristic
65	C₂×#1055	C₂³	1	1	yes	characteristic
66	#6094	—	2	4	no	—
67	C₈×D₁₃	—	2	4	no	—
68	#2932	—	13	1	no	maximal, Sylow
69	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
72	Q₈×C₁₃	—	2	2	no	—
73	D₄×C₁₃	—	2	4	no	—
74	D₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	#41	1	2	yes	—
81	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₁₃	—	2	4	no	—
83	C₂²⋊D₁₃	—	4	4	no	—
84	C₂×Dic₁₃	—	4	2	no	—
85	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
88	D₅₂	—	2	2	no	—
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	2	2	no	—
91	Dic₂₆	—	2	2	no	—
92	C₈×C₁₃	—	2	2	no	—
93	C₈×C₁₃	—	2	2	no	—
94	#1055	—	2	2	no	—
95	#1055	C₂×D₄	1	2	yes	—
96	#1055	C₂×D₄	1	2	yes	—
97	C₂×#131	—	13	1	no	—
98	C₂×#131	—	13	1	no	—
99	#532	—	13	1	no	—
100	#435	—	13	4	no	—
101	#435	—	13	4	no	—
102	#423	—	13	1	no	—
103	#408	—	13	1	no	—
104	#408	D₁₃	1	1	yes	characteristic, complemented
105	C₂×#97	—	13	1	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	—	2	4	no	—
109	C₂²×C₁₃	C₄×D₄	1	2	yes	—
110	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×D₁₃	—	2	2	no	—
112	C₂×D₁₃	—	2	4	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	Dic₁₃	—	2	2	no	—
118	Dic₁₃	—	2	2	no	—
119	Dic₁₃	—	2	2	no	—
120	C₂×#41	—	13	1	no	—
121	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
122	#131	—	26	4	no	—
123	#131	—	26	4	no	—
124	C₂×M₁₆	—	13	1	no	—
125	C₂×M₁₆	—	13	2	no	—
126	C₂×M₁₆	—	13	2	no	—
127	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	13	2	no	—
131	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
132	C₄⋊Q₈	—	13	1	no	—
133	#127	—	13	1	no	—
134	#124	—	13	1	no	—
135	#124	—	13	1	no	—
136	#121	—	13	4	no	—
137	C₄×D₄	—	13	4	no	—
138	C₂×C₄²	—	13	1	no	—
139	C₄⋊C₈	—	13	4	no	—
140	C₂²⋊C₈	—	13	4	no	—
141	C₂²⋊C₈	C₂×D₁₃	1	4	yes	—
142	#97	—	13	2	no	—
143	#97	—	13	2	no	—
144	C₂×C₁₃	—	2	2	no	—
145	C₂×C₁₃	—	4	2	no	—
146	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₁₃	#568	1	2	yes	—
148	D₁₃	—	4	2	no	—
149	#41	—	2	2	no	—
150	#41	—	2	2	no	—
151	#41	—	26	2	no	—
152	#41	—	26	2	no	—
153	C₂×Q₈	—	13	1	no	—
154	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
155	C₂×D₄	—	13	2	no	—
156	C₂×D₄	—	13	4	no	—
157	C₂×D₄	—	13	1	no	—
158	C₂×D₄	C₄×D₁₃	1	2	yes	—
159	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
160	C₂²×C₄	—	13	2	no	—
161	C₂²×C₄	—	13	2	no	—
162	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
163	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
164	M₁₆	—	2	2	no	—
165	M₁₆	—	26	4	no	—
166	M₁₆	—	26	4	no	—
167	M₁₆	—	26	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	13	4	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	26	4	no	—
174	C₂×C₈	—	26	4	no	—
175	C₂×C₈	C₂²×D₁₃	1	2	yes	—
176	C₂×C₈	C₂²×D₁₃	1	2	yes	—
177	C₄⋊C₄	—	13	4	no	—
178	C₂²⋊C₄	—	13	4	no	—
179	C₂²⋊C₄	—	13	4	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	2	no	—
183	C₁₃	#2932	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	13	2	no	—
185	C₂³	C₂×C₄×D₁₃	1	2	yes	—
186	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	26	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	26	4	no	—
192	D₄	—	26	2	no	—
193	D₄	—	52	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	2	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	26	2	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂×C₄	D₄×D₁₃	1	2	yes	—
205	C₂×C₄	#6129	1	2	yes	—
206	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	13	2	no	—
210	C₈	—	13	2	no	—
211	C₈	—	26	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	26	2	no	—
216	C₂²	—	26	4	no	—
217	C₂²	#88135	1	2	yes	—
218	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	26	2	no	—
221	C₄	—	26	2	no	—
222	C₄	—	26	2	no	—
223	C₄	#88135	1	2	yes	—
224	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
225	C₄	C₂×#6129	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	52	2	no	—
229	C₂	C₂×#88135	1	1	yes	characteristic
230	C₂	#11719455	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#408	C₂
prime radical	#11719537	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#408	D₁₃
hypocenter	C₁₃	#2932
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×#88161	C₂	semidirect
4. C₁₃×#408	C₂	semidirect
5. #11718779	C₂	semidirect
6. #11718771	C₂	semidirect
7. C₂×#88055	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718389	C₂	semidirect
62. C₂×#1055	D₄	semidirect
104. #408	D₁₃	semidirect
183. C₁₃	#2932	semidirect

Maximal subgroups

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

subgroup	index
#11719537	2
C₂×#88202	2
C₂×#88161	2
C₁₃×#408	2
#11718779	2
#11718771	2
C₂×#88055	2
#11718391	2
#11718389	2
#2932	13