Subgroups — #13123501

Order 1664.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718771	C₂	1	2	yes	maximal
4	#11718762	C₂	1	2	yes	maximal, complemented
5	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718501	C₂	1	4	yes	maximal, complemented
7	#11718496	C₂	1	2	yes	maximal, complemented
8	#11718494	C₂	1	2	yes	maximal, complemented
9	C₂×#6127	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₁₃	C₄	1	2	yes	—
13	C₄×C₂²⋊D₁₃	C₄	1	2	yes	—
14	#88180	C₂²	1	1	yes	characteristic
15	C₂×#6099	C₂²	1	2	yes	—
16	C₄×D₅₂	C₄	1	1	yes	characteristic
17	C₄×D₅₂	C₄	1	1	yes	characteristic
18	#88126	C₂²	1	2	yes	—
19	C₄²×D₁₃	C₂²	1	2	yes	—
20	C₄×Dic₂₆	C₄	1	1	yes	characteristic
21	C₄×Dic₂₆	C₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
25	#88060	C₂²	1	4	yes	—
26	#88060	C₂²	1	4	yes	—
27	#88055	C₂²	1	4	yes	—
28	C₈×Dic₁₃	C₂²	1	4	yes	—
29	C₄⋊#1055	C₂²	1	2	yes	—
30	C₄⋊#1055	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
35	#6127	—	2	2	no	—
36	#6127	—	2	2	no	—
37	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₃	C₂³	1	2	yes	—
41	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	4	no	—
43	C₂×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₈×C₁₃	C₂³	1	2	yes	—
45	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
50	#6103	C₂×C₄	1	2	yes	—
51	#6103	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	#6101	C₂×C₄	1	2	yes	—
55	#6101	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₃	C₂³	1	2	yes	—
59	#6099	—	2	4	no	—
60	C₂×#1055	C₂³	1	4	yes	—
61	#2925	—	13	1	no	maximal, Sylow
62	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	4	1	no	—
66	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#41	1	2	yes	—
72	C₂×C₄×C₁₃	#41	1	2	yes	—
73	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₁₃	—	2	4	no	—
75	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
78	D₅₂	—	2	2	no	—
79	C₄×D₁₃	—	2	4	no	—
80	C₄×D₁₃	—	2	2	no	—
81	C₄×D₁₃	—	2	2	no	—
82	C₄×D₁₃	—	4	2	no	—
83	Dic₂₆	—	2	2	no	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	—	2	2	no	—
86	#1055	—	2	4	no	—
87	C₄×#41	—	13	1	no	—
88	#433	—	13	4	no	—
89	#432	—	13	2	no	—
90	#431	—	13	2	no	—
91	#423	—	13	1	no	—
92	C₂×C₄⋊C₈	D₁₃	1	1	yes	characteristic, complemented
93	#408	—	13	2	no	—
94	#405	—	13	2	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	#117	1	2	yes	—
98	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₁₃	—	2	2	no	—
100	C₂×D₁₃	—	2	2	no	—
101	C₂×D₁₃	—	2	2	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	2	1	no	—
105	C₄×C₁₃	#117	1	2	yes	—
106	C₄×C₁₃	#131	1	2	yes	—
107	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
109	Dic₁₃	—	2	2	no	—
110	Dic₁₃	—	2	2	no	—
111	Dic₁₃	—	2	2	no	—
112	C₂×#41	—	13	1	no	—
113	C₂×M₁₆	—	13	2	no	—
114	C₂²×C₈	C₂×D₁₃	1	2	yes	—
115	C₄×Q₈	—	13	1	no	—
116	C₄×Q₈	—	13	1	no	—
117	C₄×D₄	—	13	2	no	—
118	C₄×D₄	—	13	2	no	—
119	C₄×D₄	—	13	1	no	—
120	C₄×D₄	—	13	1	no	—
121	#117	—	13	1	no	—
122	#117	—	13	2	no	—
123	C₂×C₄²	—	13	2	no	—
124	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
125	C₄⋊C₈	—	13	2	no	—
126	C₄⋊C₈	—	13	2	no	—
127	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
129	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
130	C₂²⋊C₈	—	13	4	no	—
131	C₂²⋊C₈	—	13	4	no	—
132	#97	—	13	4	no	—
133	C₄×C₈	—	13	4	no	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
136	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
137	C₂×C₁₃	#568	1	1	yes	characteristic
138	D₁₃	—	4	2	no	—
139	#41	—	26	2	no	—
140	#41	—	26	2	no	—
141	C₂×Q₈	—	13	1	no	—
142	C₂×D₄	—	13	2	no	—
143	C₂×D₄	—	13	1	no	—
144	C₂²×C₄	—	13	2	no	—
145	C₂²×C₄	—	13	2	no	—
146	C₂²×C₄	—	13	2	no	—
147	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
150	M₁₆	—	26	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	13	4	no	—
153	C₂×C₈	C₂²×D₁₃	1	2	yes	—
154	C₂×C₈	C₂²×D₁₃	1	2	yes	—
155	C₄⋊C₄	—	13	2	no	—
156	C₄⋊C₄	—	13	2	no	—
157	C₄⋊C₄	—	13	1	no	—
158	C₄⋊C₄	—	13	1	no	—
159	C₂²⋊C₄	—	13	2	no	—
160	C₂²⋊C₄	—	13	2	no	—
161	C₂²⋊C₄	—	13	2	no	—
162	C₄²	—	13	2	no	—
163	C₄²	—	13	2	no	—
164	C₄²	—	13	2	no	—
165	C₄²	C₄×D₁₃	1	1	yes	characteristic
166	C₄²	C₄×D₁₃	1	1	yes	characteristic
167	C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₁₃	#2925	1	1	yes	characteristic, complemented, Sylow
170	C₂³	—	13	2	no	—
171	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
172	Q₈	—	26	2	no	—
173	D₄	—	26	4	no	—
174	D₄	—	26	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	13	2	no	—
178	C₂×C₄	—	13	2	no	—
179	C₂×C₄	—	13	2	no	—
180	C₂×C₄	—	26	4	no	—
181	C₂×C₄	—	26	2	no	—
182	C₂×C₄	—	26	2	no	—
183	C₂×C₄	—	52	2	no	—
184	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₄	#6129	1	2	yes	—
190	C₂×C₄	#6131	1	2	yes	—
191	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	26	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	26	2	no	—
198	C₂²	—	26	2	no	—
199	C₂²	—	26	2	no	—
200	C₂²	#88146	1	2	yes	—
201	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	26	2	no	—
206	C₄	—	26	2	no	—
207	C₄	—	26	2	no	—
208	C₄	#88146	1	2	yes	—
209	C₄	#88155	1	2	yes	—
210	C₄	C₂×#6129	1	1	yes	characteristic
211	C₄	C₂×#6131	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	52	2	no	—
214	C₂	C₂×#88146	1	1	yes	characteristic
215	C₂	C₂×#88155	1	1	yes	characteristic
216	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₄⋊C₈	C₂
prime radical	C₂×#6127	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#2925
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×C₄⋊C₈	C₂	semidirect
4. #11718762	C₂	semidirect
5. #11718585	C₂	semidirect
6. #11718501	C₂	semidirect
7. #11718496	C₂	semidirect
8. #11718494	C₂	semidirect
92. C₂×C₄⋊C₈	D₁₃	semidirect
169. C₁₃	#2925	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6127	2
C₂×C₁₃×C₄⋊C₈	2
#11718771	2
#11718762	2
#11718585	2
#11718501	2
#11718496	2
#11718494	2
#2925	13