Subgroups — #13123502

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×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
2	#11718895	C₂	1	2	yes	maximal
3	#11718661	C₂	1	4	yes	maximal, complemented
4	#11718658	C₂	1	2	yes	maximal, complemented
5	#11718587	C₂	1	2	yes	maximal, complemented
6	#11718586	C₂	1	2	yes	maximal, complemented
7	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₁₃	C₄	1	2	yes	—
13	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₄	1	2	yes	—
15	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
16	C₁₃×#117	C₂²	1	1	yes	characteristic
17	C₁₃×#117	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
20	C₂×#6099	C₂²	1	2	yes	—
21	C₂²×#1055	C₂²	1	2	yes	—
22	C₂²⋊#1055	C₂²	1	4	yes	—
23	C₂²⋊#1055	C₂²	1	4	yes	—
24	C₄⋊#1055	C₂²	1	2	yes	—
25	C₄⋊#1055	C₂²	1	2	yes	—
26	C₄⋊#1055	C₂²	1	1	yes	characteristic
27	C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₄⋊#1055	C₂²	1	2	yes	—
29	#88043	C₂²	1	4	yes	—
30	C₄×#1055	C₂²	1	4	yes	—
31	C₁₃×#41	—	2	2	no	—
32	C₁₃×#41	—	2	2	no	—
33	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
47	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
48	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
49	C₄²×C₁₃	C₂×C₄	1	2	yes	—
50	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂×C₄	1	2	yes	—
52	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₄²×C₁₃	C₂³	1	2	yes	—
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	#6099	—	2	4	no	—
57	C₂×#1055	—	2	4	no	—
58	C₂×#1055	C₂³	1	2	yes	—
59	C₂×#1055	C₂³	1	2	yes	—
60	C₂×#1055	C₂³	1	4	yes	—
61	#2925	—	13	1	no	maximal, Sylow
62	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	Q₈×C₁₃	—	2	2	no	—
65	D₄×C₁₃	—	2	4	no	—
66	D₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	4	1	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#41	1	2	yes	—
82	C₂×C₄×C₁₃	#41	1	2	yes	—
83	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	#1055	—	2	2	no	—
85	#1055	—	2	2	no	—
86	#1055	—	2	4	no	—
87	C₄×#41	D₁₃	1	1	yes	characteristic
88	#433	—	13	4	no	—
89	#432	—	13	2	no	—
90	#431	—	13	2	no	—
91	#423	—	13	1	no	—
92	C₂×C₄⋊C₈	—	13	1	no	—
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	—	2	2	no	—
98	C₂²×C₁₃	—	2	2	no	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	#117	1	2	yes	—
101	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	#117	1	2	yes	—
109	C₄×C₁₃	#131	1	2	yes	—
110	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
113	C₂×M₁₆	—	13	2	no	—
114	C₂²×C₈	—	13	2	no	—
115	C₄×Q₈	Dic₁₃	1	1	yes	characteristic
116	C₄×Q₈	Dic₁₃	1	1	yes	characteristic
117	C₄×D₄	Dic₁₃	1	2	yes	—
118	C₄×D₄	Dic₁₃	1	1	yes	characteristic
119	C₄×D₄	Dic₁₃	1	2	yes	—
120	C₄×D₄	Dic₁₃	1	1	yes	characteristic
121	#117	C₂×D₁₃	1	1	yes	characteristic
122	#117	C₂×D₁₃	1	2	yes	—
123	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×D₁₃	1	2	yes	—
125	C₄⋊C₈	—	13	1	no	—
126	C₄⋊C₈	—	13	1	no	—
127	C₄⋊C₈	—	13	2	no	—
128	C₄⋊C₈	—	13	2	no	—
129	C₄⋊C₈	—	13	2	no	—
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	—	4	2	no	—
136	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
138	C₂×C₁₃	#568	1	1	yes	characteristic
139	#41	—	2	2	no	—
140	#41	—	2	2	no	—
141	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
143	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
147	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
149	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
150	M₁₆	—	26	4	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	13	4	no	—
153	C₂×C₈	—	13	2	no	—
154	C₂×C₈	—	26	4	no	—
155	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
156	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
160	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
162	C₄²	C₂×Dic₁₃	1	2	yes	—
163	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
164	C₄²	C₂×Dic₁₃	1	2	yes	—
165	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
166	C₄²	C₂²×D₁₃	1	2	yes	—
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	C₂²×Dic₁₃	1	2	yes	—
171	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#6127	1	2	yes	—
182	C₂×C₄	#6127	1	2	yes	—
183	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
185	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
188	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
191	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
192	C₈	—	26	2	no	—
193	C₈	—	26	2	no	—
194	C₈	—	26	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#88180	1	2	yes	—
201	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#88180	1	2	yes	—
209	C₄	#88202	1	2	yes	—
210	C₄	C₂×#6127	1	1	yes	characteristic
211	C₄	C₂×#6127	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#88180	1	1	yes	characteristic
215	C₂	C₂×#88202	1	1	yes	characteristic
216	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#41	C₂
prime radical	C₂×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	#2925
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
3. #11718661	C₂	semidirect
4. #11718658	C₂	semidirect
5. #11718587	C₂	semidirect
6. #11718586	C₂	semidirect
7. #11718585	C₂	semidirect
8. C₂×C₄⋊#1055	C₂	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×C₁₃×#41	2
#11718895	2
#11718661	2
#11718658	2
#11718587	2
#11718586	2
#11718585	2
C₂×C₄⋊#1055	2
#2925	13