Subgroups — #13123961

Order 1664.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#11718685	C₂	1	2	yes	maximal, complemented
3	#11718672	C₂	1	4	yes	maximal, complemented
4	D₄×#1055	C₂	1	2	yes	maximal, complemented
5	#11718587	C₂	1	2	yes	maximal, complemented
6	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊#1055	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#41	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₁₃	C₂²	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	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#117	C₄	1	1	yes	characteristic
17	C₁₃×#117	C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#6099	C₂²	1	2	yes	—
21	C₂²×#1055	C₂²	1	4	yes	—
22	C₂²⋊#1055	C₂²	1	4	yes	—
23	C₂²⋊#1055	C₂²	1	2	yes	—
24	C₄⋊#1055	C₂²	1	2	yes	—
25	C₄⋊#1055	C₂²	1	4	yes	—
26	C₄⋊#1055	C₂²	1	1	yes	characteristic
27	C₄⋊#1055	C₂²	1	2	yes	—
28	C₄⋊#1055	C₂²	1	1	yes	characteristic
29	#88043	C₂²	1	4	yes	—
30	C₄×#1055	C₂²	1	2	yes	—
31	C₁₃×#41	—	2	4	no	—
32	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
46	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
47	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
48	C₄²×C₁₃	C₂×C₄	1	2	yes	—
49	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×C₄	1	2	yes	—
51	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₄²×C₁₃	C₂³	1	2	yes	—
55	#6099	—	4	2	no	—
56	C₂×#1055	—	4	4	no	—
57	C₂×#1055	C₂³	1	2	yes	—
58	C₂×#1055	C₂³	1	2	yes	—
59	C₂×#1055	C₂³	1	4	yes	—
60	#2990	—	13	1	no	maximal, Sylow
61	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
63	Q₈×C₁₃	—	2	2	no	—
64	D₄×C₁₃	—	2	4	no	—
65	D₄×C₁₃	M₁₆	1	4	yes	complemented
66	C₂×C₄×C₁₃	—	2	4	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	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	#1055	—	2	2	no	—
84	#1055	—	2	4	no	—
85	#1055	—	2	2	no	—
86	C₄×#41	D₁₃	1	1	yes	characteristic
87	#446	—	13	2	no	—
88	#435	—	13	4	no	—
89	C₈×D₄	—	13	2	no	—
90	#432	—	13	2	no	—
91	#431	—	13	1	no	—
92	#423	—	13	1	no	—
93	C₂×C₄⋊C₈	—	13	2	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	—	2	2	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	C₂×M₁₆	1	4	yes	—
100	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₄×C₁₃	#131	1	2	yes	—
109	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×#41	Dic₁₃	1	1	yes	characteristic
112	C₂×M₁₆	—	13	2	no	—
113	C₂²×C₈	—	13	4	no	—
114	C₄×Q₈	Dic₁₃	1	1	yes	characteristic
115	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
116	C₄×D₄	Dic₁₃	1	2	yes	—
117	C₄×D₄	Dic₁₃	1	1	yes	characteristic
118	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
119	C₄×D₄	C₂×D₁₃	1	2	yes	—
120	#117	Dic₁₃	1	1	yes	characteristic
121	#117	Dic₁₃	1	2	yes	—
122	C₂×C₄²	C₂×D₁₃	1	2	yes	—
123	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
124	C₄⋊C₈	—	13	1	no	—
125	C₄⋊C₈	—	13	2	no	—
126	C₄⋊C₈	—	13	4	no	—
127	C₄⋊C₈	—	13	2	no	—
128	C₄⋊C₈	—	13	1	no	—
129	C₂²⋊C₈	—	13	2	no	—
130	C₂²⋊C₈	—	13	4	no	—
131	#97	—	13	4	no	—
132	C₄×C₈	—	13	2	no	—
133	C₂×C₁₃	—	2	4	no	—
134	C₂×C₁₃	—	4	1	no	—
135	C₂×C₁₃	#520	1	1	yes	characteristic
136	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
138	#41	—	2	4	no	—
139	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
140	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
141	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
143	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
146	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
147	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
148	M₁₆	—	52	2	no	—
149	C₂×C₈	—	13	2	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	4	no	—
152	C₂×C₈	—	52	4	no	—
153	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
154	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
155	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
156	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
158	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
159	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
160	C₄²	C₂×Dic₁₃	1	2	yes	—
161	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
162	C₄²	C₂×Dic₁₃	1	2	yes	—
163	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
164	C₄²	C₂²×D₁₃	1	1	yes	characteristic
165	C₄²	C₂²×D₁₃	1	2	yes	—
166	C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₁₃	#2990	1	1	yes	characteristic, complemented, Sylow
168	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
169	C₂³	C₂²×Dic₁₃	1	2	yes	—
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	#6099	1	4	yes	complemented
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
183	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
187	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
190	C₈	—	26	2	no	—
191	C₈	—	26	4	no	—
192	C₈	—	26	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	C₂×#6099	1	4	yes	—
199	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#6099	1	2	yes	—
207	C₄	#88202	1	2	yes	—
208	C₄	#88256	1	1	yes	characteristic
209	C₄	#88257	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	4	1	no	—
212	C₂	C₂²×#6099	1	1	yes	characteristic
213	C₂	C₂×#88202	1	1	yes	characteristic
214	C₂	#11719532	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	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	#2990
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
2. #11718685	C₂	semidirect
3. #11718672	C₂	semidirect
4. D₄×#1055	C₂	semidirect
5. #11718587	C₂	semidirect
6. #11718586	C₂	semidirect
7. #11718585	C₂	semidirect
8. C₂×C₄⋊#1055	C₂	semidirect
65. D₄×C₁₃	M₁₆	semidirect
167. C₁₃	#2990	semidirect
172. D₄	#6099	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
#11718685	2
#11718672	2
D₄×#1055	2
#11718587	2
#11718586	2
#11718585	2
C₂×C₄⋊#1055	2
#2990	13