Subgroups — #13128655

Order 1664.

1060 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#11719532	C₂	1	2	yes	maximal
3	#11719275	C₂	1	2	yes	maximal, complemented
4	Q₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
5	#11719248	C₂	1	6	yes	maximal, complemented
6	D₄×Dic₂₆	C₂	1	6	yes	maximal, complemented
7	#11719224	C₂	1	6	yes	maximal, complemented
8	#11719175	C₂	1	3	yes	maximal, complemented
9	C₂×#88123	C₂	1	3	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₆	C₂²	1	6	yes	—
12	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	3	yes	—
15	C₄×D₄×C₁₃	C₂²	1	3	yes	—
16	C₁₃×#117	C₂²	1	3	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	3	yes	—
18	Q₈×Dic₁₃	C₂²	1	4	yes	—
19	D₄×Dic₁₃	C₂²	1	12	yes	—
20	#88180	C₂²	1	6	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	6	yes	—
22	C₂×C₄⋊Dic₁₃	C₂²	1	6	yes	—
23	#88178	C₂²	1	6	yes	—
24	#88178	C₂²	1	6	yes	—
25	#88144	C₂²	1	12	yes	—
26	#88142	C₂²	1	12	yes	—
27	#88132	C₂²	1	24	yes	—
28	#88124	C₂²	1	6	yes	—
29	#88123	C₂²	1	3	yes	—
30	#88123	C₂²	1	6	yes	—
31	#88123	C₂²	1	6	yes	—
32	#88123	C₂²	1	6	yes	—
33	#88123	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₆	C₂²	1	6	yes	—
35	C₄×Dic₂₆	C₂²	1	6	yes	—
36	C₁₃×#41	Q₈	1	8	yes	complemented
37	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₁₃	C₂³	1	3	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
42	C₂²×Dic₁₃	C₂³	1	12	yes	—
43	C₂×Dic₂₆	—	4	12	no	—
44	C₂×Dic₂₆	C₂³	1	2	yes	—
45	C₂×Dic₂₆	C₂³	1	6	yes	—
46	C₂×Dic₂₆	C₂³	1	12	yes	—
47	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
48	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
49	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
50	C₄²×C₁₃	C₂³	1	3	yes	—
51	C₄²×C₁₃	C₂³	1	3	yes	—
52	C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₄²×C₁₃	C₂³	1	3	yes	—
54	C₂²⋊Dic₁₃	C₂³	1	12	yes	—
55	C₄⋊Dic₁₃	C₂³	1	2	yes	—
56	C₄⋊Dic₁₃	C₂³	1	6	yes	—
57	C₄⋊Dic₁₃	C₂³	1	6	yes	—
58	C₄⋊Dic₁₃	C₂³	1	12	yes	—
59	C₄⋊Dic₁₃	C₂³	1	12	yes	—
60	C₄⋊Dic₁₃	C₂³	1	2	yes	—
61	#6101	C₂³	1	12	yes	—
62	#6101	C₂³	1	12	yes	—
63	C₄×Dic₁₃	C₂³	1	12	yes	—
64	#3503	—	13	1	no	maximal, Sylow
65	C₂³×C₁₃	C₂⁴	1	3	yes	—
66	Q₈×C₁₃	C₂×Q₈	1	4	yes	complemented
67	D₄×C₁₃	C₂×Q₈	1	12	yes	complemented
68	C₂×C₄×C₁₃	—	2	6	no	—
69	C₂×C₄×C₁₃	—	2	6	no	—
70	C₂×C₄×C₁₃	C₂×Q₈	1	12	yes	complemented
71	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
72	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
73	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
74	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₁₃	—	4	12	no	—
78	C₂×Dic₁₃	C₂⁴	1	4	yes	—
79	C₂×Dic₁₃	C₂⁴	1	12	yes	—
80	Dic₂₆	—	2	12	no	—
81	Dic₂₆	—	2	4	no	—
82	Dic₂₆	—	4	12	no	—
83	Q₈²	—	13	2	no	—
84	#557	—	13	2	no	—
85	#554	—	13	6	no	—
86	D₄×Q₈	—	13	6	no	—
87	#544	—	13	6	no	—
88	#533	—	13	3	no	—
89	C₂×C₄⋊Q₈	—	13	3	no	—
90	#520	—	13	2	no	—
91	C₄×#41	D₁₃	1	1	yes	characteristic
92	C₂²×C₁₃	—	2	3	no	—
93	C₂²×C₁₃	—	2	3	no	—
94	C₂²×C₁₃	C₂²×Q₈	1	6	yes	—
95	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
96	C₄×C₁₃	—	2	3	no	—
97	C₄×C₁₃	—	2	3	no	—
98	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
99	C₄×C₁₃	C₂²×Q₈	1	6	yes	—
100	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
101	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
102	Dic₁₃	—	2	12	no	—
103	Dic₁₃	—	2	4	no	—
104	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
105	C₂²×Q₈	—	13	6	no	—
106	C₄⋊Q₈	—	13	12	no	—
107	C₄⋊Q₈	—	13	3	no	—
108	C₄⋊Q₈	—	13	6	no	—
109	C₄⋊Q₈	—	13	6	no	—
110	C₄⋊Q₈	—	13	6	no	—
111	C₄⋊Q₈	—	13	1	no	—
112	#125	—	13	12	no	—
113	#125	—	13	6	no	—
114	C₂²⋊Q₈	—	13	6	no	—
115	C₂²⋊Q₈	—	13	6	no	—
116	C₂²⋊Q₈	—	13	24	no	—
117	C₄×Q₈	—	13	4	no	—
118	C₄×Q₈	—	13	6	no	—
119	C₄×Q₈	—	13	6	no	—
120	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
121	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
122	C₄×D₄	—	13	12	no	—
123	C₄×D₄	C₂×D₁₃	1	3	yes	—
124	C₄×D₄	C₂×D₁₃	1	3	yes	—
125	#117	—	13	6	no	—
126	#117	C₂×D₁₃	1	3	yes	—
127	C₂×C₄⋊C₄	—	13	6	no	—
128	C₂×C₄⋊C₄	—	13	6	no	—
129	C₂×C₄²	C₂×D₁₃	1	3	yes	—
130	C₂×C₁₃	—	2	6	no	—
131	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
132	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	#41	Dic₂₆	1	8	yes	complemented
135	C₂×Q₈	—	13	2	no	—
136	C₂×Q₈	—	13	6	no	—
137	C₂×Q₈	—	13	12	no	—
138	C₂×Q₈	—	52	12	no	—
139	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
140	C₂×D₄	C₂²×D₁₃	1	3	yes	—
141	C₂²×C₄	—	13	12	no	—
142	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
143	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
144	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
145	C₄⋊C₄	—	13	6	no	—
146	C₄⋊C₄	—	13	6	no	—
147	C₄⋊C₄	—	13	12	no	—
148	C₄⋊C₄	—	13	12	no	—
149	C₄⋊C₄	—	13	2	no	—
150	C₄⋊C₄	—	13	2	no	—
151	C₄⋊C₄	—	13	12	no	—
152	C₄⋊C₄	—	13	12	no	—
153	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
154	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
155	C₂²⋊C₄	—	13	12	no	—
156	C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
157	C₄²	—	13	12	no	—
158	C₄²	C₂²×D₁₃	1	1	yes	characteristic
159	C₄²	C₂²×D₁₃	1	3	yes	—
160	C₄²	C₂²×D₁₃	1	3	yes	—
161	C₄²	C₂²×D₁₃	1	3	yes	—
162	C₁₃	#3503	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂³×D₁₃	1	3	yes	—
164	Q₈	—	26	12	no	—
165	Q₈	—	26	4	no	—
166	Q₈	—	52	12	no	—
167	Q₈	C₂×Dic₂₆	1	4	yes	complemented
168	D₄	C₂×Dic₂₆	1	12	yes	complemented
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	13	4	no	—
172	C₂×C₄	—	13	12	no	—
173	C₂×C₄	—	52	12	no	—
174	C₂×C₄	C₂×Dic₂₆	1	12	yes	complemented
175	C₂×C₄	C₂³×D₁₃	1	3	yes	—
176	C₂×C₄	C₂³×D₁₃	1	3	yes	—
177	C₂×C₄	C₂³×D₁₃	1	3	yes	—
178	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
181	C₂²	—	2	3	no	—
182	C₂²	—	2	3	no	—
183	C₂²	C₂²×Dic₂₆	1	6	yes	—
184	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
185	C₄	—	2	3	no	—
186	C₄	—	2	3	no	—
187	C₄	—	26	12	no	—
188	C₄	—	26	4	no	—
189	C₄	C₂²×Dic₂₆	1	2	yes	—
190	C₄	C₂²×Dic₂₆	1	6	yes	—
191	C₄	#88256	1	2	yes	—
192	C₄	#88257	1	2	yes	—
193	C₂	—	2	6	no	—
194	C₂	C₂³×Dic₂₆	1	1	yes	characteristic
195	C₂	C₂×#88256	1	1	yes	characteristic
196	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₄×C₁₃×#41	C₂
prime radical	C₂×C₁₃×#41	C₂²
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	#3503
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
3. #11719275	C₂	semidirect
4. Q₈×Dic₂₆	C₂	semidirect
5. #11719248	C₂	semidirect
6. D₄×Dic₂₆	C₂	semidirect
7. #11719224	C₂	semidirect
8. #11719175	C₂	semidirect
9. C₂×#88123	C₂	semidirect
36. C₁₃×#41	Q₈	semidirect
66. Q₈×C₁₃	C₂×Q₈	semidirect
67. D₄×C₁₃	C₂×Q₈	semidirect
70. C₂×C₄×C₁₃	C₂×Q₈	semidirect
134. #41	Dic₂₆	semidirect
162. C₁₃	#3503	semidirect
167. Q₈	C₂×Dic₂₆	semidirect
168. D₄	C₂×Dic₂₆	semidirect
174. C₂×C₄	C₂×Dic₂₆	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
#11719532	2
#11719275	2
Q₈×Dic₂₆	2
#11719248	2
D₄×Dic₂₆	2
#11719224	2
#11719175	2
C₂×#88123	2
#3503	13