Subgroups — #13117892

Order 1664.

552 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	2	yes	maximal
3	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₃	C₂	1	2	yes	maximal
5	#11718634	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718590	C₂	1	2	yes	maximal
8	#11718315	C₂	1	4	yes	maximal
9	C₂²×Dic₂₆	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#6101	C₂²	1	4	yes	—
19	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
20	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
21	#88123	C₄	1	8	yes	complemented
22	C₄×Dic₂₆	—	2	8	no	—
23	C₁₃×#95	C₂²	1	2	yes	—
24	#88071	C₂²	1	2	yes	—
25	#88071	C₂²	1	4	yes	—
26	#88071	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₂²×Dic₁₃	C₂³	1	4	yes	—
34	C₂²×Dic₁₃	C₂³	1	2	yes	—
35	C₂×Dic₂₆	—	2	4	no	—
36	C₂×Dic₂₆	—	2	4	no	—
37	C₂×Dic₂₆	C₂×C₄	1	8	yes	—
38	C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₁₃×C₄⋊C₄	Q₈	1	4	yes	complemented
40	C₄²×C₁₃	—	2	4	no	—
41	C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₁₃	—	2	2	no	—
43	C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
44	C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
45	C₄⋊Dic₁₃	D₄	1	4	yes	—
46	#6101	—	2	8	no	—
47	C₄×Dic₁₃	—	2	4	no	—
48	C₄×Dic₁₃	—	2	8	no	—
49	#2365	—	13	1	no	maximal, Sylow
50	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₃	#41	1	2	yes	—
65	C₂×C₄×C₁₃	#41	1	2	yes	—
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	4	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	—	2	2	no	—
73	C₂×Dic₁₃	C₂²×C₄	1	8	yes	—
74	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
75	Dic₂₆	—	2	16	no	—
76	C₂×C₄⋊Q₈	—	13	1	no	—
77	C₂×C₄×Q₈	—	13	2	no	—
78	#391	—	13	2	no	—
79	#389	—	13	2	no	—
80	#387	—	13	4	no	—
81	C₄×C₄⋊C₄	—	13	2	no	—
82	C₄×C₄⋊C₄	—	13	1	no	—
83	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
84	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
87	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
88	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
89	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
90	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	2	4	no	—
93	C₄×C₁₃	—	4	2	no	—
94	C₄×C₁₃	C₄×D₄	1	4	yes	—
95	C₄×C₁₃	C₄×Q₈	1	4	yes	complemented
96	Dic₁₃	—	2	8	no	—
97	Dic₁₃	—	2	4	no	—
98	Dic₁₃	—	4	2	no	—
99	C₂²×Q₈	—	13	2	no	—
100	C₄⋊Q₈	—	13	8	no	—
101	C₄×Q₈	—	26	8	no	—
102	C₂×C₄⋊C₄	—	13	1	no	—
103	C₂×C₄⋊C₄	—	13	2	no	—
104	C₂×C₄⋊C₄	—	13	2	no	—
105	C₂×C₄⋊C₄	—	13	4	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
109	C₂×C₄²	—	13	4	no	—
110	C₂×C₄²	—	13	2	no	—
111	C₂×C₄²	C₂×D₁₃	1	2	yes	—
112	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
113	#95	—	13	2	no	—
114	#95	—	13	4	no	—
115	#95	—	13	4	no	—
116	#95	C₂×D₁₃	1	2	yes	—
117	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₂×C₁₃	#519	1	1	yes	characteristic
120	C₂×C₁₃	#547	1	1	yes	characteristic
121	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
122	C₂×C₁₃	#556	1	1	yes	characteristic
123	C₂×C₁₃	#557	1	1	yes	characteristic
124	C₂×Q₈	—	13	8	no	—
125	C₂×Q₈	—	26	4	no	—
126	C₂×Q₈	—	26	4	no	—
127	C₂²×C₄	—	13	4	no	—
128	C₂²×C₄	—	13	2	no	—
129	C₂²×C₄	—	13	2	no	—
130	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
131	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
132	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	13	8	no	—
137	C₄⋊C₄	—	13	8	no	—
138	C₄⋊C₄	—	13	4	no	—
139	C₄⋊C₄	—	26	8	no	—
140	C₄⋊C₄	—	26	2	no	—
141	C₄⋊C₄	Dic₂₆	1	4	yes	complemented
142	C₄²	—	2	4	no	—
143	C₄²	—	26	8	no	—
144	C₄²	—	26	4	no	—
145	C₄²	C₄×D₁₃	1	4	yes	—
146	C₁₃	#2365	1	1	yes	characteristic, complemented, Sylow
147	C₂³	C₂³×D₁₃	1	1	yes	characteristic
148	Q₈	—	26	16	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	13	8	no	—
157	C₂×C₄	—	13	4	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	4	no	—
160	C₂×C₄	—	26	4	no	—
161	C₂×C₄	—	26	2	no	—
162	C₂×C₄	—	26	2	no	—
163	C₂×C₄	—	26	2	no	—
164	C₂×C₄	—	26	2	no	—
165	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
166	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
168	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
169	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
170	C₂×C₄	#6127	1	2	yes	—
171	C₂×C₄	D₄×D₁₃	1	2	yes	—
172	C₂×C₄	#6131	1	2	yes	—
173	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
174	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
175	C₂²	C₂×#6127	1	1	yes	characteristic
176	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
177	C₂²	C₂×#6131	1	1	yes	characteristic
178	C₂²	#88257	1	1	yes	characteristic
179	C₂²	#88257	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	4	2	no	—
183	C₄	—	26	8	no	—
184	C₄	—	26	4	no	—
185	C₄	—	52	2	no	—
186	C₄	C₄×Dic₂₆	1	4	yes	complemented
187	C₄	#88147	1	4	yes	—
188	C₂	C₂×C₄×Dic₂₆	1	1	yes	characteristic
189	C₂	C₂×#88147	1	1	yes	characteristic
190	C₂	#11719221	1	1	yes	characteristic
191	C₂	D₄×Dic₂₆	1	1	yes	characteristic
192	C₂	#11719256	1	1	yes	characteristic
193	C₂	#11719275	1	1	yes	characteristic
194	C₂	#11719289	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2365
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
21. #88123	C₄	semidirect
39. C₁₃×C₄⋊C₄	Q₈	semidirect
95. C₄×C₁₃	C₄×Q₈	semidirect
141. C₄⋊C₄	Dic₂₆	semidirect
146. C₁₃	#2365	semidirect
186. C₄	C₄×Dic₂₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₄×C₁₃×C₄⋊C₄	2
C₄⋊C₄×Dic₁₃	2
#11718634	2
C₄×C₄⋊Dic₁₃	2
#11718590	2
#11718315	2
#2365	13