Subgroups — #13117748

Order 1664.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88141	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#384	C₂	1	1	yes	maximal, characteristic
4	#11718637	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₁₃	C₂	1	2	yes	maximal
6	#11718634	C₂	1	2	yes	maximal
7	#11718633	C₂	1	4	yes	maximal
8	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₄	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	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×#6101	C₂²	1	4	yes	—
18	C₂×#6101	C₂²	1	4	yes	—
19	C₂×#6101	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
23	#88141	—	2	8	no	—
24	C₄×Dic₂₆	C₄	1	8	yes	complemented
25	#88071	C₂²	1	2	yes	—
26	#88071	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂³	1	2	yes	—
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₂²×Dic₁₃	C₂³	1	2	yes	—
34	C₂²×Dic₁₃	C₂³	1	2	yes	—
35	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
36	C₂×Dic₂₆	D₄	1	4	yes	—
37	C₂×Dic₂₆	Q₈	1	4	yes	—
38	C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₁₃	—	2	2	no	—
43	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₃	Q₈	1	4	yes	complemented
45	#6101	—	2	8	no	—
46	#6101	—	4	4	no	—
47	#6101	C₂×C₄	1	8	yes	—
48	C₄×Dic₁₃	—	2	4	no	—
49	C₄×Dic₁₃	—	2	8	no	—
50	C₄×Dic₁₃	C₂×C₄	1	8	yes	—
51	Q₈×C₄⋊C₄	—	13	1	no	maximal, Sylow
52	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	4	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	4	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	2	yes	—
63	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₃	#41	1	2	yes	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
73	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
74	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
75	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
76	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
77	Dic₂₆	C₄⋊C₄	1	16	yes	complemented
78	C₂×C₄×Q₈	—	13	2	no	—
79	C₂×C₄×Q₈	—	13	1	no	—
80	#389	—	13	4	no	—
81	#389	—	13	2	no	—
82	#384	—	13	2	no	—
83	#384	D₁₃	1	1	yes	characteristic
84	C₄×C₄⋊C₄	—	13	2	no	—
85	C₄×C₄⋊C₄	—	13	1	no	—
86	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
87	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
91	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	—	4	4	no	—
95	C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₄×C₁₃	C₄×Q₈	1	4	yes	complemented
97	Dic₁₃	—	2	4	no	—
98	Dic₁₃	—	2	4	no	—
99	Dic₁₃	—	4	2	no	—
100	Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
101	C₂²×Q₈	—	13	1	no	—
102	C₄×Q₈	—	13	8	no	—
103	C₄×Q₈	—	26	8	no	—
104	C₂×C₄⋊C₄	—	13	2	no	—
105	C₂×C₄⋊C₄	—	13	1	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄⋊C₄	—	13	4	no	—
108	C₂×C₄⋊C₄	—	13	1	no	—
109	C₂×C₄⋊C₄	—	13	4	no	—
110	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
113	C₂×C₄²	—	13	2	no	—
114	C₂×C₄²	—	13	4	no	—
115	C₂×C₄²	—	13	2	no	—
116	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
117	#95	—	13	4	no	—
118	#95	—	13	2	no	—
119	C₂×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₂×C₁₃	#520	1	1	yes	characteristic
122	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
123	C₂×C₁₃	#550	1	1	yes	characteristic
124	C₂×C₁₃	#557	1	1	yes	characteristic
125	C₂×C₁₃	Q₈²	1	1	yes	characteristic
126	C₂×Q₈	—	13	4	no	—
127	C₂×Q₈	—	13	4	no	—
128	C₂×Q₈	—	13	4	no	—
129	C₂²×C₄	—	13	2	no	—
130	C₂²×C₄	—	13	2	no	—
131	C₂²×C₄	—	13	2	no	—
132	C₂²×C₄	—	13	2	no	—
133	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
134	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	13	8	no	—
141	C₄⋊C₄	—	13	4	no	—
142	C₄⋊C₄	—	13	4	no	—
143	C₄⋊C₄	—	26	8	no	—
144	C₄⋊C₄	—	26	2	no	—
145	C₄⋊C₄	—	52	4	no	—
146	C₄²	—	13	8	no	—
147	C₄²	—	26	8	no	—
148	C₄²	—	26	4	no	—
149	C₄²	C₄×D₁₃	1	4	yes	—
150	C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂³×D₁₃	1	1	yes	characteristic
152	Q₈	—	13	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	13	4	no	—
159	C₂×C₄	—	13	4	no	—
160	C₂×C₄	—	13	4	no	—
161	C₂×C₄	—	13	4	no	—
162	C₂×C₄	—	13	4	no	—
163	C₂×C₄	—	26	2	no	—
164	C₂×C₄	—	26	2	no	—
165	C₂×C₄	—	26	2	no	—
166	C₂×C₄	—	26	2	no	—
167	C₂×C₄	—	26	2	no	—
168	C₂×C₄	—	26	2	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
173	C₂×C₄	D₄×D₁₃	1	2	yes	—
174	C₂×C₄	#6129	1	2	yes	—
175	C₂×C₄	Q₈×D₁₃	1	2	yes	—
176	C₂×C₄	Q₈×D₁₃	1	2	yes	—
177	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
179	C₂²	C₂×#6129	1	1	yes	characteristic
180	C₂²	#88251	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
183	C₂²	#88254	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	13	8	no	—
187	C₄	—	26	4	no	—
188	C₄	—	26	4	no	—
189	C₄	—	52	2	no	—
190	C₄	#88141	1	4	yes	complemented
191	C₄	C₄⋊C₄×D₁₃	1	4	yes	—
192	C₂	C₂×#88141	1	1	yes	characteristic
193	C₂	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
194	C₂	#11719209	1	1	yes	characteristic
195	C₂	#11719423	1	1	yes	characteristic
196	C₂	#11719426	1	1	yes	characteristic
197	C₂	#11719427	1	1	yes	characteristic
198	C₂	#11719432	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#384	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	#384	D₁₃
hypocenter	C₁₃	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
24. C₄×Dic₂₆	C₄	semidirect
44. C₄⋊Dic₁₃	Q₈	semidirect
77. Dic₂₆	C₄⋊C₄	semidirect
96. C₄×C₁₃	C₄×Q₈	semidirect
150. C₁₃	Q₈×C₄⋊C₄	semidirect
190. C₄	#88141	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88141	2
C₂×C₄×Dic₂₆	2
C₁₃×#384	2
#11718637	2
C₄⋊C₄×Dic₁₃	2
#11718634	2
#11718633	2
C₄×C₄⋊Dic₁₃	2
Q₈×C₄⋊C₄	13