Subgroups — #13120206

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	2	yes	maximal
4	#11718645	C₂	1	4	yes	maximal
5	#11718634	C₂	1	2	yes	maximal
6	#11718596	C₂	1	1	yes	maximal, characteristic
7	#11718590	C₂	1	2	yes	maximal
8	#11718589	C₂	1	2	yes	maximal
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₁₃	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	4	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
23	C₄×Dic₂₆	—	2	4	no	—
24	#88071	C₂²	1	4	yes	—
25	#88071	C₂²	1	4	yes	—
26	#88071	C₂²	1	4	yes	—
27	C₂×Q₈×C₁₃	—	2	2	no	—
28	C₂×Q₈×C₁₃	—	2	2	no	—
29	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
30	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₂³	1	4	yes	—
36	C₂²×Dic₁₃	C₂³	1	2	yes	—
37	C₂²×Dic₁₃	C₂³	1	2	yes	—
38	C₂×Dic₂₆	—	2	2	no	—
39	C₂×Dic₂₆	—	2	2	no	—
40	C₂×Dic₂₆	Q₈	1	4	yes	complemented
41	C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₄²×C₁₃	—	2	4	no	—
44	C₄²×C₁₃	D₄	1	4	yes	—
45	C₄⋊Dic₁₃	—	2	2	no	—
46	C₄⋊Dic₁₃	—	2	4	no	—
47	C₄⋊Dic₁₃	—	2	4	no	—
48	C₄⋊Dic₁₃	—	4	2	no	—
49	#6101	—	2	4	no	—
50	#6101	—	4	4	no	—
51	C₄×Dic₁₃	—	2	4	no	—
52	#2582	—	13	1	no	maximal, Sylow
53	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
54	Q₈×C₁₃	—	2	8	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₃	#41	1	2	yes	—
69	C₂×C₄×C₁₃	#41	1	2	yes	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	—	2	2	no	—
73	C₂×Dic₁₃	—	2	4	no	—
74	C₂×Dic₁₃	—	2	4	no	—
75	C₂×Dic₁₃	—	2	2	no	—
76	C₂×Dic₁₃	—	2	4	no	—
77	C₂×Dic₁₃	—	2	2	no	—
78	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
79	Dic₂₆	—	2	8	no	—
80	C₂×C₄×Q₈	—	13	1	no	—
81	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
82	#398	—	13	4	no	—
83	#391	—	13	2	no	—
84	#391	—	13	2	no	—
85	#389	—	13	2	no	—
86	#389	—	13	2	no	—
87	#384	—	13	1	no	—
88	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
92	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
93	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
95	C₄×C₁₃	—	2	4	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	4	2	no	—
98	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
99	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
100	Dic₁₃	—	2	4	no	—
101	Dic₁₃	—	4	2	no	—
102	Dic₁₃	—	4	4	no	—
103	C₂²×Q₈	—	13	1	no	—
104	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
105	C₄×Q₈	—	2	4	no	—
106	C₄×Q₈	—	26	4	no	—
107	C₂×C₄⋊C₄	—	13	2	no	—
108	C₂×C₄⋊C₄	—	13	1	no	—
109	C₂×C₄⋊C₄	—	13	2	no	—
110	C₂×C₄⋊C₄	—	13	2	no	—
111	C₂×C₄⋊C₄	—	13	4	no	—
112	C₂×C₄⋊C₄	—	13	2	no	—
113	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
115	C₂×C₄²	—	13	2	no	—
116	C₂×C₄²	C₂×D₁₃	1	2	yes	—
117	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
118	#95	—	13	4	no	—
119	#95	—	13	4	no	—
120	#95	—	13	4	no	—
121	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₁₃	#537	1	1	yes	characteristic
124	C₂×C₁₃	#557	1	1	yes	characteristic
125	C₂×C₁₃	#557	1	1	yes	characteristic
126	C₂×C₁₃	Q₈²	1	1	yes	characteristic
127	C₂×C₁₃	#559	1	1	yes	characteristic
128	C₂×Q₈	—	2	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	13	4	no	—
131	C₂×Q₈	—	26	2	no	—
132	C₂×Q₈	—	26	2	no	—
133	C₂×Q₈	Dic₂₆	1	4	yes	complemented
134	C₂²×C₄	—	13	2	no	—
135	C₂²×C₄	—	13	4	no	—
136	C₂²×C₄	—	13	2	no	—
137	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
138	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
139	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	26	4	no	—
145	C₄⋊C₄	—	26	4	no	—
146	C₄⋊C₄	—	26	4	no	—
147	C₄⋊C₄	—	26	2	no	—
148	C₄⋊C₄	—	52	4	no	—
149	C₄⋊C₄	—	52	2	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	26	4	no	—
152	C₄²	C₂²⋊D₁₃	1	4	yes	—
153	C₁₃	#2582	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂³×D₁₃	1	1	yes	characteristic
155	Q₈	—	2	8	no	—
156	Q₈	—	26	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	13	4	no	—
165	C₂×C₄	—	26	2	no	—
166	C₂×C₄	—	26	2	no	—
167	C₂×C₄	—	26	4	no	—
168	C₂×C₄	—	26	4	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	—	26	4	no	—
171	C₂×C₄	—	26	2	no	—
172	C₂×C₄	—	26	2	no	—
173	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
174	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
175	C₂×C₄	#6127	1	2	yes	—
176	C₂×C₄	Q₈×D₁₃	1	2	yes	—
177	C₂×C₄	#6131	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
180	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
181	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
182	C₂²	C₂×#6127	1	1	yes	characteristic
183	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
184	C₂²	C₂×#6131	1	1	yes	characteristic
185	C₂²	#88256	1	1	yes	characteristic
186	C₂²	#88257	1	1	yes	characteristic
187	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	26	4	no	—
192	C₄	—	52	2	no	—
193	C₄	—	52	4	no	—
194	C₄	#88178	1	4	yes	—
195	C₄	#88200	1	4	yes	—
196	C₂	Q₈×Dic₂₆	1	1	yes	characteristic
197	C₂	#11719272	1	1	yes	characteristic
198	C₂	#11719275	1	1	yes	characteristic
199	C₂	#11719290	1	1	yes	characteristic
200	C₂	C₂×#88178	1	1	yes	characteristic
201	C₂	C₂×#88200	1	1	yes	characteristic
202	C₂	#11719538	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₄×Q₈×C₁₃	C₂
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#2582
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
29. C₂×Q₈×C₁₃	Q₈	semidirect
40. C₂×Dic₂₆	Q₈	semidirect
133. C₂×Q₈	Dic₂₆	semidirect
153. C₁₃	#2582	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
C₂×C₄×Dic₂₆	2
#11718891	2
#11718645	2
#11718634	2
#11718596	2
#11718590	2
#11718589	2
#2582	13