Subgroups — #15768748

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	2	yes	maximal
2	C₂×#11718144	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×#6099	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₃	—	2	4	no	—
8	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
10	#11718585	—	2	4	no	—
11	#11718585	C₄	1	8	yes	—
12	C₂×C₄⋊#1055	—	2	4	no	—
13	#11718144	—	2	8	no	—
14	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	—	2	4	no	—
17	C₂²×C₈×C₁₃	—	2	4	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₃	D₄	1	4	yes	—
20	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	2	yes	—
25	C₂×C₄²×C₁₃	D₄	1	4	yes	—
26	C₂×#6099	—	2	4	no	—
27	C₂×#6099	—	2	8	no	—
28	C₂×#6099	—	2	8	no	—
29	C₂×#6099	C₂×C₄	1	8	yes	—
30	C₂²×#1055	—	2	4	no	—
31	C₄×C₈×C₁₃	—	2	4	no	—
32	C₄×C₈×C₁₃	—	2	4	no	—
33	C₄⋊#1055	—	2	16	no	—
34	#10844	—	13	1	no	maximal, Sylow
35	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	—	2	8	no	—
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₁₃	#41	1	4	yes	—
47	C₂×C₈×C₁₃	—	2	4	no	—
48	C₂×C₈×C₁₃	—	2	4	no	—
49	C₂×C₈×C₁₃	—	2	8	no	—
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
52	C₄²×C₁₃	C₄²	1	2	yes	—
53	C₄²×C₁₃	C₄²	1	2	yes	—
54	C₄²×C₁₃	C₂²⋊C₄	1	8	yes	—
55	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
57	#6099	—	2	16	no	—
58	#6099	—	4	8	no	—
59	C₂×#1055	—	2	8	no	—
60	C₂×#1055	—	4	4	no	—
61	C₂×#1055	—	4	4	no	—
62	C₂×#423	—	13	2	no	—
63	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
64	C₂×#341	—	13	4	no	—
65	C₂³×C₁₃	—	2	4	no	—
66	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₁₃	#125	1	2	yes	—
69	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	16	no	—
72	C₂×C₄×C₁₃	#95	1	4	yes	—
73	C₂×C₄×C₁₃	#95	1	4	yes	—
74	C₂×C₄×C₁₃	#108	1	8	yes	—
75	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₁₃	#117	1	2	yes	—
80	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
82	C₂×C₄×C₁₃	#121	1	4	yes	—
83	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₄×C₁₃	#124	1	2	yes	—
85	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₈×C₁₃	—	2	8	no	—
88	#1055	—	4	8	no	—
89	C₂²×M₁₆	—	13	4	no	—
90	C₂³×C₈	C₂×D₁₃	1	2	yes	—
91	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
92	#423	—	13	8	no	—
93	#423	—	26	4	no	—
94	C₂×C₄⋊C₈	—	26	4	no	—
95	C₂×C₄×C₈	—	2	4	no	—
96	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
97	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
98	#341	—	26	8	no	—
99	C₂²×C₁₃	—	2	4	no	—
100	C₂²×C₁₃	—	2	4	no	—
101	C₂²×C₁₃	—	2	4	no	—
102	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
103	C₂²×C₁₃	#382	1	1	yes	characteristic
104	C₂²×C₁₃	#384	1	1	yes	characteristic
105	C₂²×C₁₃	#389	1	4	yes	—
106	C₂²×C₁₃	C₂×#108	1	4	yes	—
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	#341	1	8	yes	complemented
109	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
110	C₄×C₁₃	#380	1	2	yes	—
111	C₄×C₁₃	#390	1	2	yes	—
112	C₄×C₁₃	#391	1	2	yes	—
113	C₂³×C₄	Dic₂₆	1	2	yes	—
114	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×M₁₆	—	13	8	no	—
116	C₂×M₁₆	—	26	4	no	—
117	C₂×M₁₆	—	26	8	no	—
118	C₂×M₁₆	—	26	8	no	—
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	26	4	no	—
122	C₂²×C₈	Dic₂₆	1	4	yes	—
123	C₂²×C₈	D₅₂	1	4	yes	—
124	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
125	C₂×C₄²	C₄×D₁₃	1	4	yes	—
126	C₂×C₄²	D₅₂	1	2	yes	—
127	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
128	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
130	C₄⋊C₈	—	26	16	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	2	4	no	—
133	C₂×C₁₃	—	2	4	no	—
134	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
135	C₂×C₁₃	C₂×#341	1	2	yes	—
136	C₂×C₁₃	#1837	1	2	yes	—
137	C₂×C₁₃	#1848	1	2	yes	—
138	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	#6101	1	8	yes	—
141	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
142	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
143	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
145	C₂²×C₄	C₂×D₅₂	1	2	yes	—
146	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
147	C₂²×C₄	#6127	1	4	yes	—
148	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
150	M₁₆	—	26	16	no	—
151	M₁₆	—	52	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	26	8	no	—
157	C₂×C₈	—	52	4	no	—
158	C₂×C₈	—	52	4	no	—
159	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
160	C₄²	C₄×Dic₁₃	1	2	yes	—
161	C₄²	C₄×Dic₁₃	1	2	yes	—
162	C₄²	#6103	1	8	yes	—
163	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
164	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
165	C₁₃	#10844	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	2	4	no	—
167	C₂³	#88123	1	2	yes	—
168	C₂³	#88124	1	2	yes	—
169	C₂³	C₂×#6101	1	2	yes	—
170	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	#88059	1	8	yes	—
174	C₂×C₄	#88071	1	4	yes	—
175	C₂×C₄	#88071	1	4	yes	—
176	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
177	C₂×C₄	#88123	1	1	yes	characteristic
178	C₂×C₄	C₄×D₅₂	1	4	yes	—
179	C₂×C₄	#88128	1	1	yes	characteristic
180	C₂×C₄	#88129	1	2	yes	—
181	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
182	C₂×C₄	#88178	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
184	C₂×C₄	#88180	1	2	yes	—
185	C₂×C₄	C₂×#6103	1	2	yes	—
186	C₂×C₄	#88184	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
188	C₈	—	2	8	no	—
189	C₈	—	52	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	#11718589	1	4	yes	—
194	C₂²	#11718596	1	1	yes	characteristic
195	C₂²	#11718597	1	1	yes	characteristic
196	C₂²	C₂×#88059	1	4	yes	—
197	C₂²	C₂×#88071	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	#11718144	1	8	yes	complemented
200	C₄	#11718590	1	2	yes	—
201	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
202	C₄	#11718600	1	2	yes	—
203	C₄	#11718873	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	#13113066	1	1	yes	characteristic
206	C₂	C₂×#11718144	1	2	yes	—
207	C₂	#13114974	1	2	yes	—
208	C₂	#13115008	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#10844
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
108. C₄×C₁₃	#341	semidirect
165. C₁₃	#10844	semidirect
199. C₄	#11718144	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂×#11718144	2
C₂²×C₄×C₈×C₁₃	2
#10844	13