Subgroups — #15753388

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊#1055	C₂	1	2	yes	maximal
2	C₂×#11718047	C₂	1	2	yes	maximal, complemented
3	#13112663	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	—	2	4	no	—
8	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊#1055	C₄	1	4	yes	—
12	C₂×C₄×#1055	C₂²	1	4	yes	—
13	#11718116	C₄	1	4	yes	—
14	#11718047	—	2	4	no	—
15	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	4	no	—
18	C₂²×C₈×C₁₃	—	2	4	no	—
19	C₂²×C₈×C₁₃	C₈	1	8	yes	complemented
20	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	—	2	2	no	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	8	yes	—
28	C₄×C₈×C₁₃	—	2	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	D₄	1	4	yes	—
31	C₂²⋊#1055	—	2	8	no	—
32	C₄×#1055	—	2	4	no	—
33	C₄×#1055	—	2	4	no	—
34	#8144	—	13	1	no	maximal, Sylow
35	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	—	2	1	no	—
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	16	no	—
50	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
54	C₄²×C₁₃	—	2	1	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	4	no	—
57	C₄²×C₁₃	C₂×D₄	1	2	yes	—
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₂×#1055	—	2	8	no	—
60	C₂×#1055	—	2	8	no	—
61	C₂×#1055	—	2	8	no	—
62	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
63	C₄×C₂²⋊C₈	—	13	2	no	—
64	C₂×#322	—	13	2	no	—
65	#1273	—	13	2	no	—
66	C₂³×C₁₃	—	2	1	no	—
67	C₂³×C₁₃	—	2	1	no	—
68	C₂³×C₁₃	—	2	2	no	—
69	C₂³×C₁₃	C₄×C₈	1	2	yes	—
70	C₂³×C₁₃	#97	1	4	yes	—
71	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	16	no	—
79	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
80	C₂×C₄×C₁₃	#97	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
87	C₂×C₄×C₁₃	#131	1	4	yes	—
88	C₈×C₁₃	—	2	4	no	—
89	C₈×C₁₃	C₂²⋊C₈	1	8	yes	complemented
90	#1055	—	4	8	no	—
91	C₂³×C₈	Dic₁₃	1	2	yes	—
92	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
93	C₂×C₂²⋊C₈	—	13	4	no	—
94	C₂×C₄×C₈	—	2	4	no	—
95	C₂×C₄×C₈	—	13	4	no	—
96	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	#336	—	13	4	no	—
100	#322	—	26	4	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	#322	1	4	yes	complemented
106	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₁₃	C₂×#97	1	2	yes	—
109	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₁₃	#405	1	2	yes	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
114	C₄×C₁₃	#431	1	2	yes	—
115	C₄×C₁₃	#435	1	8	yes	—
116	C₂³×C₄	C₄×D₁₃	1	2	yes	—
117	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	13	8	no	—
121	C₂²×C₈	#1055	1	8	yes	complemented
122	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
123	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₄×D₁₃	1	2	yes	—
127	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	C₂²⋊C₈	—	26	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	26	4	no	—
133	C₄×C₈	—	26	4	no	—
134	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
135	C₂×C₁₃	—	2	4	no	—
136	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
137	C₂×C₁₃	#1450	1	1	yes	characteristic
138	C₂×C₁₃	#1451	1	2	yes	—
139	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
140	C₂×C₁₃	#1752	1	2	yes	—
141	C₂⁴	C₄×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	#6094	1	8	yes	—
148	C₂²×C₄	C₂×#1055	1	4	yes	—
149	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	16	no	—
156	C₂×C₈	—	26	8	no	—
157	C₂×C₈	—	26	8	no	—
158	C₂×C₈	—	26	8	no	—
159	C₂×C₈	C₂×#1055	1	4	yes	—
160	C₂×C₈	C₂×#1055	1	4	yes	—
161	C₂×C₈	#6099	1	4	yes	—
162	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	1	no	—
166	C₄²	#6127	1	2	yes	—
167	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
168	C₁₃	#8144	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	2	2	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	1	no	—
172	C₂³	C₄×#1055	1	2	yes	—
173	C₂³	#88055	1	4	yes	—
174	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	16	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	C₄×#1055	1	2	yes	—
183	C₂×C₄	#88055	1	4	yes	—
184	C₂×C₄	C₂×#6094	1	4	yes	—
185	C₂×C₄	#88155	1	4	yes	—
186	C₂×C₄	C₂²×#1055	1	2	yes	—
187	C₂×C₄	C₂×#6099	1	2	yes	—
188	C₂×C₄	#88180	1	1	yes	characteristic
189	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
190	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	—	52	8	no	—
193	C₈	C₂²⋊#1055	1	8	yes	complemented
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	#11718047	1	4	yes	complemented
199	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
200	C₂²	C₄×#6099	1	1	yes	characteristic
201	C₂²	C₂×#88055	1	2	yes	—
202	C₂²	#11718762	1	2	yes	—
203	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	#11718586	1	2	yes	—
207	C₄	#11718772	1	8	yes	—
208	C₄	C₂×C₂²⋊#1055	1	2	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₂×#11718047	1	1	yes	characteristic
211	C₂	#13113119	1	1	yes	characteristic
212	C₂	#13113125	1	2	yes	—
213	C₂	C₄×C₂²⋊#1055	1	1	yes	characteristic
214	C₂	#13114612	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#8144
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718047	C₂	semidirect
3. #13112663	C₂	semidirect
19. C₂²×C₈×C₁₃	C₈	semidirect
89. C₈×C₁₃	C₂²⋊C₈	semidirect
105. C₂²×C₁₃	#322	semidirect
121. C₂²×C₈	#1055	semidirect
168. C₁₃	#8144	semidirect
193. C₈	C₂²⋊#1055	semidirect
198. C₂²	#11718047	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#1055	2
C₂×#11718047	2
#13112663	2
C₂²×C₄×C₈×C₁₃	2
#8144	13