Subgroups — #16807064

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131203	C₂	1	8	yes	maximal, complemented
2	#13131170	C₂	1	2	yes	maximal, complemented
3	#13131065	C₂	1	2	yes	maximal
4	C₄×#88106	C₂	1	2	yes	maximal, complemented
5	C₂×#11718369	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	#11718369	C₄	1	4	yes	—
15	#11718369	C₂²	1	4	yes	—
16	#11718260	C₂²	1	8	yes	complemented
17	#11718259	C₂²	1	8	yes	—
18	#11718231	C₂²	1	4	yes	complemented
19	C₄×#6092	C₂²	1	4	yes	complemented
20	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#6103	C₈	1	2	yes	—
25	C₂×#6101	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂³	1	1	yes	characteristic
29	C₂×#6094	—	2	4	no	—
30	C₂×#6094	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₁₃	—	2	4	no	—
32	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₁₃	C₂³	1	2	yes	—
34	#88126	C₈	1	4	yes	—
35	#88126	C₂×C₄	1	4	yes	—
36	#88106	—	2	8	no	—
37	C₂×#6092	C₂³	1	8	yes	—
38	C₁₃×#97	C₂×C₄	1	4	yes	—
39	#88055	C₂×C₄	1	4	yes	—
40	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₈×Dic₁₃	C₂³	1	2	yes	—
42	C₄×#1055	C₂×C₄	1	2	yes	—
43	C₄×#1055	C₂³	1	2	yes	—
44	#33419	—	13	1	no	maximal, Sylow
45	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₄×D₁₃	—	2	2	no	—
51	C₂×C₄×D₁₃	—	2	2	no	—
52	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
53	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
58	C₄²×C₁₃	C₂×C₈	1	2	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
60	#6103	C₂×C₈	1	8	yes	—
61	#6101	C₂×C₈	1	8	yes	—
62	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
63	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
64	C₂×#1055	M₁₆	1	4	yes	—
65	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
66	C₂×#1055	C₂²×C₄	1	2	yes	—
67	C₂×#1055	#41	1	4	yes	—
68	C₂×#1055	C₂⁴	1	1	yes	characteristic
69	#6094	—	2	8	no	—
70	C₈×D₁₃	—	2	4	no	—
71	C₈×D₁₃	—	2	4	no	—
72	#6092	—	2	8	no	—
73	C₂×#405	—	13	1	no	—
74	#2164	—	13	8	no	—
75	#2150	—	13	2	no	—
76	#2112	—	13	2	no	—
77	C₄×M₃₂	—	13	2	no	—
78	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂²×D₁₃	—	2	2	no	—
80	C₂²×D₁₃	—	4	2	no	—
81	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	#117	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
91	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
92	C₄×D₁₃	—	2	2	no	—
93	C₄×D₁₃	—	2	2	no	—
94	C₄×D₁₃	—	2	4	no	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	2	no	—
97	#1055	#117	1	2	yes	—
98	#1055	C₂×M₁₆	1	4	yes	—
99	#1055	C₂×#41	1	2	yes	—
100	C₂²×M₁₆	—	13	1	no	—
101	C₂³×C₈	—	13	1	no	—
102	C₂×#117	—	13	1	no	—
103	C₂×M₃₂	—	13	4	no	—
104	#405	—	13	4	no	—
105	#405	—	13	4	no	—
106	C₂×#97	—	13	1	no	—
107	C₂×#97	#261	1	1	yes	characteristic
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₄⋊C₁₆	—	13	8	no	—
111	C₂²⋊C₁₆	—	13	8	no	—
112	#346	—	13	4	no	—
113	C₄×C₁₆	—	13	4	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	#431	1	2	yes	—
116	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
117	C₂×D₁₃	—	2	4	no	—
118	C₂×D₁₃	—	2	2	no	—
119	C₂×D₁₃	—	4	4	no	—
120	C₄×C₁₃	—	2	4	no	—
121	C₄×C₁₃	#431	1	2	yes	—
122	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
123	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
124	Dic₁₃	—	2	4	no	—
125	Dic₁₃	—	2	2	no	—
126	C₂³×C₄	—	13	1	no	—
127	C₂×M₁₆	—	13	4	no	—
128	C₂×M₁₆	—	26	4	no	—
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	2	no	—
132	C₂²×C₈	—	13	2	no	—
133	C₂²×C₈	—	26	4	no	—
134	C₂²×C₈	C₂×#261	1	1	yes	characteristic
135	C₂²×C₈	C₂×#261	1	1	yes	characteristic
136	#117	—	13	4	no	—
137	#117	—	13	4	no	—
138	C₂×C₄⋊C₄	—	13	2	no	—
139	C₂×C₂²⋊C₄	—	13	2	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	C₂×#261	1	1	yes	characteristic
142	M₃₂	—	26	8	no	—
143	C₂×C₁₆	—	13	8	no	—
144	#97	—	13	4	no	—
145	#97	C₂×#261	1	4	yes	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	13	2	no	—
150	C₂×C₁₃	—	2	2	no	—
151	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
152	C₂×C₁₃	#3407	1	2	yes	—
153	D₁₃	—	4	2	no	—
154	C₂⁴	—	13	1	no	—
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	2	no	—
157	C₂²×C₄	—	13	2	no	—
158	C₂²×C₄	—	13	2	no	—
159	C₂²×C₄	—	26	2	no	—
160	C₂²×C₄	—	26	2	no	—
161	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
162	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
163	M₁₆	—	26	8	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	13	4	no	—
167	C₂×C₈	—	13	4	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	26	4	no	—
172	C₂×C₈	—	26	4	no	—
173	C₂×C₈	C₂²×#261	1	2	yes	—
174	C₂×C₈	C₂²×#261	1	2	yes	—
175	C₄⋊C₄	—	13	8	no	—
176	C₂²⋊C₄	—	13	8	no	—
177	C₄²	—	13	2	no	—
178	C₄²	—	13	2	no	—
179	C₄²	D₁₃⋊C₈	1	2	yes	—
180	C₄²	C₂²×#261	1	2	yes	—
181	C₁₆	—	26	8	no	—
182	C₁₃	#33419	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	13	2	no	—
184	C₂³	—	26	2	no	—
185	C₂³	—	52	2	no	—
186	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	13	4	no	—
189	C₂×C₄	—	13	2	no	—
190	C₂×C₄	—	26	2	no	—
191	C₂×C₄	—	26	4	no	—
192	C₂×C₄	—	26	2	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	52	2	no	—
195	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
196	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
197	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
198	C₂×C₄	#88237	1	2	yes	—
199	C₂×C₄	C₂³×#261	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	13	4	no	—
203	C₈	—	13	2	no	—
204	C₈	—	13	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	26	4	no	—
207	C₂²	—	26	2	no	—
208	C₂²	—	52	4	no	—
209	C₂²	#11719049	1	2	yes	—
210	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	26	4	no	—
213	C₄	—	26	2	no	—
214	C₄	#11719049	1	2	yes	—
215	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
216	C₄	C₂×#88237	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	52	2	no	—
219	C₂	C₂×#11719049	1	1	yes	characteristic
220	C₂	#13131805	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×#97	C₄
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#33419
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
1. #13131203	C₂	semidirect
2. #13131170	C₂	semidirect
4. C₄×#88106	C₂	semidirect
16. #11718260	C₂²	semidirect
18. #11718231	C₂²	semidirect
19. C₄×#6092	C₂²	semidirect
182. C₁₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
#13131203	2
#13131170	2
#13131065	2
C₄×#88106	2
C₂×#11718369	2
#33419	13