Subgroups — #16789348

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6092	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#6092	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718257	C₂	1	2	yes	maximal, complemented
4	#13129875	C₂	1	2	yes	maximal, complemented
5	#13114610	C₂	1	1	yes	maximal, characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂²×#6092	—	2	4	no	—
8	C₂²×#6092	C₄	1	4	yes	complemented
9	C₂²×#6092	C₂²	1	1	yes	characteristic
10	C₂²×#6092	C₂²	1	2	yes	—
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
13	C₄×C₂²⋊Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
15	C₂²⋊#6092	C₄	1	4	yes	complemented
16	#11718257	—	2	4	no	—
17	#11718142	C₄	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂²×#1055	—	2	2	no	—
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂³	1	1	yes	characteristic
28	C₂×#6092	—	2	2	no	—
29	C₂×#6092	—	2	2	no	—
30	C₂×#6092	—	2	2	no	—
31	C₂×#6092	—	2	8	no	—
32	C₂×#6092	C₂×C₄	1	4	yes	—
33	C₂×#6092	C₂×C₄	1	2	yes	—
34	C₂×#6092	C₂×C₄	1	2	yes	—
35	C₂×#6092	D₄	1	4	yes	—
36	C₁₃×C₂²⋊C₈	C₈	1	4	yes	—
37	#88071	C₈	1	2	yes	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	C₁₆×C₂²⋊C₄	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₁₃	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	4	yes	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
52	C₂²⋊Dic₁₃	C₁₆	1	8	yes	complemented
53	C₄×Dic₁₃	—	2	2	no	—
54	C₄×Dic₁₃	—	2	2	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	4	no	—
59	C₂×#1055	C₄²	1	2	yes	—
60	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
61	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
62	C₂×#1055	C₂×D₄	1	2	yes	—
63	C₂×#1055	#41	1	2	yes	—
64	#6092	—	2	2	no	—
65	#6092	—	2	4	no	—
66	#6092	C₂²⋊C₄	1	4	yes	complemented
67	C₂³×C₁₆	—	13	1	no	—
68	C₂×C₂²⋊C₁₆	—	13	1	no	—
69	C₂×C₄×C₁₆	—	13	2	no	—
70	C₈×C₂²⋊C₄	—	13	1	no	—
71	#1374	—	13	2	no	—
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	C₄×C₈	1	2	yes	—
75	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
76	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	4	no	—
81	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	2	yes	—
86	C₂×Dic₁₃	—	2	4	no	—
87	C₂×Dic₁₃	—	2	4	no	—
88	C₂×Dic₁₃	C₂×C₁₆	1	8	yes	complemented
89	C₈×C₁₃	—	2	4	no	—
90	#1055	—	2	2	no	—
91	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	#1055	#117	1	1	yes	characteristic
93	#1055	C₄×D₄	1	2	yes	—
94	C₂³×C₈	—	13	1	no	—
95	C₂²×C₁₆	—	13	1	no	—
96	C₂²×C₁₆	—	13	2	no	—
97	C₂²×C₁₆	—	13	1	no	—
98	C₂²×C₁₆	—	13	4	no	—
99	C₂²×C₁₆	—	26	4	no	—
100	C₂×C₂²⋊C₈	#261	1	1	yes	characteristic
101	C₂×C₄×C₈	—	13	2	no	—
102	C₄×C₂²⋊C₄	—	13	1	no	—
103	C₂²⋊C₁₆	—	13	4	no	—
104	C₄×C₁₆	—	26	4	no	—
105	#336	—	13	2	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	8	no	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	C₄×C₁₆	1	4	yes	complemented
110	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂²×C₁₃	C₈×D₄	1	2	yes	—
112	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
113	C₂²×C₁₃	#504	1	2	yes	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	#405	1	1	yes	characteristic
117	C₄×C₁₃	C₈×D₄	1	2	yes	—
118	Dic₁₃	—	2	8	no	—
119	C₂³×C₄	C₂×#261	1	1	yes	characteristic
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	—	13	2	no	—
122	C₂²×C₈	—	13	2	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	—	26	2	no	—
125	C₂²×C₈	—	26	2	no	—
126	C₂²×C₈	C₂×#261	1	2	yes	—
127	C₂×C₂²⋊C₄	—	13	2	no	—
128	C₂×C₄²	—	13	2	no	—
129	C₂×C₁₆	—	13	2	no	—
130	C₂×C₁₆	—	13	4	no	—
131	C₂×C₁₆	—	13	4	no	—
132	C₂×C₁₆	—	13	2	no	—
133	C₂×C₁₆	—	26	8	no	—
134	C₂×C₁₆	—	26	2	no	—
135	C₂×C₁₆	—	26	2	no	—
136	C₂×C₁₆	—	26	2	no	—
137	C₂²⋊C₈	#1057	1	4	yes	—
138	C₄×C₈	—	26	2	no	—
139	C₄×C₈	—	26	2	no	—
140	#95	—	13	2	no	—
141	C₂×C₁₃	—	2	4	no	—
142	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
144	C₂×C₁₃	#2108	1	1	yes	characteristic
145	C₂×C₁₃	D₄×C₁₆	1	4	yes	—
146	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	13	4	no	—
150	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₄×#261	1	2	yes	—
152	C₂²×C₄	C₂×#1057	1	2	yes	—
153	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	13	1	no	—
157	C₂×C₈	—	13	1	no	—
158	C₂×C₈	—	13	2	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	13	2	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	—	26	4	no	—
163	C₂×C₈	—	26	2	no	—
164	C₂×C₈	—	26	2	no	—
165	C₂×C₈	C₂×#1057	1	4	yes	—
166	C₂²⋊C₄	—	13	8	no	—
167	C₄²	—	26	2	no	—
168	C₄²	—	26	2	no	—
169	C₁₆	—	13	4	no	—
170	C₁₆	—	26	2	no	—
171	C₁₆	—	26	4	no	—
172	C₁₃	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	D₁₃⋊C₁₆	1	4	yes	—
176	C₂³	C₄×#1057	1	2	yes	—
177	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	13	8	no	—
183	C₂×C₄	—	26	4	no	—
184	C₂×C₄	—	26	4	no	—
185	C₂×C₄	C₄×#1057	1	2	yes	—
186	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
187	C₂×C₄	#88238	1	2	yes	—
188	C₂×C₄	D₄×#261	1	2	yes	—
189	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	13	1	no	—
192	C₈	—	13	2	no	—
193	C₈	—	13	1	no	—
194	C₈	—	26	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	#11718257	1	4	yes	complemented
199	C₂²	#11719063	1	2	yes	—
200	C₂²	C₂×D₁₃⋊C₁₆	1	2	yes	—
201	C₂²	#11719096	1	2	yes	—
202	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	26	8	no	—
205	C₄	#11719062	1	1	yes	characteristic
206	C₄	C₂²⋊C₄×#261	1	1	yes	characteristic
207	C₄	D₄×#1057	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₂²⋊C₄×#1057	1	1	yes	characteristic
210	C₂	C₂×#11718257	1	1	yes	characteristic
211	C₂	#13131047	1	1	yes	characteristic
212	C₂	#13131214	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₂×C₁₃×C₂²⋊C₈	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₂²⋊C₈	#261
hypocenter	C₁₃	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718257	C₂	semidirect
4. #13129875	C₂	semidirect
8. C₂²×#6092	C₄	semidirect
15. C₂²⋊#6092	C₄	semidirect
52. C₂²⋊Dic₁₃	C₁₆	semidirect
66. #6092	C₂²⋊C₄	semidirect
88. C₂×Dic₁₃	C₂×C₁₆	semidirect
109. C₂²×C₁₃	C₄×C₁₆	semidirect
172. C₁₃	C₁₆×C₂²⋊C₄	semidirect
198. C₂²	#11718257	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#6092	2
C₂×C₂²⋊#6092	2
C₂×#11718257	2
#13129875	2
#13114610	2
C₁₆×C₂²⋊C₄	13