Subgroups — #16790254

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718233	C₂	1	2	yes	maximal
2	C₂×#11718261	C₂	1	2	yes	maximal
3	#13131170	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88106	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718261	—	2	4	no	—
14	#11718260	—	2	4	no	—
15	#11718233	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	—	2	2	no	—
20	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	#88106	—	2	4	no	—
25	#88106	—	2	4	no	—
26	C₂×#6092	—	2	4	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₈×Dic₁₃	C₈	1	2	yes	—
30	C₈×Dic₁₃	C₈	1	2	yes	—
31	C₈×Dic₁₃	D₄	1	2	yes	—
32	C₈×Dic₁₃	D₄	1	2	yes	—
33	C₈×Dic₁₃	Q₈	1	2	yes	—
34	C₈×Dic₁₃	Q₈	1	2	yes	—
35	C₄×#1055	D₄	1	2	yes	—
36	C₄×#1055	Q₈	1	2	yes	—
37	#12900	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₃	—	2	1	no	—
41	C₂²×Dic₁₃	M₁₆	1	2	yes	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
48	C₄²×C₁₃	—	2	1	no	—
49	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₄×Dic₁₃	—	2	8	no	—
51	C₄×Dic₁₃	—	2	1	no	—
52	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂×#1055	C₄⋊C₄	1	2	yes	—
57	C₂×#1055	C₂×C₈	1	2	yes	—
58	C₂×#1055	M₁₆	1	2	yes	—
59	C₂×#1055	C₂×D₄	1	1	yes	characteristic
60	C₂×#1055	C₂×D₄	1	1	yes	characteristic
61	C₂×#1055	C₂×D₄	1	1	yes	characteristic
62	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
63	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
64	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
65	#6092	—	4	4	no	—
66	C₂×#364	—	13	2	no	—
67	C₂×#364	—	13	2	no	—
68	#2150	—	13	2	no	—
69	C₄²×C₈	—	13	1	no	—
70	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×Dic₁₃	—	2	4	no	—
80	C₂×Dic₁₃	—	2	2	no	—
81	C₂×Dic₁₃	—	2	4	no	—
82	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
83	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
86	#1055	C₄⋊C₈	1	4	yes	—
87	#1055	#127	1	1	yes	characteristic
88	#1055	C₄⋊Q₈	1	1	yes	characteristic
89	#1055	C₄⋊Q₈	1	1	yes	characteristic
90	#1055	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×M₃₂	—	13	4	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	C₂×C₄×C₈	—	13	1	no	—
95	C₂×C₄×C₈	—	13	1	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₂×C₄×C₈	#261	1	1	yes	characteristic
98	C₄³	—	13	1	no	—
99	#364	—	26	4	no	—
100	#364	—	26	4	no	—
101	C₄⋊C₁₆	—	26	4	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₁₃	#423	1	1	yes	characteristic
105	C₂²×C₁₃	#423	1	1	yes	characteristic
106	C₄×C₁₃	—	2	4	no	—
107	C₄×C₁₃	#384	1	1	yes	characteristic
108	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₁₃	#423	1	1	yes	characteristic
111	Dic₁₃	—	2	4	no	—
112	Dic₁₃	—	2	2	no	—
113	Dic₁₃	#364	1	4	yes	complemented
114	C₂²×C₈	—	13	1	no	—
115	C₂²×C₈	—	13	1	no	—
116	C₂²×C₈	C₂×#261	1	1	yes	characteristic
117	C₂²×C₈	C₂×#261	1	1	yes	characteristic
118	C₂×C₄²	—	13	1	no	—
119	C₂×C₄²	—	13	1	no	—
120	C₂×C₄²	—	26	2	no	—
121	C₂×C₄²	C₂×#261	1	1	yes	characteristic
122	M₃₂	—	26	4	no	—
123	M₃₂	—	26	4	no	—
124	C₂×C₁₆	—	26	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	13	2	no	—
127	C₄×C₈	—	13	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₄×C₈	—	13	2	no	—
130	C₄×C₈	—	13	2	no	—
131	C₄×C₈	—	13	2	no	—
132	C₄×C₈	—	13	2	no	—
133	C₄×C₈	—	13	2	no	—
134	C₄×C₈	—	26	2	no	—
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	—	2	1	no	—
137	C₂×C₁₃	#1842	1	1	yes	characteristic
138	C₂×C₁₃	C₂×#364	1	2	yes	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	13	2	no	—
141	C₂²×C₄	—	26	1	no	—
142	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	13	1	no	—
148	C₂×C₈	—	13	1	no	—
149	C₂×C₈	—	13	2	no	—
150	C₂×C₈	—	13	1	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	1	no	—
154	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
155	C₂×C₈	C₄⋊#261	1	2	yes	—
156	C₂×C₈	C₄⋊#261	1	2	yes	—
157	C₂×C₈	#6122	1	2	yes	—
158	C₄²	—	2	1	no	—
159	C₄²	—	13	1	no	—
160	C₄²	—	13	1	no	—
161	C₄²	—	13	2	no	—
162	C₄²	—	13	2	no	—
163	C₄²	—	26	8	no	—
164	C₄²	—	26	1	no	—
165	C₄²	C₄⋊#261	1	2	yes	—
166	C₁₆	—	52	4	no	—
167	C₁₃	#12900	1	1	yes	characteristic, complemented, Sylow
168	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	13	2	no	—
172	C₂×C₄	—	13	2	no	—
173	C₂×C₄	—	26	4	no	—
174	C₂×C₄	—	26	4	no	—
175	C₂×C₄	—	26	2	no	—
176	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
177	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
181	C₂×C₄	C₂×#6122	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	13	4	no	—
184	C₈	—	13	1	no	—
185	C₈	—	13	1	no	—
186	C₈	—	13	1	no	—
187	C₈	—	13	1	no	—
188	C₈	#88112	1	4	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
191	C₂²	#11719053	1	1	yes	characteristic
192	C₂²	#11719129	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	13	4	no	—
195	C₄	—	26	4	no	—
196	C₄	—	26	2	no	—
197	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
198	C₄	#11719059	1	1	yes	characteristic
199	C₄	C₂×#88112	1	1	yes	characteristic
200	C₄	#11719129	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#13130769	1	1	yes	characteristic
204	C₂	D₁₃⋊#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#12900
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
113. Dic₁₃	#364	semidirect
167. C₁₃	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718233	2
C₂×#11718261	2
#13131170	2
C₄×C₈×Dic₁₃	2
#12900	13