Subgroups — #16789570

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 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	#13131042	C₂	1	2	yes	maximal
3	#13129881	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	#11718258	—	2	2	no	—
14	#11718257	—	2	2	no	—
15	#11718233	C₄	1	8	yes	complemented
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	C₂×C₄	1	8	yes	—
26	C₂×#6092	—	2	4	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₈×Dic₁₃	C₈	1	4	yes	—
30	C₈×Dic₁₃	C₈	1	4	yes	—
31	C₈×Dic₁₃	D₄	1	1	yes	characteristic
32	C₈×Dic₁₃	D₄	1	1	yes	characteristic
33	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
34	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
35	C₄×#1055	C₂×C₄	1	4	yes	—
36	C₄×#364	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	2	1	no	—
40	C₂²×Dic₁₃	M₁₆	1	2	yes	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×C₈	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	C₄²	1	2	yes	—
50	C₄×Dic₁₃	—	2	8	no	—
51	C₄×Dic₁₃	—	2	1	no	—
52	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
55	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
56	C₂×#1055	C₄²	1	2	yes	—
57	C₂×#1055	C₂×C₈	1	2	yes	—
58	C₂×#1055	C₂×C₈	1	2	yes	—
59	C₂×#1055	C₂²×C₄	1	2	yes	—
60	C₂×#1055	C₂×D₄	1	1	yes	characteristic
61	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
62	C₂×#1055	#41	1	2	yes	—
63	#6092	—	2	8	no	—
64	C₂×#364	—	13	2	no	—
65	C₄×M₃₂	—	13	2	no	—
66	C₄²×C₈	—	13	1	no	—
67	#1376	—	13	2	no	—
68	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₁₃	—	2	4	no	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	—	2	4	no	—
80	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
81	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
82	C₈×C₁₃	—	2	2	no	—
83	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
84	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
85	#1055	C₄×C₈	1	4	yes	—
86	#1055	C₄×D₄	1	2	yes	—
87	#1055	C₄×Q₈	1	2	yes	—
88	C₂×M₃₂	—	13	4	no	—
89	C₂×C₄×C₈	—	13	1	no	—
90	C₂×C₄×C₈	—	13	1	no	—
91	C₂×C₄×C₈	—	13	1	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	C₂×C₄×C₈	#261	1	1	yes	characteristic
95	C₄³	—	13	1	no	—
96	#364	—	13	8	no	—
97	#346	—	26	2	no	—
98	C₄×C₁₆	—	26	2	no	—
99	C₂²×C₁₃	—	2	2	no	—
100	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₃	#431	1	1	yes	characteristic
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
106	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₁₃	#431	1	1	yes	characteristic
108	Dic₁₃	—	2	4	no	—
109	Dic₁₃	—	2	2	no	—
110	Dic₁₃	#364	1	4	yes	complemented
111	C₂²×C₈	—	13	1	no	—
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	C₂×#261	1	1	yes	characteristic
114	C₂²×C₈	C₂×#261	1	1	yes	characteristic
115	C₂×C₄²	—	13	1	no	—
116	C₂×C₄²	—	13	1	no	—
117	C₂×C₄²	—	26	2	no	—
118	C₂×C₄²	C₂×#261	1	1	yes	characteristic
119	M₃₂	—	13	8	no	—
120	M₃₂	—	26	4	no	—
121	C₂×C₁₆	—	26	4	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	13	4	no	—
124	C₄×C₈	—	13	4	no	—
125	C₄×C₈	—	13	1	no	—
126	C₄×C₈	—	13	1	no	—
127	C₄×C₈	—	13	1	no	—
128	C₄×C₈	—	13	1	no	—
129	C₄×C₈	—	13	4	no	—
130	C₄×C₈	—	26	2	no	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	—	2	1	no	—
133	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
134	C₂×C₁₃	C₂×#364	1	2	yes	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	13	2	no	—
137	C₂²×C₄	—	26	1	no	—
138	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	13	2	no	—
142	C₂×C₈	—	13	2	no	—
143	C₂×C₈	—	13	2	no	—
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	—	13	1	no	—
146	C₂×C₈	—	13	1	no	—
147	C₂×C₈	—	13	2	no	—
148	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
149	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
150	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
151	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
152	C₂×C₈	#6122	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	13	2	no	—
155	C₄²	—	13	1	no	—
156	C₄²	—	13	1	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	26	8	no	—
159	C₄²	—	26	1	no	—
160	C₄²	C₄×#261	1	2	yes	—
161	C₁₆	—	26	8	no	—
162	C₁₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
163	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	13	2	no	—
167	C₂×C₄	—	13	2	no	—
168	C₂×C₄	—	26	4	no	—
169	C₂×C₄	—	26	4	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
175	C₂×C₄	#88237	1	1	yes	characteristic
176	C₂×C₄	C₂×#6122	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	13	4	no	—
179	C₈	—	13	2	no	—
180	C₈	—	13	2	no	—
181	C₈	#88112	1	2	yes	—
182	C₈	#88112	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	#11719049	1	1	yes	characteristic
185	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
186	C₂²	C₄×#6122	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	13	4	no	—
189	C₄	—	26	4	no	—
190	C₄	—	26	2	no	—
191	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
192	C₄	C₄×C₄⋊#261	1	1	yes	characteristic
193	C₄	C₂×#88112	1	1	yes	characteristic
194	C₄	#11719125	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₄×#88112	1	1	yes	characteristic
198	C₂	D₁₃⋊#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄×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	C₄×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
15. #11718233	C₄	semidirect
110. Dic₁₃	#364	semidirect
162. C₁₃	C₄×#364	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
#13131042	2
#13129881	2
C₄×C₈×Dic₁₃	2
C₄×#364	13