Subgroups — #15780949

Order 3328.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
3	#13115569	C₂	1	2	yes	maximal
4	#13112928	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88058	C₂²	1	2	yes	—
11	#11718589	—	2	2	no	—
12	C₄×C₁₃×C₁₆	—	2	2	no	—
13	#11718143	—	2	2	no	—
14	#11718096	—	2	2	no	—
15	#11718096	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×#6101	—	4	2	no	—
22	C₂×C₁₃×C₁₆	—	2	1	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
26	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
27	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
28	C₄×C₈×C₁₃	—	2	1	no	—
29	C₄×C₈×C₁₃	—	2	1	no	—
30	#88071	—	4	2	no	—
31	#88058	—	2	2	no	—
32	#88058	—	2	2	no	—
33	#88058	—	2	2	no	—
34	#88058	C₂×C₄	1	4	yes	—
35	#13165	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	4	2	no	—
40	C₂²×Dic₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
49	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
50	C₄²×C₁₃	—	2	1	no	—
51	C₄²×C₁₃	—	2	1	no	—
52	C₄⋊Dic₁₃	—	2	4	no	—
53	C₄⋊Dic₁₃	—	4	2	no	—
54	C₄⋊Dic₁₃	—	4	2	no	—
55	C₄⋊Dic₁₃	—	4	2	no	—
56	#6101	—	8	4	no	—
57	C₁₃×C₁₆	—	2	2	no	—
58	C₁₃×C₁₆	C₄⋊C₄	1	4	yes	complemented
59	C₂×#366	—	13	1	no	—
60	C₂×#366	—	13	1	no	—
61	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
62	#1942	—	13	2	no	—
63	#1380	—	13	2	no	—
64	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
70	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
71	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
77	C₂×Dic₁₃	—	4	4	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₂×Dic₁₃	—	8	2	no	—
80	C₂×Dic₁₃	—	8	2	no	—
81	C₂×Dic₁₃	—	8	2	no	—
82	C₂×Dic₁₃	—	8	2	no	—
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₈×C₁₃	#125	1	1	yes	characteristic
88	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
89	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
90	C₂×#107	—	13	2	no	—
91	C₂×#107	—	13	2	no	—
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	#389	—	26	2	no	—
94	#366	—	13	4	no	—
95	#366	—	26	2	no	—
96	C₄×C₁₆	—	2	2	no	—
97	#340	—	26	2	no	—
98	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₁₃	#466	1	1	yes	characteristic
100	C₂²×C₁₃	#500	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₁₃	#508	1	1	yes	characteristic
104	C₂²×C₁₃	#508	1	1	yes	characteristic
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	#389	1	1	yes	characteristic
108	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
109	C₄×C₁₃	#467	1	1	yes	characteristic
110	C₄×C₁₃	#499	1	1	yes	characteristic
111	Dic₁₃	—	8	4	no	—
112	Dic₁₃	—	16	2	no	—
113	C₂²×C₈	D₅₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	26	2	no	—
116	C₂×C₄⋊C₄	—	26	2	no	—
117	C₂×C₄⋊C₄	—	52	2	no	—
118	C₂×C₄²	D₅₂	1	1	yes	characteristic
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	Dic₂₆	1	2	yes	—
122	C₂×C₁₆	Dic₂₆	1	2	yes	—
123	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
125	#107	—	13	4	no	—
126	#107	—	26	2	no	—
127	#107	—	26	2	no	—
128	#107	—	26	2	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	#95	—	52	2	no	—
132	C₂×C₁₃	#1942	1	1	yes	characteristic
133	C₂×C₁₃	C₄×D₁₆	1	1	yes	characteristic
134	C₂×C₁₃	C₄×Q₃₂	1	1	yes	characteristic
135	C₂×C₁₃	#2215	1	1	yes	characteristic
136	C₂×C₁₃	#2216	1	1	yes	characteristic
137	C₂×C₁₃	#2252	1	1	yes	characteristic
138	C₂×C₁₃	#2253	1	1	yes	characteristic
139	C₂²×C₄	—	52	2	no	—
140	C₂²×C₄	—	52	2	no	—
141	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
142	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	Dic₅₂	1	2	yes	—
147	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
150	C₂×C₈	#6127	1	1	yes	characteristic
151	C₂×C₈	#6127	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	26	4	no	—
154	C₄⋊C₄	—	52	2	no	—
155	C₄⋊C₄	—	52	2	no	—
156	C₄⋊C₄	—	52	2	no	—
157	C₄⋊C₄	—	104	4	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₁₆	—	2	2	no	—
161	C₁₆	#6101	1	4	yes	complemented
162	C₁₃	#13165	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	52	4	no	—
169	C₂×C₄	—	104	2	no	—
170	C₂×C₄	—	104	2	no	—
171	C₂×C₄	—	104	2	no	—
172	C₂×C₄	—	104	2	no	—
173	C₂×C₄	—	104	2	no	—
174	C₂×C₄	D₂₀₈	1	2	yes	—
175	C₂×C₄	Dic₁₀₄	1	2	yes	—
176	C₂×C₄	#88123	1	1	yes	characteristic
177	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
178	C₂×C₄	#88158	1	1	yes	characteristic
179	C₂×C₄	#88158	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
181	C₂×C₄	#88184	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₄×Dic₂₆	1	1	yes	characteristic
184	C₈	#88124	1	1	yes	characteristic
185	C₈	C₂×#6101	1	1	yes	characteristic
186	C₈	#88178	1	1	yes	characteristic
187	C₂²	#11718343	1	1	yes	characteristic
188	C₂²	C₄×D₁₀₄	1	1	yes	characteristic
189	C₂²	C₂×D₂₀₈	1	1	yes	characteristic
190	C₂²	#11718565	1	1	yes	characteristic
191	C₂²	#11718565	1	1	yes	characteristic
192	C₂²	C₂×Dic₁₀₄	1	1	yes	characteristic
193	C₂²	#11718776	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	104	4	no	—
197	C₄	—	208	2	no	—
198	C₄	#11718344	1	1	yes	characteristic
199	C₄	C₄×Dic₅₂	1	1	yes	characteristic
200	C₄	#11718589	1	1	yes	characteristic
201	C₄	#11718777	1	1	yes	characteristic
202	C₂	#13115569	1	1	yes	characteristic
203	C₂	C₄×D₂₀₈	1	1	yes	characteristic
204	C₂	C₄×Dic₁₀₄	1	1	yes	characteristic
205	C₂	#13116922	1	1	yes	characteristic
206	C₂	#13116927	1	1	yes	characteristic
207	C₂	#13117111	1	1	yes	characteristic
208	C₂	#13117115	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#13165
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. #11718096	C₄	semidirect
58. C₁₃×C₁₆	C₄⋊C₄	semidirect
161. C₁₆	#6101	semidirect
162. C₁₃	#13165	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718096	2
#13115569	2
#13112928	2
C₂×C₄×C₁₃×C₁₆	2
#13165	13