Subgroups — #15761950

Order 3328.

570 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	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718138	C₂	1	4	yes	maximal
4	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#384	—	2	1	no	—
8	C₁₃×#384	—	2	1	no	—
9	C₁₃×#384	—	2	2	no	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	—	2	2	no	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	—	2	1	no	—
15	#11718138	—	2	8	no	—
16	#11718116	—	2	1	no	—
17	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
18	C₁₃×C₄⋊Q₈	—	2	4	no	—
19	C₁₃×C₄⋊Q₈	—	2	4	no	—
20	C₁₃×C₄⋊Q₈	—	4	8	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×#1055	—	2	2	no	—
31	C₂²×#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	4	no	—
34	C₄⋊#1055	—	4	4	no	—
35	C₄⋊#1055	D₄	1	4	yes	—
36	C₄×#1055	—	4	2	no	—
37	#9691	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	—	2	8	no	—
39	C₂×Q₈×C₁₃	—	4	8	no	—
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	4	8	no	—
47	C₁₃×C₄⋊C₄	—	4	8	no	—
48	C₁₃×C₄⋊C₄	—	4	4	no	—
49	C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₁₃×C₄⋊C₄	—	4	2	no	—
51	C₄²×C₁₃	—	2	4	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	SD₁₆	1	2	yes	—
54	C₄²×C₁₃	SD₁₆	1	4	yes	—
55	C₄²×C₁₃	Q₁₆	1	4	yes	—
56	C₄²×C₁₃	Q₁₆	1	2	yes	—
57	C₄²×C₁₃	C₂×D₄	1	2	yes	—
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	4	2	no	—
61	C₂×#1055	—	4	2	no	—
62	C₂×#1055	—	4	2	no	—
63	C₂×#1055	—	4	4	no	—
64	#2868	D₁₃	1	1	yes	characteristic
65	#1842	—	13	1	no	—
66	C₄×C₄⋊C₈	—	13	1	no	—
67	C₂×#333	—	13	4	no	—
68	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	Q₈×C₁₃	—	8	8	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	—	4	8	no	—
75	C₂×C₄×C₁₃	#100	1	4	yes	—
76	C₂×C₄×C₁₃	#103	1	4	yes	—
77	C₂×C₄×C₁₃	#103	1	4	yes	—
78	C₂×C₄×C₁₃	#103	1	4	yes	—
79	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	#121	1	2	yes	—
81	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	#137	1	2	yes	—
88	#1055	—	4	4	no	—
89	#1055	—	8	2	no	—
90	C₂×C₄⋊Q₈	Dic₁₃	1	4	yes	—
91	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
92	C₂×C₄⋊C₈	—	13	2	no	—
93	C₂×C₄⋊C₈	—	13	2	no	—
94	C₂×C₄⋊C₈	—	26	2	no	—
95	C₂×C₄×C₈	—	26	1	no	—
96	#384	—	2	1	no	—
97	#384	—	2	1	no	—
98	#384	—	2	2	no	—
99	C₄³	C₂×D₁₃	1	1	yes	characteristic
100	#336	—	26	1	no	—
101	#333	—	26	8	no	—
102	C₂²×C₁₃	#390	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×#100	1	2	yes	—
104	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×#103	1	2	yes	—
106	C₂²×C₁₃	#419	1	1	yes	characteristic
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	8	4	no	—
110	C₄×C₁₃	#333	1	8	yes	complemented
111	C₄×C₁₃	#461	1	2	yes	—
112	C₄×C₁₃	#462	1	2	yes	—
113	C₄×C₁₃	#487	1	2	yes	—
114	C₄×C₁₃	#492	1	1	yes	characteristic
115	C₄×C₁₃	#494	1	1	yes	characteristic
116	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
117	C₂²×C₈	—	26	2	no	—
118	C₂²×C₈	—	26	2	no	—
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	2	4	no	—
121	C₄⋊Q₈	—	4	8	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊C₈	—	13	4	no	—
132	C₄⋊C₈	—	26	4	no	—
133	C₄⋊C₈	—	26	2	no	—
134	C₄⋊C₈	—	52	4	no	—
135	C₄×C₈	—	52	2	no	—
136	C₂×C₁₃	C₂×#333	1	2	yes	—
137	C₂×C₁₃	#1548	1	2	yes	—
138	C₂×C₁₃	#1956	1	1	yes	characteristic
139	C₂×C₁₃	#1965	1	1	yes	characteristic
140	C₂×C₁₃	#1981	1	1	yes	characteristic
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	4	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
149	C₂×C₈	—	26	4	no	—
150	C₂×C₈	—	52	2	no	—
151	C₂×C₈	—	52	2	no	—
152	C₂×C₈	—	52	2	no	—
153	C₂×C₈	—	52	4	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	#6106	1	2	yes	—
162	C₄²	#6106	1	4	yes	—
163	C₄²	#6107	1	4	yes	—
164	C₄²	#6107	1	2	yes	—
165	C₄²	D₄×D₁₃	1	2	yes	—
166	C₄²	#6129	1	2	yes	—
167	C₁₃	#9691	1	1	yes	characteristic, complemented, Sylow
168	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	#88073	1	4	yes	—
176	C₂×C₄	#88075	1	4	yes	—
177	C₂×C₄	#88075	1	4	yes	—
178	C₂×C₄	#88075	1	4	yes	—
179	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
180	C₂×C₄	#88190	1	1	yes	characteristic
181	C₂×C₄	#88192	1	2	yes	—
182	C₂×C₄	#88194	1	1	yes	characteristic
183	C₂×C₄	C₂×#6106	1	2	yes	—
184	C₂×C₄	C₂×#6106	1	1	yes	characteristic
185	C₂×C₄	#88196	1	2	yes	—
186	C₂×C₄	C₂×#6107	1	2	yes	—
187	C₂×C₄	C₂×#6107	1	1	yes	characteristic
188	C₈	—	52	4	no	—
189	C₈	—	104	2	no	—
190	C₂²	C₂×#88073	1	2	yes	—
191	C₂²	#11718882	1	1	yes	characteristic
192	C₂²	C₂×#88075	1	1	yes	characteristic
193	C₂²	C₂×#88075	1	2	yes	—
194	C₂²	#11718886	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	8	4	no	—
198	C₄	#11718138	1	8	yes	complemented
199	C₄	#11718742	1	2	yes	—
200	C₄	#11718744	1	2	yes	—
201	C₄	#11718746	1	1	yes	characteristic
202	C₄	#11718751	1	2	yes	—
203	C₄	#11718753	1	1	yes	characteristic
204	C₂	C₂×#11718138	1	2	yes	—
205	C₂	#13113517	1	2	yes	—
206	C₂	#13115639	1	1	yes	characteristic
207	C₂	#13115677	1	1	yes	characteristic
208	C₂	#13115760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2868	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2868	D₁₃
hypocenter	C₁₃	#9691
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
110. C₄×C₁₃	#333	semidirect
167. C₁₃	#9691	semidirect
198. C₄	#11718138	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
C₄×C₄⋊#1055	2
C₂×#11718138	2
C₁₃×#2868	2
#9691	13