Subgroups — #15762934

Order 3328.

458 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88124	C₂	1	1	yes	maximal, characteristic
2	#13114988	C₂	1	1	yes	maximal, characteristic
3	#13112676	C₂	1	4	yes	maximal
4	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#88124	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	#11718597	C₄	1	1	yes	characteristic
9	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	#11718594	—	2	2	no	—
11	C₄×#6101	—	2	2	no	—
12	#11718589	C₄	1	2	yes	—
13	C₂×C₄⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	2	no	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂×#6101	—	2	2	no	—
25	C₂×#6101	—	2	2	no	—
26	C₂×C₄×Dic₁₃	—	2	2	no	—
27	C₂²×#1055	—	2	2	no	—
28	#88124	—	2	2	no	—
29	#88124	C₈	1	4	yes	complemented
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	Q₈	1	4	yes	—
32	#88071	—	2	2	no	—
33	#88071	—	2	2	no	—
34	#88071	—	2	2	no	—
35	C₄⋊#1055	—	2	4	no	—
36	C₄⋊#1055	—	4	4	no	—
37	#9872	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	4	4	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	4	2	no	—
51	C₄²×C₁₃	C₂×C₈	1	2	yes	—
52	C₄²×C₁₃	M₁₆	1	2	yes	—
53	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
54	C₄²×C₁₃	#41	1	2	yes	—
55	C₄⋊Dic₁₃	—	2	2	no	—
56	C₄⋊Dic₁₃	—	2	4	no	—
57	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
58	#6101	—	2	8	no	—
59	#6101	—	4	4	no	—
60	C₄×Dic₁₃	—	4	2	no	—
61	C₄×Dic₁₃	—	4	2	no	—
62	C₂×#1055	—	4	2	no	—
63	C₂×#1055	—	4	4	no	—
64	C₄×#125	—	13	1	no	—
65	#1842	—	13	1	no	—
66	#1744	D₁₃	1	1	yes	characteristic
67	#1279	—	13	4	no	—
68	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	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	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
76	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#131	1	2	yes	—
83	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₄×C₁₃	#137	1	2	yes	—
85	C₂×Dic₁₃	—	2	4	no	—
86	C₂×Dic₁₃	—	4	2	no	—
87	C₂×Dic₁₃	—	4	2	no	—
88	C₂×Dic₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₈×C₁₃	—	4	4	no	—
92	#1055	—	8	2	no	—
93	C₂×#125	—	13	1	no	—
94	C₂×C₄⋊C₈	—	13	2	no	—
95	C₂×C₄⋊C₈	—	26	2	no	—
96	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
97	#389	—	13	2	no	—
98	#387	—	26	2	no	—
99	#382	—	13	1	no	—
100	C₄×C₄⋊C₄	—	13	2	no	—
101	C₄×C₄⋊C₄	—	26	2	no	—
102	C₄³	C₂×D₁₃	1	1	yes	characteristic
103	#336	—	2	2	no	—
104	C₂²×C₁₃	#391	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₂²×C₁₃	#408	1	1	yes	characteristic
107	C₂²×C₁₃	#417	1	2	yes	—
108	C₂²×C₁₃	#421	1	2	yes	—
109	C₄×C₁₃	—	2	4	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₄×C₁₃	C₈×Q₈	1	2	yes	—
113	C₄×C₁₃	#446	1	2	yes	—
114	C₄×C₁₃	#490	1	1	yes	characteristic
115	C₄×C₁₃	#491	1	1	yes	characteristic
116	C₄×C₁₃	#501	1	2	yes	—
117	Dic₁₃	—	4	4	no	—
118	Dic₁₃	—	8	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	26	2	no	—
121	#125	—	13	4	no	—
122	#125	—	26	2	no	—
123	C₂×C₄⋊C₄	—	13	2	no	—
124	C₂×C₄⋊C₄	—	13	2	no	—
125	C₂×C₄⋊C₄	—	26	2	no	—
126	C₂×C₄⋊C₄	—	26	2	no	—
127	C₂×C₄²	—	26	2	no	—
128	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
129	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
130	C₂×C₄²	D₅₂	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
132	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
133	C₄⋊C₈	—	26	4	no	—
134	C₄⋊C₈	—	52	4	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	Dic₂₆	1	4	yes	—
137	#95	—	26	2	no	—
138	#95	—	26	2	no	—
139	#95	—	26	2	no	—
140	C₂×C₁₃	#1477	1	2	yes	—
141	C₂×C₁₃	#1495	1	2	yes	—
142	C₂×C₁₃	#1984	1	1	yes	characteristic
143	C₂×C₁₃	#1992	1	1	yes	characteristic
144	C₂×C₁₃	#1995	1	1	yes	characteristic
145	C₂²×C₄	—	26	2	no	—
146	C₂²×C₄	—	26	2	no	—
147	C₂²×C₄	#6103	1	2	yes	—
148	C₂²×C₄	#6103	1	2	yes	—
149	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	52	2	no	—
155	C₂×C₈	—	52	4	no	—
156	C₄⋊C₄	—	13	4	no	—
157	C₄⋊C₄	—	26	2	no	—
158	C₄⋊C₄	—	26	8	no	—
159	C₄⋊C₄	—	26	4	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	52	2	no	—
166	C₄²	—	52	2	no	—
167	C₄²	C₈×D₁₃	1	2	yes	—
168	C₄²	#6094	1	2	yes	—
169	C₄²	C₂×Dic₂₆	1	2	yes	—
170	C₄²	#6127	1	2	yes	—
171	C₁₃	#9872	1	1	yes	characteristic, complemented, Sylow
172	C₂³	C₂×#6103	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	26	4	no	—
180	C₂×C₄	—	52	2	no	—
181	C₂×C₄	—	52	2	no	—
182	C₂×C₄	—	52	2	no	—
183	C₂×C₄	—	52	2	no	—
184	C₂×C₄	—	52	2	no	—
185	C₂×C₄	#88060	1	4	yes	—
186	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
187	C₂×C₄	#88123	1	1	yes	characteristic
188	C₂×C₄	#88129	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂×#6094	1	1	yes	characteristic
191	C₂×C₄	#88155	1	2	yes	—
192	C₂×C₄	#88162	1	2	yes	—
193	C₂×C₄	#88163	1	2	yes	—
194	C₂×C₄	#88178	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	104	2	no	—
197	C₂²	#11718590	1	1	yes	characteristic
198	C₂²	#11718666	1	2	yes	—
199	C₂²	C₂×#88060	1	1	yes	characteristic
200	C₂²	#11718771	1	1	yes	characteristic
201	C₂²	#11718794	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	52	4	no	—
206	C₄	—	104	2	no	—
207	C₄	C₈×Dic₂₆	1	2	yes	—
208	C₄	#11718340	1	2	yes	—
209	C₄	#11718363	1	2	yes	—
210	C₄	#11718364	1	1	yes	characteristic
211	C₄	#11718374	1	1	yes	characteristic
212	C₂	#13113224	1	2	yes	—
213	C₂	#13113302	1	2	yes	—
214	C₂	#13115772	1	1	yes	characteristic
215	C₂	#13115805	1	1	yes	characteristic
216	C₂	#13115817	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#9872
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
29. #88124	C₈	semidirect
171. C₁₃	#9872	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88124	2
#13114988	2
#13112676	2
C₁₃×#1744	2
#9872	13