Subgroups — #15762743

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	#13115797	C₂	1	2	yes	maximal
3	#13115638	C₂	1	2	yes	maximal
4	#13114964	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88058	—	2	2	no	—
11	C₂×#88056	—	2	4	no	—
12	#11718596	C₂²	1	2	yes	—
13	#11718343	—	2	2	no	—
14	#11718343	C₄	1	4	yes	complemented
15	C₁₃×#322	—	2	2	no	—
16	#11718143	—	2	2	no	—
17	#11718143	—	2	4	no	—
18	C₂²×Dic₂₆	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	2	yes	—
20	C₂²×C₈×C₁₃	D₄	1	2	yes	—
21	C₂²×C₈×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	#88123	—	2	2	no	—
29	#88123	—	2	2	no	—
30	#88123	—	2	2	no	—
31	#88123	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₃	—	2	1	no	—
33	C₄×C₈×C₁₃	—	2	1	no	—
34	C₄×C₈×C₁₃	—	2	1	no	—
35	C₄×C₈×C₁₃	—	2	1	no	—
36	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₃	D₄	1	2	yes	—
38	#88058	—	2	8	no	—
39	#88056	—	4	8	no	—
40	#9834	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	4	2	no	—
45	C₂²×Dic₁₃	—	4	2	no	—
46	C₂×Dic₂₆	—	2	4	no	—
47	C₂×Dic₂₆	—	4	4	no	—
48	C₂×Dic₂₆	—	4	4	no	—
49	C₂×Dic₂₆	—	4	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	4	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
57	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₃	#41	1	1	yes	characteristic
60	C₄²×C₁₃	#41	1	1	yes	characteristic
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	—	2	4	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	4	no	—
66	C₄⋊Dic₁₃	—	4	4	no	—
67	C₄⋊Dic₁₃	—	4	4	no	—
68	C₂×#500	—	13	1	no	—
69	#1990	—	13	2	no	—
70	#1956	—	13	2	no	—
71	#1834	—	13	1	no	—
72	C₂×#322	D₁₃	1	1	yes	characteristic
73	C₂³×C₁₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₁₃	#135	1	2	yes	—
82	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₄×C₁₃	#137	1	4	yes	—
84	C₂×Dic₁₃	—	4	4	no	—
85	C₂×Dic₁₃	—	8	2	no	—
86	C₂×Dic₁₃	—	8	2	no	—
87	C₂×Dic₁₃	—	8	2	no	—
88	C₂×Dic₁₃	—	8	2	no	—
89	C₂×Dic₁₃	—	8	2	no	—
90	Dic₂₆	—	4	8	no	—
91	Dic₂₆	—	8	4	no	—
92	C₈×C₁₃	—	2	4	no	—
93	C₈×C₁₃	—	2	4	no	—
94	C₈×C₁₃	—	4	2	no	—
95	C₂×C₄⋊Q₈	—	13	2	no	—
96	#500	—	13	4	no	—
97	#500	—	26	2	no	—
98	C₂×#107	—	26	2	no	—
99	C₂×#103	—	26	4	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
103	#384	—	13	2	no	—
104	#340	—	26	2	no	—
105	#340	—	26	4	no	—
106	#322	—	2	2	no	—
107	C₂²×C₁₃	#390	1	1	yes	characteristic
108	C₂²×C₁₃	#494	1	1	yes	characteristic
109	C₂²×C₁₃	#495	1	1	yes	characteristic
110	C₂²×C₁₃	#496	1	2	yes	—
111	C₂²×C₁₃	#497	1	2	yes	—
112	C₄×C₁₃	#419	1	2	yes	—
113	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
114	C₄×C₁₃	#442	1	2	yes	—
115	C₄×C₁₃	#467	1	2	yes	—
116	C₄×C₁₃	#469	1	2	yes	—
117	C₄×C₁₃	#491	1	2	yes	—
118	Dic₁₃	—	8	4	no	—
119	Dic₁₃	—	16	2	no	—
120	C₂²×Q₈	—	26	2	no	—
121	C₂²×C₈	D₅₂	1	2	yes	—
122	C₂²×C₈	D₅₂	1	2	yes	—
123	C₂²×C₈	D₅₂	1	2	yes	—
124	C₄⋊Q₈	—	13	4	no	—
125	C₄⋊Q₈	—	26	2	no	—
126	C₄⋊Q₈	—	26	2	no	—
127	C₄⋊Q₈	—	26	2	no	—
128	C₂×C₄⋊C₄	—	26	2	no	—
129	C₂×C₄⋊C₄	—	26	2	no	—
130	C₂×C₄⋊C₄	—	26	2	no	—
131	C₂×C₄⋊C₄	—	26	2	no	—
132	C₂×C₄⋊C₄	—	26	2	no	—
133	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
134	#107	—	26	8	no	—
135	#103	—	52	8	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	C₄×D₁₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₁₃	#1719	1	2	yes	—
143	C₂×C₁₃	#1723	1	2	yes	—
144	C₂×C₁₃	#1960	1	1	yes	characteristic
145	C₂×C₁₃	#1969	1	1	yes	characteristic
146	C₂×C₁₃	#1972	1	1	yes	characteristic
147	C₂×Q₈	—	26	4	no	—
148	C₂×Q₈	—	52	2	no	—
149	C₂×Q₈	—	52	4	no	—
150	C₂×Q₈	—	52	4	no	—
151	C₂²×C₄	—	52	2	no	—
152	C₂²×C₄	—	52	2	no	—
153	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	Dic₅₂	1	4	yes	—
162	C₂×C₈	#6103	1	4	yes	—
163	C₄⋊C₄	—	26	4	no	—
164	C₄⋊C₄	—	26	4	no	—
165	C₄⋊C₄	—	52	2	no	—
166	C₄⋊C₄	—	52	2	no	—
167	C₄⋊C₄	—	52	4	no	—
168	C₄⋊C₄	—	52	4	no	—
169	C₄⋊C₄	—	52	4	no	—
170	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₄²	#6127	1	1	yes	characteristic
172	C₄²	#6127	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
174	C₁₃	#9834	1	1	yes	characteristic, complemented, Sylow
175	C₂³	#88128	1	1	yes	characteristic
176	Q₈	—	52	8	no	—
177	Q₈	—	104	4	no	—
178	C₂×C₄	—	52	4	no	—
179	C₂×C₄	—	104	2	no	—
180	C₂×C₄	—	104	2	no	—
181	C₂×C₄	—	104	2	no	—
182	C₂×C₄	—	104	2	no	—
183	C₂×C₄	—	104	2	no	—
184	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
185	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
186	C₂×C₄	#88129	1	1	yes	characteristic
187	C₂×C₄	#88158	1	2	yes	—
188	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
189	C₂×C₄	#88162	1	4	yes	—
190	C₂×C₄	#88163	1	4	yes	—
191	C₂×C₄	C₂×#6103	1	1	yes	characteristic
192	C₂×C₄	#88184	1	1	yes	characteristic
193	C₂×C₄	#88184	1	1	yes	characteristic
194	C₈	—	2	4	no	—
195	C₈	—	2	4	no	—
196	C₈	—	4	2	no	—
197	C₂²	#11718357	1	1	yes	characteristic
198	C₂²	#11718360	1	1	yes	characteristic
199	C₂²	#11718373	1	2	yes	—
200	C₂²	#11718376	1	2	yes	—
201	C₂²	#11718600	1	1	yes	characteristic
202	C₄	—	104	4	no	—
203	C₄	—	208	2	no	—
204	C₄	C₄×Dic₅₂	1	2	yes	—
205	C₄	#11718364	1	2	yes	—
206	C₄	#11718372	1	2	yes	—
207	C₄	#11718777	1	2	yes	—
208	C₄	#11718781	1	2	yes	—
209	C₄	#11718796	1	2	yes	—
210	C₂	#13114489	1	2	yes	—
211	C₂	#13114513	1	2	yes	—
212	C₂	#13115656	1	1	yes	characteristic
213	C₂	#13115697	1	1	yes	characteristic
214	C₂	#13115712	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#9834
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
14. #11718343	C₄	semidirect
174. C₁₃	#9834	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718343	2
#13115797	2
#13115638	2
#13114964	2
C₂×C₁₃×#322	2
#9834	13