Subgroups — #15781138

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718156	C₂	1	4	yes	maximal
2	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
3	#13114996	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	—	2	1	no	—
6	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₁₃×#494	—	2	2	no	—
9	C₁₃×#494	C₄	1	4	yes	complemented
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88058	—	2	1	no	—
12	C₂×#88058	C₂²	1	2	yes	—
13	C₂²×#6090	C₂²	1	2	yes	—
14	#11718596	—	2	1	no	—
15	#11718156	—	2	8	no	—
16	C₄×#6090	—	2	2	no	—
17	C₂²×Q₈×C₁₃	—	2	2	no	—
18	C₂×C₁₃×Q₁₆	—	2	4	no	—
19	C₂×C₁₃×Q₁₆	—	2	4	no	—
20	C₂×C₁₃×Q₁₆	—	2	2	no	—
21	C₂×C₁₃×Q₁₆	—	2	4	no	—
22	C₂×C₁₃×Q₁₆	—	4	2	no	—
23	C₂×C₁₃×Q₁₆	—	4	2	no	—
24	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	—	2	4	no	—
28	C₁₃×C₄⋊Q₈	—	4	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	4	2	no	—
33	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₃	D₄	1	2	yes	—
35	#88058	—	2	4	no	—
36	#88058	—	4	2	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	2	2	no	—
39	C₂×#6090	D₄	1	4	yes	—
40	#13203	—	13	1	no	maximal, Sylow
41	C₂×Q₈×C₁₃	—	2	4	no	—
42	C₂×Q₈×C₁₃	—	4	2	no	—
43	C₂×Q₈×C₁₃	—	4	2	no	—
44	C₂×Q₈×C₁₃	—	4	2	no	—
45	C₂×Q₈×C₁₃	—	4	2	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	D₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₁₃×Q₁₆	—	2	8	no	—
52	C₁₃×Q₁₆	—	4	8	no	—
53	C₁₃×Q₁₆	—	4	4	no	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃	D₈	1	2	yes	—
56	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	4	8	no	—
62	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₃	SD₁₆	1	2	yes	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	8	4	no	—
67	#6090	—	2	4	no	—
68	C₂×#494	D₁₃	1	1	yes	characteristic
69	C₂×#358	—	13	4	no	—
70	C₂×C₄×C₁₆	—	13	1	no	—
71	#1845	—	13	1	no	—
72	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₁₃	—	4	4	no	—
74	Q₈×C₁₃	—	4	4	no	—
75	Q₈×C₁₃	—	8	2	no	—
76	Q₈×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	—	4	4	no	—
78	C₂×C₄×C₁₃	—	8	2	no	—
79	C₂×C₄×C₁₃	—	8	2	no	—
80	C₂×C₄×C₁₃	#102	1	4	yes	—
81	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
82	C₂×C₄×C₁₃	Q₃₂	1	4	yes	—
83	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×Dic₁₃	—	8	2	no	—
90	C₂×Dic₁₃	—	8	4	no	—
91	C₈×C₁₃	#102	1	4	yes	—
92	C₈×C₁₃	C₄×D₄	1	2	yes	—
93	C₈×C₁₃	#121	1	2	yes	—
94	C₂²×Q₁₆	—	2	1	no	—
95	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
96	C₂×C₄⋊Q₈	—	2	1	no	—
97	C₂²×C₁₆	—	13	2	no	—
98	#494	—	2	2	no	—
99	#494	Dic₁₃	1	4	yes	complemented
100	C₂×#107	—	13	2	no	—
101	C₂×#107	—	26	1	no	—
102	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
103	#384	—	26	1	no	—
104	#358	—	26	8	no	—
105	C₄×C₁₆	—	26	2	no	—
106	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
107	C₂²×C₁₃	#486	1	1	yes	characteristic
108	C₂²×C₁₃	#493	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
110	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
111	C₄×C₁₃	—	8	4	no	—
112	C₄×C₁₃	#358	1	8	yes	complemented
113	C₄×C₁₃	#390	1	1	yes	characteristic
114	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
115	C₄×C₁₃	#486	1	1	yes	characteristic
116	C₄×C₁₃	#492	1	1	yes	characteristic
117	Dic₁₃	—	16	2	no	—
118	C₂²×Q₈	—	2	2	no	—
119	C₂×Q₁₆	—	2	4	no	—
120	C₂×Q₁₆	—	2	4	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	2	no	—
123	C₂×Q₁₆	—	4	2	no	—
124	C₂×Q₁₆	—	4	2	no	—
125	C₂×Q₁₆	C₂×Dic₁₃	1	4	yes	—
126	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	4	no	—
129	C₄⋊Q₈	—	4	2	no	—
130	C₂×C₄⋊C₄	—	4	2	no	—
131	C₂×C₄⋊C₄	—	26	2	no	—
132	C₂×C₄⋊C₄	—	52	2	no	—
133	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
134	C₂×C₁₆	—	13	4	no	—
135	C₂×C₁₆	—	26	2	no	—
136	C₂×C₁₆	—	26	2	no	—
137	#107	—	26	4	no	—
138	#107	—	52	2	no	—
139	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
140	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
141	C₂×C₁₃	#1957	1	1	yes	characteristic
142	C₂×C₁₃	C₂×#358	1	2	yes	—
143	C₂×C₁₃	#2241	1	2	yes	—
144	C₂×C₁₃	#2247	1	1	yes	characteristic
145	C₂×C₁₃	#2248	1	1	yes	characteristic
146	C₂×Q₈	—	2	4	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	52	2	no	—
154	C₂²×C₄	#6104	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
156	Q₁₆	—	2	8	no	—
157	Q₁₆	—	4	8	no	—
158	Q₁₆	—	4	4	no	—
159	C₂×C₈	#6104	1	2	yes	—
160	C₂×C₈	#6105	1	2	yes	—
161	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
162	C₂×C₈	D₄×D₁₃	1	2	yes	—
163	C₂×C₈	#6129	1	2	yes	—
164	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	52	2	no	—
168	C₄⋊C₄	—	52	2	no	—
169	C₄⋊C₄	—	104	4	no	—
170	C₄²	#6105	1	2	yes	—
171	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
172	C₁₆	—	26	4	no	—
173	C₁₃	#13203	1	1	yes	characteristic, complemented, Sylow
174	C₂³	C₂×#6104	1	1	yes	characteristic
175	Q₈	—	4	4	no	—
176	Q₈	—	4	4	no	—
177	Q₈	—	8	2	no	—
178	Q₈	—	8	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	104	2	no	—
183	C₂×C₄	—	104	4	no	—
184	C₂×C₄	#88068	1	4	yes	—
185	C₂×C₄	#88069	1	4	yes	—
186	C₂×C₄	#88072	1	4	yes	—
187	C₂×C₄	C₂×#6104	1	1	yes	characteristic
188	C₂×C₄	C₂×#6105	1	1	yes	characteristic
189	C₂×C₄	C₂×#6105	1	1	yes	characteristic
190	C₂×C₄	#88190	1	1	yes	characteristic
191	C₂×C₄	#88194	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₈	#88072	1	4	yes	—
194	C₈	D₄×Dic₁₃	1	2	yes	—
195	C₈	#88192	1	2	yes	—
196	C₂²	#11718731	1	1	yes	characteristic
197	C₂²	#11718734	1	1	yes	characteristic
198	C₂²	C₂×#88068	1	2	yes	—
199	C₂²	C₂×#88069	1	2	yes	—
200	C₂²	C₂×#88072	1	1	yes	characteristic
201	C₄	—	8	4	no	—
202	C₄	—	208	2	no	—
203	C₄	#11718156	1	8	yes	complemented
204	C₄	#11718731	1	1	yes	characteristic
205	C₄	#11718737	1	1	yes	characteristic
206	C₄	C₂×#88072	1	1	yes	characteristic
207	C₄	#11718882	1	1	yes	characteristic
208	C₂	#13115643	1	1	yes	characteristic
209	C₂	C₂×#11718156	1	2	yes	—
210	C₂	#13117063	1	2	yes	—
211	C₂	#13117088	1	1	yes	characteristic
212	C₂	#13117093	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#13203
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
9. C₁₃×#494	C₄	semidirect
99. #494	Dic₁₃	semidirect
112. C₄×C₁₃	#358	semidirect
173. C₁₃	#13203	semidirect
203. C₄	#11718156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718156	2
C₂×C₄×#6090	2
#13114996	2
C₂×C₁₃×#494	2
#13203	13