Subgroups — #15781521

Order 3328.

414 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×C₄×#6090	C₂	1	1	yes	maximal, characteristic
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13112927	C₂	1	4	yes	maximal
4	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
6	C₁₃×#500	—	2	2	no	—
7	C₁₃×#500	C₄	1	4	yes	complemented
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₁₃×#107	C₂²	1	2	yes	—
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	C₄×#6090	—	2	2	no	—
16	C₂²×Q₈×C₁₃	—	4	1	no	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	—	2	4	no	—
20	C₁₃×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	4	1	no	—
26	C₂×C₄⋊Dic₁₃	—	4	1	no	—
27	C₁₃×#107	—	2	4	no	—
28	C₁₃×#107	—	2	2	no	—
29	C₁₃×#107	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃	D₄	1	2	yes	—
32	#88058	—	2	4	no	—
33	#88058	—	4	2	no	—
34	C₂×#6090	—	2	2	no	—
35	C₂×#6090	—	2	2	no	—
36	C₂×#6090	Q₈	1	4	yes	complemented
37	#13282	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	—	4	4	no	—
39	C₂×Q₈×C₁₃	—	8	2	no	—
40	C₂×Q₈×C₁₃	—	8	2	no	—
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₁₃	#41	1	2	yes	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	4	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₁₃	SD₁₆	1	2	yes	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	C₄⋊Dic₁₃	—	8	2	no	—
61	C₄⋊Dic₁₃	—	8	2	no	—
62	#6090	—	2	4	no	—
63	C₂×#500	D₁₃	1	1	yes	characteristic
64	C₂×C₄×C₁₆	—	13	1	no	—
65	#1845	—	13	1	no	—
66	#1380	—	13	4	no	—
67	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₁₃	—	8	8	no	—
69	C₂×C₄×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	—	8	2	no	—
71	C₂×C₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	#102	1	2	yes	—
73	C₂×C₄×C₁₃	#102	1	2	yes	—
74	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
75	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
76	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
77	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×Dic₁₃	—	8	2	no	—
85	C₂×Dic₁₃	—	8	2	no	—
86	C₂×Dic₁₃	—	8	2	no	—
87	C₈×C₁₃	#103	1	4	yes	—
88	C₈×C₁₃	C₄×Q₈	1	2	yes	—
89	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄⋊Q₈	—	2	1	no	—
91	C₂²×C₁₆	—	13	2	no	—
92	#500	—	2	2	no	—
93	#500	Dic₁₃	1	4	yes	complemented
94	C₂×#107	—	2	1	no	—
95	C₂×#107	—	13	2	no	—
96	C₂×#107	—	26	1	no	—
97	C₂×#107	C₂×D₁₃	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	#384	—	26	1	no	—
100	C₄×C₁₆	—	26	2	no	—
101	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
102	C₂²×C₁₃	#486	1	1	yes	characteristic
103	C₂²×C₁₃	#500	1	1	yes	characteristic
104	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
108	C₄×C₁₃	—	8	4	no	—
109	C₄×C₁₃	#357	1	4	yes	—
110	C₄×C₁₃	#358	1	4	yes	—
111	C₄×C₁₃	#391	1	1	yes	characteristic
112	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
113	C₄×C₁₃	#487	1	1	yes	characteristic
114	C₄×C₁₃	#498	1	1	yes	characteristic
115	Dic₁₃	—	16	2	no	—
116	C₂²×Q₈	—	4	1	no	—
117	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	4	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄⋊C₄	—	26	2	no	—
124	C₂×C₄⋊C₄	—	52	1	no	—
125	C₂×C₄⋊C₄	—	52	1	no	—
126	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
127	C₂×C₁₆	—	13	4	no	—
128	C₂×C₁₆	—	26	2	no	—
129	C₂×C₁₆	—	26	2	no	—
130	#107	—	2	4	no	—
131	#107	—	2	2	no	—
132	#107	—	26	4	no	—
133	#107	—	52	2	no	—
134	#107	C₂×Dic₁₃	1	4	yes	—
135	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₁₃	#1990	1	1	yes	characteristic
138	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
139	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
140	C₂×C₁₃	#2240	1	1	yes	characteristic
141	C₂×C₁₃	#2241	1	1	yes	characteristic
142	C₂×C₁₃	#2252	1	1	yes	characteristic
143	C₂×C₁₃	#2254	1	1	yes	characteristic
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	8	2	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	52	2	no	—
149	C₂²×C₄	#6104	1	1	yes	characteristic
150	C₂²×C₄	#6104	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
152	C₂×C₈	#6106	1	2	yes	—
153	C₂×C₈	#6107	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
155	C₂×C₈	Q₈×D₁₃	1	2	yes	—
156	C₂×C₈	#6131	1	2	yes	—
157	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₄⋊C₄	—	52	2	no	—
164	C₄⋊C₄	—	104	2	no	—
165	C₄⋊C₄	—	104	2	no	—
166	C₄²	#6105	1	2	yes	—
167	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
168	C₁₆	—	26	4	no	—
169	C₁₃	#13282	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂×#6104	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	104	2	no	—
176	C₂×C₄	—	104	2	no	—
177	C₂×C₄	—	104	2	no	—
178	C₂×C₄	#88066	1	2	yes	—
179	C₂×C₄	#88067	1	2	yes	—
180	C₂×C₄	#88068	1	2	yes	—
181	C₂×C₄	#88069	1	2	yes	—
182	C₂×C₄	#88072	1	2	yes	—
183	C₂×C₄	#88072	1	2	yes	—
184	C₂×C₄	C₂×#6105	1	1	yes	characteristic
185	C₂×C₄	C₂×#6106	1	1	yes	characteristic
186	C₂×C₄	C₂×#6107	1	1	yes	characteristic
187	C₂×C₄	#88198	1	1	yes	characteristic
188	C₂×C₄	#88201	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
190	C₈	#88075	1	4	yes	—
191	C₈	Q₈×Dic₁₃	1	2	yes	—
192	C₈	#88200	1	2	yes	—
193	C₂²	#11718739	1	1	yes	characteristic
194	C₂²	#11718749	1	1	yes	characteristic
195	C₂²	C₂×#88066	1	1	yes	characteristic
196	C₂²	C₂×#88067	1	1	yes	characteristic
197	C₂²	C₂×#88068	1	1	yes	characteristic
198	C₂²	C₂×#88069	1	1	yes	characteristic
199	C₂²	C₂×#88072	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	208	2	no	—
202	C₄	#11718154	1	4	yes	—
203	C₄	#11718156	1	4	yes	—
204	C₄	#11718740	1	1	yes	characteristic
205	C₄	#11718754	1	1	yes	characteristic
206	C₄	C₂×#88075	1	1	yes	characteristic
207	C₄	#11718891	1	1	yes	characteristic
208	C₂	#13115798	1	1	yes	characteristic
209	C₂	C₂×#11718154	1	1	yes	characteristic
210	C₂	C₂×#11718156	1	1	yes	characteristic
211	C₂	#13117061	1	1	yes	characteristic
212	C₂	#13117065	1	1	yes	characteristic
213	C₂	#13117109	1	1	yes	characteristic
214	C₂	#13117116	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×#500	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#13282
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
7. C₁₃×#500	C₄	semidirect
36. C₂×#6090	Q₈	semidirect
93. #500	Dic₁₃	semidirect
169. C₁₃	#13282	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6090	2
#13114996	2
#13112927	2
C₂×C₁₃×#500	2
#13282	13