Subgroups — #15781525

Order 3328.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
2	#13115000	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	C₂²	1	2	yes	—
12	C₂×#88057	—	2	1	no	—
13	C₂²×#6090	C₂²	1	2	yes	—
14	#11718597	—	2	1	no	—
15	#11718053	—	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	C₁₃×#107	—	2	4	no	—
26	C₁₃×#107	—	2	2	no	—
27	C₁₃×#107	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	#88071	—	4	1	no	—
31	#88071	—	4	1	no	—
32	#88058	—	2	4	no	—
33	#88057	—	4	2	no	—
34	C₂×#6090	—	2	2	no	—
35	C₂×#6090	—	2	2	no	—
36	C₂×#6090	Q₈	1	4	yes	complemented
37	#13283	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	—	4	4	no	—
39	C₂×Q₈×C₁₃	—	8	4	no	—
40	C₂²×C₄×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	4	2	no	—
45	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₃	#41	1	2	yes	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₁₃	SD₁₆	1	2	yes	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	#6090	—	2	4	no	—
60	C₂×#500	D₁₃	1	1	yes	characteristic
61	C₂×#346	—	13	1	no	—
62	#1846	—	13	1	no	—
63	#1380	—	13	4	no	—
64	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
65	Q₈×C₁₃	—	8	8	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	4	4	no	—
69	C₂×C₄×C₁₃	—	8	4	no	—
70	C₂×C₄×C₁₃	#102	1	2	yes	—
71	C₂×C₄×C₁₃	#102	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₁₃	—	8	2	no	—
79	C₂×Dic₁₃	—	8	2	no	—
80	C₂×Dic₁₃	—	8	2	no	—
81	C₈×C₁₃	#103	1	4	yes	—
82	C₈×C₁₃	C₄×Q₈	1	2	yes	—
83	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄⋊Q₈	—	2	1	no	—
85	C₂²×C₁₆	—	13	2	no	—
86	#500	—	2	2	no	—
87	#500	Dic₁₃	1	4	yes	complemented
88	C₂×#107	—	2	1	no	—
89	C₂×#107	—	13	2	no	—
90	C₂×#107	C₂×D₁₃	1	2	yes	—
91	C₂×#106	—	26	1	no	—
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	#382	—	26	1	no	—
94	#346	—	26	2	no	—
95	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
96	C₂²×C₁₃	#486	1	1	yes	characteristic
97	C₂²×C₁₃	#500	1	1	yes	characteristic
98	C₂²×C₁₃	#509	1	2	yes	—
99	C₂²×C₁₃	#510	1	2	yes	—
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	8	4	no	—
103	C₄×C₁₃	#391	1	1	yes	characteristic
104	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
105	C₄×C₁₃	#487	1	1	yes	characteristic
106	C₄×C₁₃	#498	1	1	yes	characteristic
107	Dic₁₃	—	16	2	no	—
108	C₂²×Q₈	—	4	1	no	—
109	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
111	C₄⋊Q₈	—	2	4	no	—
112	C₄⋊Q₈	—	4	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	4	1	no	—
115	C₂×C₄⋊C₄	—	26	2	no	—
116	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
117	C₂×C₁₆	—	13	4	no	—
118	C₂×C₁₆	—	26	2	no	—
119	C₂×C₁₆	—	26	2	no	—
120	#107	—	2	4	no	—
121	#107	—	2	2	no	—
122	#107	—	26	4	no	—
123	#107	C₂×Dic₁₃	1	4	yes	—
124	#106	—	52	2	no	—
125	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
126	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
127	#95	—	52	1	no	—
128	#95	—	52	1	no	—
129	C₂×C₁₃	#1990	1	1	yes	characteristic
130	C₂×C₁₃	#2140	1	1	yes	characteristic
131	C₂×C₁₃	#2141	1	1	yes	characteristic
132	C₂×C₁₃	#2243	1	2	yes	—
133	C₂×C₁₃	#2255	1	2	yes	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	52	2	no	—
138	C₂²×C₄	#6104	1	1	yes	characteristic
139	C₂²×C₄	#6104	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
141	C₂×C₈	#6106	1	2	yes	—
142	C₂×C₈	#6107	1	2	yes	—
143	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
144	C₂×C₈	Q₈×D₁₃	1	2	yes	—
145	C₂×C₈	#6131	1	2	yes	—
146	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	52	2	no	—
152	C₄⋊C₄	—	52	2	no	—
153	C₄²	#6105	1	2	yes	—
154	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
155	C₁₆	—	26	4	no	—
156	C₁₃	#13283	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂×#6104	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	104	2	no	—
164	C₂×C₄	—	104	2	no	—
165	C₂×C₄	—	104	2	no	—
166	C₂×C₄	#88072	1	2	yes	—
167	C₂×C₄	#88072	1	2	yes	—
168	C₂×C₄	C₂×#6105	1	1	yes	characteristic
169	C₂×C₄	C₂×#6106	1	1	yes	characteristic
170	C₂×C₄	C₂×#6107	1	1	yes	characteristic
171	C₂×C₄	#88198	1	1	yes	characteristic
172	C₂×C₄	#88201	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
174	C₈	#88075	1	4	yes	—
175	C₈	Q₈×Dic₁₃	1	2	yes	—
176	C₈	#88200	1	2	yes	—
177	C₂²	#11718739	1	1	yes	characteristic
178	C₂²	#11718749	1	1	yes	characteristic
179	C₂²	#11718808	1	2	yes	—
180	C₂²	#11718840	1	2	yes	—
181	C₂²	C₂×#88072	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	208	2	no	—
186	C₄	#11718740	1	1	yes	characteristic
187	C₄	#11718754	1	1	yes	characteristic
188	C₄	C₂×#88075	1	1	yes	characteristic
189	C₄	#11718891	1	1	yes	characteristic
190	C₂	#13115798	1	1	yes	characteristic
191	C₂	#13116552	1	1	yes	characteristic
192	C₂	#13116556	1	1	yes	characteristic
193	C₂	#13117075	1	2	yes	—
194	C₂	#13117119	1	2	yes	—

	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	#13283
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
87. #500	Dic₁₃	semidirect
156. C₁₃	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718053	2
#13115000	2
#13112927	2
C₂×C₁₃×#500	2
#13283	13