Subgroups — #15778328

Order 3328.

358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718094	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₁₃	C₂	1	2	yes	maximal
4	#13112927	C₂	1	2	yes	maximal
5	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	C₂²	1	2	yes	—
8	C₂×#88058	C₂²	1	2	yes	—
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₁₃	—	2	2	no	—
12	C₁₃×#367	C₄	1	4	yes	complemented
13	#11718121	—	2	2	no	—
14	#11718096	C₄	1	4	yes	complemented
15	#11718094	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄⋊Dic₁₃	—	2	2	no	—
19	C₂×C₄×Dic₁₃	—	4	2	no	—
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₁₃×#107	—	2	2	no	—
26	C₁₃×#107	C₂×C₄	1	4	yes	—
27	#88071	—	4	2	no	—
28	#88058	—	2	2	no	—
29	#88058	C₂×C₄	1	4	yes	—
30	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₃	Q₈	1	2	yes	—
32	C₂×#6090	—	2	1	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	C₂×C₄	1	2	yes	—
35	#12674	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	4	2	no	—
39	C₂²×Dic₁₃	D₈	1	2	yes	—
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	4	2	no	—
48	C₄⋊Dic₁₃	—	2	4	no	—
49	C₄⋊Dic₁₃	—	4	2	no	—
50	C₄×Dic₁₃	—	8	4	no	—
51	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₁₃	Q₁₆	1	2	yes	—
53	C₂×#1055	—	2	1	no	—
54	C₂×#1055	—	2	1	no	—
55	C₂×#1055	C₄⋊C₄	1	2	yes	—
56	C₁₃×C₁₆	C₄²	1	4	yes	complemented
57	#6090	—	2	2	no	—
58	C₂×#367	D₁₃	1	1	yes	characteristic
59	C₂×#366	—	13	1	no	—
60	C₂×#346	—	13	1	no	—
61	C₄×#107	—	13	2	no	—
62	#1380	—	13	2	no	—
63	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	4	4	no	—
65	C₂×C₄×C₁₃	—	8	2	no	—
66	C₂×C₄×C₁₃	—	8	2	no	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
73	C₂×Dic₁₃	—	2	2	no	—
74	C₂×Dic₁₃	—	2	2	no	—
75	C₂×Dic₁₃	—	4	4	no	—
76	C₂×Dic₁₃	—	8	2	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	#107	1	4	yes	—
79	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
80	C₈×C₁₃	#117	1	1	yes	characteristic
81	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	#1055	—	2	2	no	—
84	C₂²×C₁₆	—	13	1	no	—
85	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
86	C₂×#107	—	13	2	no	—
87	C₂×#107	C₂×D₁₃	1	2	yes	—
88	C₂×C₄×C₈	—	13	1	no	—
89	C₄×C₄⋊C₄	—	26	2	no	—
90	#367	Dic₁₃	1	4	yes	complemented
91	#366	—	13	4	no	—
92	#346	—	13	4	no	—
93	#340	—	26	2	no	—
94	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
95	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
96	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₁₃	#509	1	2	yes	—
98	C₂²×C₁₃	#510	1	2	yes	—
99	C₄×C₁₃	—	8	4	no	—
100	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₁₃	#427	1	1	yes	characteristic
102	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
103	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
104	Dic₁₃	—	2	4	no	—
105	Dic₁₃	—	8	4	no	—
106	C₂²×C₈	—	13	1	no	—
107	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	26	2	no	—
110	C₂×C₄²	—	13	1	no	—
111	C₂×C₄²	—	52	2	no	—
112	C₂×C₁₆	—	13	2	no	—
113	C₂×C₁₆	—	26	1	no	—
114	C₂×C₁₆	—	26	1	no	—
115	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
116	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
118	#107	—	2	2	no	—
119	#107	—	13	4	no	—
120	#107	—	26	2	no	—
121	#107	C₂×Dic₁₃	1	4	yes	—
122	C₄×C₈	—	13	2	no	—
123	C₄×C₈	—	13	2	no	—
124	#95	—	52	2	no	—
125	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
126	C₂×C₁₃	#2159	1	2	yes	—
127	C₂×C₁₃	#2175	1	2	yes	—
128	C₂×C₁₃	#2176	1	1	yes	characteristic
129	C₂×C₁₃	#2177	1	1	yes	characteristic
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	13	2	no	—
132	C₂²×C₄	—	52	2	no	—
133	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
134	C₂×C₈	—	13	2	no	—
135	C₂×C₈	—	26	1	no	—
136	C₂×C₈	—	26	1	no	—
137	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
139	C₂×C₈	#6129	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
141	C₂×C₈	#6131	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	26	4	no	—
146	C₄⋊C₄	—	52	2	no	—
147	C₄²	—	13	2	no	—
148	C₄²	—	13	2	no	—
149	C₄²	—	104	4	no	—
150	C₁₆	—	26	2	no	—
151	C₁₆	C₄×Dic₁₃	1	4	yes	complemented
152	C₁₃	#12674	1	1	yes	characteristic, complemented, Sylow
153	C₂³	D₈×D₁₃	1	1	yes	characteristic
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	13	4	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	2	no	—
160	C₂×C₄	—	52	4	no	—
161	C₂×C₄	—	104	2	no	—
162	C₂×C₄	—	104	2	no	—
163	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
164	C₂×C₄	#88147	1	1	yes	characteristic
165	C₂×C₄	#88166	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
167	C₂×C₄	#88173	1	1	yes	characteristic
168	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
169	C₈	—	26	2	no	—
170	C₈	#88141	1	1	yes	characteristic
171	C₈	#88146	1	1	yes	characteristic
172	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
173	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
174	C₂²	#11718533	1	1	yes	characteristic
175	C₂²	D₁₃×#107	1	1	yes	characteristic
176	C₂²	#11718575	1	2	yes	—
177	C₂²	#11718576	1	2	yes	—
178	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
179	C₄	—	8	4	no	—
180	C₄	—	26	4	no	—
181	C₄	—	104	4	no	—
182	C₄	#11718534	1	1	yes	characteristic
183	C₄	#11718541	1	1	yes	characteristic
184	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
185	C₄	Q₁₆×Dic₁₃	1	1	yes	characteristic
186	C₂	#107×Dic₁₃	1	1	yes	characteristic
187	C₂	#13116623	1	2	yes	—
188	C₂	#13116701	1	2	yes	—
189	C₂	#13116706	1	1	yes	characteristic
190	C₂	#13116711	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₃
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#367	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#367	D₁₃
hypocenter	C₁₃	#12674
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
12. C₁₃×#367	C₄	semidirect
14. #11718096	C₄	semidirect
15. #11718094	C₄	semidirect
56. C₁₃×C₁₆	C₄²	semidirect
90. #367	Dic₁₃	semidirect
151. C₁₆	C₄×Dic₁₃	semidirect
152. C₁₃	#12674	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718096	2
C₂×#11718094	2
#107×Dic₁₃	2
#13112927	2
C₂×C₁₃×#367	2
#12674	13