Subgroups — #15754103

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₁₃	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718071	C₂	1	2	yes	maximal
4	#13112671	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	—	2	2	no	—
9	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
11	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	#11718142	—	2	2	no	—
15	#11718071	C₄	1	8	yes	complemented
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₃	—	2	2	no	—
29	C₈×Dic₁₃	—	2	4	no	—
30	C₈×Dic₁₃	—	4	4	no	—
31	C₈×Dic₁₃	C₈	1	8	yes	complemented
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	C₂×C₄	1	4	yes	—
34	C₄×#1055	C₂×C₄	1	4	yes	—
35	C₄×#335	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
40	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	4	no	—
44	C₂×C₈×C₁₃	—	4	2	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
47	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
48	C₄²×C₁₃	#41	1	2	yes	—
49	C₄×Dic₁₃	—	2	2	no	—
50	C₄×Dic₁₃	—	2	4	no	—
51	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
52	C₄×Dic₁₃	M₁₆	1	4	yes	—
53	C₄×Dic₁₃	D₈	1	2	yes	—
54	C₄×Dic₁₃	Q₁₆	1	2	yes	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	4	no	—
57	C₂×#1055	—	4	2	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	C₄²	1	2	yes	—
60	C₂×#1055	C₄²	1	2	yes	—
61	C₂×#1055	C₂×C₈	1	4	yes	—
62	C₄×C₄⋊C₈	—	13	2	no	—
63	C₄²×C₈	—	13	1	no	—
64	C₂×#335	—	13	2	no	—
65	#1277	—	13	1	no	—
66	#1277	D₁₃	1	1	yes	characteristic
67	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#135	1	2	yes	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	—	2	2	no	—
80	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
81	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
82	C₂×Dic₁₃	#107	1	4	yes	—
83	C₂×Dic₁₃	#108	1	4	yes	—
84	C₈×C₁₃	—	2	4	no	—
85	C₈×C₁₃	—	4	4	no	—
86	#1055	—	2	2	no	—
87	#1055	—	4	4	no	—
88	#1055	C₄×C₈	1	4	yes	complemented
89	C₂×C₄⋊C₈	—	13	2	no	—
90	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
91	C₂×C₄×C₈	—	13	2	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	2	no	—
94	C₂×C₄×C₈	—	26	2	no	—
95	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
96	C₄³	—	13	1	no	—
97	#336	—	26	2	no	—
98	#335	—	13	8	no	—
99	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
101	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
102	C₂²×C₁₃	#427	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×#108	1	2	yes	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
106	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
107	C₄×C₁₃	#431	1	2	yes	—
108	C₄×C₁₃	C₄×D₈	1	2	yes	—
109	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
110	Dic₁₃	—	2	4	no	—
111	Dic₁₃	#335	1	8	yes	complemented
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	13	1	no	—
114	C₂²×C₈	—	13	1	no	—
115	C₂²×C₈	—	26	2	no	—
116	C₂²×C₈	C₄×D₁₃	1	2	yes	—
117	C₂×C₄²	—	13	2	no	—
118	C₂×C₄²	—	13	1	no	—
119	C₂×C₄²	—	13	1	no	—
120	C₂×C₄²	—	13	2	no	—
121	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	13	4	no	—
124	C₄⋊C₈	—	26	2	no	—
125	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	13	8	no	—
128	C₄×C₈	—	13	4	no	—
129	C₄×C₈	—	26	4	no	—
130	C₄×C₈	—	52	4	no	—
131	C₂×C₁₃	C₂×#335	1	2	yes	—
132	C₂×C₁₃	#1564	1	2	yes	—
133	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
134	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
135	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
136	C₂²×C₄	—	13	2	no	—
137	C₂²×C₄	—	13	2	no	—
138	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	—	13	2	no	—
149	C₂×C₈	—	26	2	no	—
150	C₂×C₈	—	26	4	no	—
151	C₂×C₈	—	52	2	no	—
152	C₂×C₈	—	52	2	no	—
153	C₂×C₈	C₈×D₁₃	1	4	yes	—
154	C₄²	—	13	4	no	—
155	C₄²	—	13	4	no	—
156	C₄²	—	13	2	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	26	4	no	—
159	C₄²	—	26	2	no	—
160	C₄²	C₂×C₄×D₁₃	1	2	yes	—
161	C₄²	#6129	1	2	yes	—
162	C₁₃	C₄×#335	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	13	4	no	—
166	C₂×C₄	—	13	4	no	—
167	C₂×C₄	—	13	4	no	—
168	C₂×C₄	—	13	4	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
172	C₂×C₄	#88131	1	1	yes	characteristic
173	C₂×C₄	#88135	1	2	yes	—
174	C₂×C₄	#88141	1	2	yes	—
175	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
176	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
177	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
178	C₂×C₄	#88170	1	2	yes	—
179	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	4	4	no	—
182	C₈	—	13	4	no	—
183	C₈	—	26	2	no	—
184	C₈	—	52	4	no	—
185	C₂²	#11718310	1	1	yes	characteristic
186	C₂²	D₁₃×C₄⋊C₈	1	2	yes	—
187	C₂²	#11718521	1	1	yes	characteristic
188	C₂²	D₁₃×#107	1	1	yes	characteristic
189	C₂²	D₁₃×#108	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	13	8	no	—
192	C₄	—	26	4	no	—
193	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
194	C₄	#11718368	1	1	yes	characteristic
195	C₄	#11718379	1	2	yes	—
196	C₄	#11718417	1	2	yes	—
197	C₄	#11718451	1	2	yes	—
198	C₂	D₁₃×#335	1	2	yes	—
199	C₂	#13113581	1	2	yes	—
200	C₂	#13114684	1	1	yes	characteristic
201	C₂	#13114728	1	1	yes	characteristic
202	C₂	#13114737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1277	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1277	D₁₃
hypocenter	C₁₃	C₄×#335
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
15. #11718071	C₄	semidirect
31. C₈×Dic₁₃	C₈	semidirect
88. #1055	C₄×C₈	semidirect
111. Dic₁₃	#335	semidirect
162. C₁₃	C₄×#335	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₁₃	2
C₄×C₈×Dic₁₃	2
C₂×#11718071	2
#13112671	2
C₁₃×#1277	2
C₄×#335	13