Subgroups — #15778227

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×#11718093	C₂	1	2	yes	maximal
2	C₂×#11718094	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13112914	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	#11718094	—	2	4	no	—
16	#11718093	C₄	1	8	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	C₈	1	4	yes	—
32	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
33	C₈×Dic₁₃	D₄	1	2	yes	—
34	C₈×Dic₁₃	Q₈	1	2	yes	—
35	C₂×#6090	—	2	2	no	—
36	C₂×#6090	—	2	2	no	—
37	C₂×#6090	C₂×C₄	1	4	yes	—
38	C₄×#1055	—	2	2	no	—
39	#12653	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
44	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	#41	1	2	yes	—
52	C₄²×C₁₃	—	2	1	no	—
53	C₄²×C₁₃	—	2	1	no	—
54	C₄×Dic₁₃	—	2	4	no	—
55	C₄×Dic₁₃	—	2	4	no	—
56	C₄×Dic₁₃	C₄²	1	2	yes	—
57	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₁₃	M₁₆	1	2	yes	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	C₄²	1	2	yes	—
63	C₂×#1055	C₄⋊C₄	1	2	yes	—
64	C₂×#1055	C₂×C₈	1	2	yes	—
65	C₂×#1055	M₁₆	1	2	yes	—
66	C₁₃×C₁₆	—	2	4	no	—
67	#6090	—	2	4	no	—
68	C₂×C₄⋊C₁₆	—	13	2	no	—
69	C₂×#346	—	13	2	no	—
70	C₄²×C₈	—	13	1	no	—
71	#1374	—	13	1	no	—
72	#1374	D₁₃	1	1	yes	characteristic
73	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
87	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
88	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
89	C₂×Dic₁₃	M₃₂	1	4	yes	—
90	C₈×C₁₃	—	2	2	no	—
91	C₈×C₁₃	C₄×D₄	1	2	yes	—
92	C₈×C₁₃	C₄×Q₈	1	2	yes	—
93	#1055	—	2	2	no	—
94	#1055	C₄⋊C₈	1	4	yes	—
95	C₂²×C₁₆	—	13	2	no	—
96	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
103	C₄³	—	13	1	no	—
104	C₄⋊C₁₆	—	13	8	no	—
105	#346	—	26	4	no	—
106	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₁₃	#431	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
110	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₁₃	#431	1	1	yes	characteristic
116	Dic₁₃	—	2	2	no	—
117	Dic₁₃	—	2	4	no	—
118	Dic₁₃	#346	1	4	yes	complemented
119	C₂²×C₈	—	13	1	no	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₄²	—	13	1	no	—
124	C₂×C₄²	—	13	1	no	—
125	C₂×C₄²	—	13	2	no	—
126	C₂×C₄²	—	13	2	no	—
127	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	13	4	no	—
131	C₂×C₁₆	—	26	2	no	—
132	C₂×C₁₆	—	26	2	no	—
133	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	13	4	no	—
136	C₄×C₈	—	13	4	no	—
137	C₄×C₈	—	13	2	no	—
138	C₄×C₈	—	13	2	no	—
139	C₄×C₈	—	26	2	no	—
140	C₄×C₈	—	26	2	no	—
141	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
143	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
144	C₂×C₁₃	#2150	1	2	yes	—
145	C₂×C₁₃	#2163	1	2	yes	—
146	C₂²×C₄	—	13	1	no	—
147	C₂²×C₄	—	13	2	no	—
148	C₂²×C₄	—	13	1	no	—
149	C₂²×C₄	C₈×D₁₃	1	2	yes	—
150	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	13	2	no	—
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	26	1	no	—
158	C₂×C₈	—	26	1	no	—
159	C₂×C₈	C₈×D₁₃	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
161	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
162	C₂×C₈	#6129	1	2	yes	—
163	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	13	2	no	—
167	C₄²	—	13	2	no	—
168	C₄²	—	13	2	no	—
169	C₄²	—	13	2	no	—
170	C₄²	—	26	4	no	—
171	C₄²	—	26	4	no	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	26	4	no	—
174	C₁₃	#12653	1	1	yes	characteristic, complemented, Sylow
175	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	13	4	no	—
179	C₂×C₄	—	13	2	no	—
180	C₂×C₄	—	13	4	no	—
181	C₂×C₄	—	13	2	no	—
182	C₂×C₄	—	26	2	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	—	26	2	no	—
185	C₂×C₄	#88038	1	4	yes	—
186	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
187	C₂×C₄	#88131	1	1	yes	characteristic
188	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
190	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	13	4	no	—
194	C₈	—	26	2	no	—
195	C₈	#88135	1	2	yes	—
196	C₈	#88141	1	2	yes	—
197	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
198	C₂²	#11718379	1	1	yes	characteristic
199	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
200	C₂²	C₂×#88038	1	2	yes	—
201	C₂²	D₁₃×M₃₂	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	13	4	no	—
204	C₄	—	26	2	no	—
205	C₄	—	26	4	no	—
206	C₄	#11718310	1	1	yes	characteristic
207	C₄	#11718368	1	1	yes	characteristic
208	C₄	#11718379	1	1	yes	characteristic
209	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
210	C₂	#13114684	1	1	yes	characteristic
211	C₂	D₁₃×#346	1	1	yes	characteristic
212	C₂	C₄×#88038	1	1	yes	characteristic
213	C₂	#13116590	1	2	yes	—
214	C₂	#13116636	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#1374	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1374	D₁₃
hypocenter	C₁₃	#12653
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
16. #11718093	C₄	semidirect
118. Dic₁₃	#346	semidirect
174. C₁₃	#12653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718093	2
C₂×#11718094	2
C₄×C₈×Dic₁₃	2
#13112914	2
C₁₃×#1374	2
#12653	13