Subgroups — #15781038

Order 3328.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116630	C₂	1	2	yes	maximal
2	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
3	#13114715	C₂	1	1	yes	maximal, characteristic
4	#13112950	C₂	1	2	yes	maximal
5	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	—	2	1	no	—
7	C₂×C₁₃×#108	C₂²	1	2	yes	—
8	C₁₃×#423	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88059	C₂²	1	1	yes	characteristic
11	#11718767	C₂²	1	1	yes	characteristic
12	#11718762	—	2	1	no	—
13	C₂×#88052	C₂²	1	2	yes	—
14	#11718115	—	2	1	no	—
15	#11718085	—	2	4	no	—
16	#11718055	C₄	1	2	yes	complemented
17	#11718055	C₄	1	2	yes	complemented
18	C₂×C₁₃×M₁₆	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	#88180	—	2	1	no	—
23	C₂×#6099	—	2	1	no	—
24	C₁₃×#108	—	2	2	no	—
25	C₁₃×#108	—	2	2	no	—
26	C₁₃×#108	—	4	1	no	—
27	C₁₃×#108	—	4	1	no	—
28	C₁₃×C₄⋊C₈	—	4	2	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	#88059	—	2	1	no	—
34	#88059	D₄	1	2	yes	—
35	#88058	—	2	1	no	—
36	#88057	Q₈	1	2	yes	—
37	#88055	—	4	1	no	—
38	C₈×Dic₁₃	—	4	1	no	—
39	#88052	—	2	2	no	—
40	#88052	C₂×C₄	1	4	yes	—
41	C₂×#6090	—	2	2	no	—
42	#13182	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₁₃×M₁₆	—	4	2	no	—
46	C₁₃×M₁₆	—	4	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
60	C₄²×C₁₃	—	2	1	no	—
61	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	—	4	1	no	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	4	1	no	—
66	#6099	—	2	2	no	—
67	#6099	—	4	1	no	—
68	C₂×#1055	—	4	1	no	—
69	#6090	—	2	4	no	—
70	#2161	—	13	2	no	—
71	C₂×#364	—	13	1	no	—
72	#1848	D₁₃	1	1	yes	characteristic
73	#1772	—	13	1	no	—
74	#1389	—	13	2	no	—
75	C₂³×C₁₃	#137	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×Dic₁₃	—	4	2	no	—
86	C₈×C₁₃	—	8	2	no	—
87	C₈×C₁₃	#107	1	1	yes	characteristic
88	C₈×C₁₃	#107	1	1	yes	characteristic
89	C₈×C₁₃	#107	1	1	yes	characteristic
90	C₈×C₁₃	#107	1	1	yes	characteristic
91	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
92	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
93	C₈×C₁₃	#125	1	1	yes	characteristic
94	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
95	#1055	—	4	2	no	—
96	C₂×M₃₂	—	13	2	no	—
97	C₂×#108	—	2	1	no	—
98	C₂×#108	—	13	1	no	—
99	C₂×#108	C₂×D₁₃	1	2	yes	—
100	#427	—	13	1	no	—
101	#423	—	2	1	no	—
102	#405	—	26	1	no	—
103	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
104	#368	—	26	4	no	—
105	#364	—	13	2	no	—
106	#364	—	13	2	no	—
107	#339	—	26	1	no	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#428	1	1	yes	characteristic
110	C₂²×C₁₃	#462	1	1	yes	characteristic
111	C₂²×C₁₃	#474	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	4	1	no	—
114	C₄×C₁₃	#389	1	1	yes	characteristic
115	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
116	C₄×C₁₃	#459	1	1	yes	characteristic
117	C₄×C₁₃	#477	1	1	yes	characteristic
118	Dic₁₃	—	8	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	26	1	no	—
121	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	#117	—	26	1	no	—
124	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
125	M₃₂	—	13	4	no	—
126	M₃₂	—	26	2	no	—
127	C₂×C₁₆	—	26	2	no	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#108	—	4	1	no	—
131	#108	—	4	1	no	—
132	#108	—	13	2	no	—
133	#108	—	26	1	no	—
134	#107	—	26	1	no	—
135	#106	—	13	2	no	—
136	C₄⋊C₈	—	4	2	no	—
137	#97	—	52	1	no	—
138	C₄×C₈	—	52	1	no	—
139	C₄×C₈	Dic₂₆	1	1	yes	characteristic
140	C₄×C₈	Dic₂₆	1	1	yes	characteristic
141	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
142	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1947	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	M₁₆	—	4	2	no	—
151	M₁₆	—	26	2	no	—
152	M₁₆	—	52	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	52	1	no	—
155	C₂×C₈	#6101	1	1	yes	characteristic
156	C₂×C₈	#6101	1	1	yes	characteristic
157	C₂×C₈	#6104	1	1	yes	characteristic
158	C₂×C₈	#6104	1	1	yes	characteristic
159	C₂×C₈	#6107	1	1	yes	characteristic
160	C₂×C₈	#6107	1	1	yes	characteristic
161	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
163	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
164	C₂×C₈	#6129	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
166	C₂×C₈	#6131	1	1	yes	characteristic
167	C₄⋊C₄	—	26	2	no	—
168	C₂²⋊C₄	—	52	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	52	1	no	—
171	C₄²	#6101	1	1	yes	characteristic
172	C₄²	#6101	1	1	yes	characteristic
173	C₁₆	—	26	4	no	—
174	C₁₃	#13182	1	1	yes	characteristic, complemented, Sylow
175	C₂³	#88196	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	52	2	no	—
180	C₂×C₄	C₂×#6101	1	1	yes	characteristic
181	C₂×C₄	C₂×#6104	1	1	yes	characteristic
182	C₂×C₄	#88186	1	1	yes	characteristic
183	C₂×C₄	#88192	1	1	yes	characteristic
184	C₂×C₄	C₂×#6107	1	1	yes	characteristic
185	C₂×C₄	#88200	1	1	yes	characteristic
186	C₈	—	8	2	no	—
187	C₈	—	52	2	no	—
188	C₈	#88047	1	1	yes	characteristic
189	C₈	#88047	1	1	yes	characteristic
190	C₈	#88047	1	1	yes	characteristic
191	C₈	#88047	1	1	yes	characteristic
192	C₈	#88141	1	1	yes	characteristic
193	C₈	#88142	1	1	yes	characteristic
194	C₈	#88144	1	1	yes	characteristic
195	C₈	#88147	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#11718625	1	1	yes	characteristic
198	C₂²	#11718748	1	1	yes	characteristic
199	C₂²	#11718751	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	104	2	no	—
203	C₄	C₂×#88047	1	1	yes	characteristic
204	C₄	#11718634	1	1	yes	characteristic
205	C₄	#11718733	1	1	yes	characteristic
206	C₄	#11718752	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#13115592	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718733
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂×C₁₃	#1947
prime radical	C₂³×C₁₃	#137
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#13182
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
16. #11718055	C₄	semidirect
17. #11718055	C₄	semidirect
174. C₁₃	#13182	semidirect

Maximal subgroups

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

subgroup	index
#13116630	2
C₂×#11718055	2
#13114715	2
#13112950	2
C₁₃×#1848	2
#13182	13