Subgroups — #15781008

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718081	C₂	1	2	yes	maximal
2	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
3	#13114996	C₂	1	1	yes	maximal, characteristic
4	#13112927	C₂	1	2	yes	maximal
5	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
10	C₂×#88058	—	2	1	no	—
11	C₂×#88058	C₂²	1	2	yes	—
12	C₂²×#6090	C₂²	1	2	yes	—
13	#11718596	—	2	1	no	—
14	C₁₃×#340	—	2	1	no	—
15	#11718081	—	2	4	no	—
16	C₄⋊#6090	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
21	C₂×C₄²×C₁₃	—	4	1	no	—
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	4	1	no	—
25	C₂×C₄⋊Dic₁₃	—	4	1	no	—
26	C₁₃×#107	—	2	1	no	—
27	C₁₃×#107	—	2	1	no	—
28	C₁₃×#107	D₄	1	2	yes	—
29	C₁₃×#107	Q₈	1	2	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃	Q₈	1	2	yes	—
32	C₁₃×#95	—	4	1	no	—
33	#88058	—	2	2	no	—
34	#88058	—	2	2	no	—
35	#88058	—	4	2	no	—
36	C₂×#6090	—	2	2	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	C₂×C₄	1	4	yes	—
39	#13176	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	—	4	1	no	—
41	C₂²×C₄×C₁₃	—	4	1	no	—
42	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	4	1	no	—
58	C₁₃×C₄⋊C₄	—	4	1	no	—
59	C₄²×C₁₃	—	8	2	no	—
60	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₁₃	Q₁₆	1	2	yes	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	8	2	no	—
65	C₄⋊Dic₁₃	—	8	2	no	—
66	#6090	—	2	4	no	—
67	C₂×#366	—	13	2	no	—
68	C₂×C₄⋊C₁₆	—	13	1	no	—
69	#1845	—	13	1	no	—
70	C₄×#107	D₁₃	1	1	yes	characteristic
71	#1380	—	13	2	no	—
72	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	8	1	no	—
78	C₂×C₄×C₁₃	—	8	1	no	—
79	C₂×C₄×C₁₃	—	8	1	no	—
80	C₂×C₄×C₁₃	—	8	1	no	—
81	C₂×C₄×C₁₃	#107	1	2	yes	—
82	C₂×C₄×C₁₃	#107	1	2	yes	—
83	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
91	C₂×Dic₁₃	—	8	2	no	—
92	C₂×Dic₁₃	—	8	2	no	—
93	C₂×Dic₁₃	—	8	2	no	—
94	C₈×C₁₃	—	2	2	no	—
95	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
96	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₁₃	#125	1	1	yes	characteristic
98	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₂²×C₁₆	—	13	2	no	—
100	C₂×#107	—	13	2	no	—
101	C₂×#107	—	26	1	no	—
102	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
103	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	#384	—	26	1	no	—
106	C₄×C₄⋊C₄	—	2	1	no	—
107	#366	—	26	4	no	—
108	C₄⋊C₁₆	—	13	4	no	—
109	#340	—	2	1	no	—
110	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
111	C₂²×C₁₃	#459	1	1	yes	characteristic
112	C₂²×C₁₃	#474	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
115	C₂²×C₁₃	#509	1	1	yes	characteristic
116	C₂²×C₁₃	#510	1	1	yes	characteristic
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	8	2	no	—
119	C₄×C₁₃	—	8	2	no	—
120	C₄×C₁₃	#366	1	4	yes	complemented
121	C₄×C₁₃	#389	1	1	yes	characteristic
122	C₄×C₁₃	#428	1	1	yes	characteristic
123	C₄×C₁₃	#462	1	1	yes	characteristic
124	C₄×C₁₃	#477	1	1	yes	characteristic
125	Dic₁₃	—	16	2	no	—
126	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	26	2	no	—
131	C₂×C₄⋊C₄	—	52	1	no	—
132	C₂×C₄⋊C₄	—	52	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂×C₁₆	—	13	4	no	—
136	C₂×C₁₆	—	26	2	no	—
137	C₂×C₁₆	—	26	2	no	—
138	#107	—	2	1	no	—
139	#107	—	2	1	no	—
140	#107	—	26	2	no	—
141	#107	—	26	2	no	—
142	#107	—	52	2	no	—
143	#107	Dic₂₆	1	2	yes	—
144	#107	D₅₂	1	2	yes	—
145	C₄×C₈	Dic₂₆	1	2	yes	—
146	C₄×C₈	C₄×D₁₃	1	2	yes	—
147	#95	—	4	1	no	—
148	C₂×C₁₃	#1947	1	1	yes	characteristic
149	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
150	C₂×C₁₃	#2159	1	1	yes	characteristic
151	C₂×C₁₃	#2206	1	1	yes	characteristic
152	C₂×C₁₃	#2209	1	1	yes	characteristic
153	C₂×C₁₃	#2224	1	1	yes	characteristic
154	C₂×C₁₃	#2227	1	1	yes	characteristic
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	—	52	2	no	—
158	C₂²×C₄	#6104	1	1	yes	characteristic
159	C₂²×C₄	#6104	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	#6101	1	2	yes	—
164	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
167	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
168	C₂×C₈	#6127	1	1	yes	characteristic
169	C₂×C₈	#6127	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	52	2	no	—
175	C₄⋊C₄	—	52	2	no	—
176	C₄⋊C₄	—	104	2	no	—
177	C₄⋊C₄	—	104	2	no	—
178	C₄²	—	8	2	no	—
179	C₄²	#6101	1	2	yes	—
180	C₄²	#6107	1	2	yes	—
181	C₁₆	—	26	4	no	—
182	C₁₃	#13176	1	1	yes	characteristic, complemented, Sylow
183	C₂³	C₂×#6104	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	8	1	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	104	2	no	—
193	C₂×C₄	—	104	2	no	—
194	C₂×C₄	—	104	2	no	—
195	C₂×C₄	#88047	1	2	yes	—
196	C₂×C₄	#88047	1	2	yes	—
197	C₂×C₄	#88066	1	2	yes	—
198	C₂×C₄	#88069	1	2	yes	—
199	C₂×C₄	C₂×#6101	1	1	yes	characteristic
200	C₂×C₄	#88178	1	1	yes	characteristic
201	C₂×C₄	#88184	1	1	yes	characteristic
202	C₂×C₄	C₂×#6107	1	1	yes	characteristic
203	C₂×C₄	#88203	1	1	yes	characteristic
204	C₂×C₄	#88205	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₄×Dic₂₆	1	1	yes	characteristic
207	C₈	#88123	1	1	yes	characteristic
208	C₈	#88124	1	1	yes	characteristic
209	C₈	C₄×D₅₂	1	1	yes	characteristic
210	C₂²	C₂×#88047	1	1	yes	characteristic
211	C₂²	#11718668	1	1	yes	characteristic
212	C₂²	#11718676	1	1	yes	characteristic
213	C₂²	C₂×#88066	1	1	yes	characteristic
214	C₂²	C₂×#88069	1	1	yes	characteristic
215	C₂²	#11718854	1	1	yes	characteristic
216	C₂²	#11718856	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	208	2	no	—
221	C₄	#11718081	1	4	yes	complemented
222	C₄	#11718589	1	1	yes	characteristic
223	C₄	#11718656	1	1	yes	characteristic
224	C₄	#11718682	1	1	yes	characteristic
225	C₄	#11718692	1	1	yes	characteristic
226	C₂	#13115594	1	1	yes	characteristic
227	C₂	C₂×#11718081	1	1	yes	characteristic
228	C₂	#13116625	1	1	yes	characteristic
229	C₂	#13116871	1	1	yes	characteristic
230	C₂	#13116886	1	1	yes	characteristic
231	C₂	#13116973	1	1	yes	characteristic
232	C₂	#13116988	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×#107	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×#107	D₁₃
hypocenter	C₁₃	#13176
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
16. C₄⋊#6090	C₄	semidirect
120. C₄×C₁₃	#366	semidirect
182. C₁₃	#13176	semidirect
221. C₄	#11718081	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718081	2
C₂×C₄⋊#6090	2
#13114996	2
#13112927	2
C₄×C₁₃×#107	2
#13176	13