Subgroups — #15761472

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115555	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
4	#13112677	C₂	1	2	yes	maximal
5	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#106	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#106	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄⋊#1055	—	2	2	no	—
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₁₃×#340	C₄	1	2	yes	—
15	#11718116	—	2	2	no	—
16	#11718050	—	2	2	no	—
17	#11718047	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	1	no	—
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	D₄	1	1	yes	characteristic
27	C₂²×#1055	D₄	1	1	yes	characteristic
28	C₁₃×#106	—	2	2	no	—
29	C₁₃×#106	C₈	1	4	yes	complemented
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	Q₈	1	2	yes	—
32	C₁₃×#95	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄⋊#1055	—	2	2	no	—
35	C₄⋊#1055	—	4	4	no	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	—	2	1	no	—
39	C₄×#1055	Q₈	1	2	yes	—
40	#9596	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
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	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
56	C₄²×C₁₃	—	4	2	no	—
57	C₄²×C₁₃	—	4	2	no	—
58	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₁₃	#41	1	1	yes	characteristic
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	—	4	2	no	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	—	4	2	no	—
70	C₂×#1055	—	4	2	no	—
71	#1939	—	13	2	no	—
72	C₄×#106	D₁₃	1	1	yes	characteristic
73	C₂×#335	—	13	1	no	—
74	C₂×#322	—	13	1	no	—
75	#1279	—	13	2	no	—
76	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	4	2	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	—	4	2	no	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄×Q₈	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	#125	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
94	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₄×C₁₃	#137	1	2	yes	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	C₄⋊C₈	1	4	yes	complemented
98	#1055	—	4	2	no	—
99	#1055	—	4	1	no	—
100	#1055	—	4	1	no	—
101	#1055	—	8	2	no	—
102	C₂×#106	Dic₁₃	1	1	yes	characteristic
103	C₂×#106	Dic₁₃	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	13	2	no	—
105	C₂×C₄⋊C₈	—	26	2	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
109	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
110	#340	Dic₁₃	1	2	yes	—
111	#336	—	26	2	no	—
112	#335	—	26	2	no	—
113	#322	—	26	2	no	—
114	C₂²×C₁₃	#389	1	1	yes	characteristic
115	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
116	C₂²×C₁₃	#436	1	1	yes	characteristic
117	C₂²×C₁₃	#440	1	2	yes	—
118	C₂²×C₁₃	#444	1	2	yes	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	4	4	no	—
122	C₄×C₁₃	—	8	2	no	—
123	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₄×C₁₃	#424	1	1	yes	characteristic
125	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
126	C₄×C₁₃	#446	1	1	yes	characteristic
127	C₄×C₁₃	#469	1	1	yes	characteristic
128	C₄×C₁₃	#470	1	1	yes	characteristic
129	C₄×C₁₃	#501	1	2	yes	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	26	1	no	—
133	C₂²×C₈	—	26	2	no	—
134	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
140	#106	—	2	2	no	—
141	#106	#1055	1	4	yes	complemented
142	C₄⋊C₈	—	26	2	no	—
143	C₄⋊C₈	—	26	2	no	—
144	C₄⋊C₈	—	52	4	no	—
145	C₄×C₈	—	13	2	no	—
146	C₄×C₈	—	26	1	no	—
147	C₄×C₈	—	26	1	no	—
148	C₄×C₈	—	26	1	no	—
149	C₄×C₈	Dic₂₆	1	2	yes	—
150	C₄×C₈	D₅₂	1	2	yes	—
151	#95	—	2	2	no	—
152	C₂×C₁₃	SD₁₆⋊C₈	1	2	yes	—
153	C₂×C₁₃	#1592	1	2	yes	—
154	C₂×C₁₃	#1939	1	1	yes	characteristic
155	C₂×C₁₃	#1944	1	1	yes	characteristic
156	C₂×C₁₃	#1946	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
160	C₂²×C₄	#6129	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	26	1	no	—
165	C₂×C₈	—	26	1	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	26	2	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	—	52	2	no	—
171	C₂×C₈	—	52	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	C₂×#1055	1	2	yes	—
174	C₂×C₈	#6099	1	2	yes	—
175	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
176	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₂×#1055	1	4	yes	—
181	C₄²	—	4	2	no	—
182	C₄²	—	4	2	no	—
183	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
184	C₄²	C₂×D₅₂	1	1	yes	characteristic
185	C₄²	Q₈×D₁₃	1	1	yes	characteristic
186	C₄²	#6131	1	1	yes	characteristic
187	C₁₃	#9596	1	1	yes	characteristic, complemented, Sylow
188	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#88142	1	1	yes	characteristic
197	C₂×C₄	#88144	1	1	yes	characteristic
198	C₂×C₄	#88149	1	1	yes	characteristic
199	C₂×C₄	#88151	1	1	yes	characteristic
200	C₂×C₄	#88168	1	2	yes	—
201	C₂×C₄	#88169	1	2	yes	—
202	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
203	C₂×C₄	C₂×#6099	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
205	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
206	C₂×C₄	#88202	1	1	yes	characteristic
207	C₂×C₄	#88202	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	52	2	no	—
210	C₈	—	52	1	no	—
211	C₈	—	52	1	no	—
212	C₈	—	104	2	no	—
213	C₈	C₄⋊#1055	1	4	yes	complemented
214	C₂²	#11718646	1	1	yes	characteristic
215	C₂²	D₄×#1055	1	1	yes	characteristic
216	C₂²	#11718673	1	1	yes	characteristic
217	C₂²	#11718825	1	2	yes	—
218	C₂²	#11718858	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	#11718517	1	2	yes	—
224	C₄	#11718526	1	1	yes	characteristic
225	C₄	#11718529	1	1	yes	characteristic
226	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
227	C₄	#11718660	1	1	yes	characteristic
228	C₄	Q₈×#1055	1	1	yes	characteristic
229	C₄	#11718685	1	1	yes	characteristic
230	C₂	SD₁₆⋊#1055	1	2	yes	—
231	C₂	#13113739	1	2	yes	—
232	C₂	#13115555	1	1	yes	characteristic
233	C₂	#13115580	1	1	yes	characteristic
234	C₂	#13115590	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#106	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#106	D₁₃
hypocenter	C₁₃	#9596
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
29. C₁₃×#106	C₈	semidirect
97. C₈×C₁₃	C₄⋊C₈	semidirect
141. #106	#1055	semidirect
187. C₁₃	#9596	semidirect
213. C₈	C₄⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
#13115555	2
C₂×#11718050	2
C₂×#11718047	2
#13112677	2
C₄×C₁₃×#106	2
#9596	13