Subgroups — #15761454

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115552	C₂	1	2	yes	maximal
2	C₄×#88057	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	#13112675	C₂	1	2	yes	maximal
5	C₂×C₁₃×#334	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×#88057	C₄	1	1	yes	characteristic
9	C₂×#88057	C₄	1	1	yes	characteristic
10	C₂×#88054	—	2	2	no	—
11	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₁₃×#334	—	2	2	no	—
15	#11718143	C₄	1	2	yes	—
16	#11718142	—	2	2	no	—
17	C₈×#1055	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₁₃	—	2	2	no	—
25	C₂²×#1055	—	2	1	no	—
26	C₂²×#1055	D₄	1	2	yes	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	#88071	—	2	2	no	—
32	#88057	—	2	2	no	—
33	#88057	C₈	1	4	yes	complemented
34	#88054	—	4	4	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	Q₈	1	2	yes	—
39	#9593	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	2	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₃	#41	1	1	yes	characteristic
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
62	C₄×Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	C₂×#1055	—	2	4	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	—	2	2	no	—
68	C₂×#1055	SD₁₆	1	4	yes	—
69	#1939	—	13	2	no	—
70	C₄×#106	—	13	1	no	—
71	C₂×#334	D₁₃	1	1	yes	characteristic
72	C₂×C₈²	—	13	1	no	—
73	#1279	—	13	2	no	—
74	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	#135	1	2	yes	—
88	C₂×Dic₁₃	—	2	4	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	4	2	no	—
94	C₈×C₁₃	—	2	2	no	—
95	C₈×C₁₃	—	8	2	no	—
96	C₈×C₁₃	C₄⋊C₈	1	4	yes	complemented
97	#1055	—	2	4	no	—
98	#1055	—	2	4	no	—
99	C₂×#106	—	13	1	no	—
100	C₂×#106	—	13	1	no	—
101	C₂×C₄⋊C₈	—	26	2	no	—
102	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	13	2	no	—
107	#340	—	13	2	no	—
108	#336	—	26	2	no	—
109	#334	—	2	2	no	—
110	C₈²	—	26	2	no	—
111	C₂²×C₁₃	#389	1	1	yes	characteristic
112	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
113	C₂²×C₁₃	#436	1	1	yes	characteristic
114	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
115	C₂²×C₁₃	#443	1	2	yes	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₄×C₁₃	#424	1	1	yes	characteristic
120	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
121	C₄×C₁₃	#446	1	1	yes	characteristic
122	C₄×C₁₃	#465	1	2	yes	—
123	C₄×C₁₃	#498	1	1	yes	characteristic
124	C₄×C₁₃	#499	1	1	yes	characteristic
125	Dic₁₃	—	4	4	no	—
126	Dic₁₃	—	8	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	13	2	no	—
129	C₂²×C₈	—	26	1	no	—
130	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	13	2	no	—
133	C₂×C₄⋊C₄	—	13	2	no	—
134	C₂×C₄²	—	26	2	no	—
135	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
136	#106	—	13	4	no	—
137	#106	—	26	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	52	4	no	—
141	C₄×C₈	—	13	2	no	—
142	C₄×C₈	—	26	1	no	—
143	C₄×C₈	—	26	1	no	—
144	C₄×C₈	—	26	1	no	—
145	C₄×C₈	Dic₂₆	1	2	yes	—
146	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
147	#95	—	26	2	no	—
148	C₂×C₁₃	C₈×SD₁₆	1	2	yes	—
149	C₂×C₁₃	#1590	1	2	yes	—
150	C₂×C₁₃	#1939	1	1	yes	characteristic
151	C₂×C₁₃	#1941	1	1	yes	characteristic
152	C₂×C₁₃	#1945	1	1	yes	characteristic
153	C₂²×C₄	—	26	2	no	—
154	C₂²×C₄	—	26	2	no	—
155	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
156	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	#6131	1	1	yes	characteristic
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	26	4	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	26	2	no	—
168	C₂×C₈	C₈×D₁₃	1	2	yes	—
169	C₂×C₈	#6094	1	2	yes	—
170	C₂×C₈	#6101	1	2	yes	—
171	C₂×C₈	#6101	1	2	yes	—
172	C₄⋊C₄	—	13	4	no	—
173	C₄⋊C₄	—	26	2	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₄²	—	52	2	no	—
177	C₄²	—	52	2	no	—
178	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
179	C₄²	#6129	1	1	yes	characteristic
180	C₄²	Q₈×D₁₃	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₁₃	#9593	1	1	yes	characteristic, complemented, Sylow
183	C₂³	#88147	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	26	4	no	—
186	C₂×C₄	—	52	2	no	—
187	C₂×C₄	—	52	2	no	—
188	C₂×C₄	—	52	2	no	—
189	C₂×C₄	—	52	2	no	—
190	C₂×C₄	—	52	2	no	—
191	C₂×C₄	#88141	1	1	yes	characteristic
192	C₂×C₄	#88142	1	1	yes	characteristic
193	C₂×C₄	#88144	1	1	yes	characteristic
194	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂×#6094	1	1	yes	characteristic
196	C₂×C₄	#88161	1	1	yes	characteristic
197	C₂×C₄	#88161	1	1	yes	characteristic
198	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
199	C₂×C₄	#88170	1	2	yes	—
200	C₂×C₄	C₂×#6101	1	1	yes	characteristic
201	C₂×C₄	#88192	1	1	yes	characteristic
202	C₂×C₄	#88200	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	8	2	no	—
205	C₈	—	26	4	no	—
206	C₈	—	26	4	no	—
207	C₈	#88054	1	4	yes	complemented
208	C₂²	#11718495	1	1	yes	characteristic
209	C₂²	#11718499	1	1	yes	characteristic
210	C₂²	#11718513	1	2	yes	—
211	C₂²	#11718556	1	2	yes	—
212	C₂²	#11718634	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	52	4	no	—
216	C₄	—	104	2	no	—
217	C₄	#11718491	1	1	yes	characteristic
218	C₄	#11718492	1	1	yes	characteristic
219	C₄	#11718516	1	1	yes	characteristic
220	C₄	#11718519	1	1	yes	characteristic
221	C₄	C₂×#88054	1	1	yes	characteristic
222	C₄	#11718780	1	1	yes	characteristic
223	C₄	#11718832	1	2	yes	—
224	C₂	#13113635	1	2	yes	—
225	C₂	#13113725	1	2	yes	—
226	C₂	#13115552	1	1	yes	characteristic
227	C₂	#13115562	1	1	yes	characteristic
228	C₂	#13115582	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#334	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#334	D₁₃
hypocenter	C₁₃	#9593
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
33. #88057	C₈	semidirect
96. C₈×C₁₃	C₄⋊C₈	semidirect
182. C₁₃	#9593	semidirect
207. C₈	#88054	semidirect

Maximal subgroups

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

subgroup	index
#13115552	2
C₄×#88057	2
C₂×C₈×#1055	2
#13112675	2
C₂×C₁₃×#334	2
#9593	13