Subgroups — #261×#364 #16789572

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#364	C₂	1	1	yes	maximal, characteristic
2	M₃₂×#261	C₂	1	2	yes	maximal
3	C₄×C₈×#261	C₂	1	1	yes	maximal, characteristic
4	#13129879	C₂	1	2	yes	maximal
5	D₁₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#261	C₄	1	1	yes	characteristic
7	C₂×C₈×#261	C₄	1	1	yes	characteristic
8	C₂×C₈×#261	C₂²	1	1	yes	characteristic
9	C₂×C₈×#261	C₂²	1	1	yes	characteristic
10	D₁₃⋊M₃₂	C₂²	1	2	yes	—
11	C₄²×#261	C₄	1	1	yes	characteristic
12	C₄×D₁₃⋊C₈	C₄	1	1	yes	characteristic
13	D₁₃×M₃₂	C₂²	1	2	yes	—
14	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
15	#11718261	C₄	1	2	yes	complemented
16	#11718233	C₄	1	2	yes	complemented
17	#11718216	—	2	2	no	—
18	C₁₆×#261	—	2	2	no	—
19	C₁₃×#364	C₄	1	1	yes	characteristic, complemented
20	#11718103	C₄	1	2	yes	complemented
21	#11718055	C₄	1	1	yes	characteristic, complemented
22	C₂×C₄×#261	—	2	2	no	—
23	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
25	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×#1057	—	2	2	no	—
31	#88106	C₂×C₄	1	4	yes	—
32	C₈×#261	—	2	1	no	—
33	C₈×#261	—	2	1	no	—
34	C₈×#261	C₈	1	4	yes	—
35	C₈×#261	C₈	1	4	yes	—
36	C₈×#261	D₄	1	1	yes	characteristic
37	C₈×#261	D₄	1	1	yes	characteristic
38	C₈×#261	Q₈	1	1	yes	characteristic
39	C₈×#261	Q₈	1	1	yes	characteristic
40	#88098	—	2	2	no	—
41	D₁₃⋊C₁₆	—	2	2	no	—
42	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
46	#88052	C₂×C₄	1	2	yes	—
47	C₄×#1055	C₂×C₄	1	1	yes	characteristic
48	#88038	—	2	2	no	—
49	C₁₆×D₁₃	—	2	2	no	—
50	C₄×#364	—	13	1	no	maximal, Sylow
51	C₂²×#261	—	2	1	no	—
52	C₂²×#261	M₁₆	1	2	yes	—
53	C₂×C₄×D₁₃	—	2	1	no	—
54	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×#1057	—	2	1	no	—
56	C₂×#1057	M₁₆	1	2	yes	—
57	C₄×#261	—	2	8	no	—
58	C₄×#261	—	2	1	no	—
59	C₄×#261	C₄⋊C₄	1	1	yes	characteristic
60	C₄×#261	C₄⋊C₄	1	1	yes	characteristic
61	C₄×#261	C₂×C₈	1	2	yes	—
62	C₄×#261	C₂×C₈	1	2	yes	—
63	D₁₃⋊C₈	—	2	1	no	—
64	D₁₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
65	D₁₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
66	D₁₃⋊C₈	C₂×C₈	1	2	yes	—
67	D₁₃⋊C₈	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₄²	1	1	yes	characteristic
71	C₄×Dic₁₃	—	2	1	no	—
72	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
73	C₂×#1055	C₄²	1	1	yes	characteristic
74	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
75	C₈×D₁₃	C₂×C₈	1	2	yes	—
76	C₈×D₁₃	C₂×C₈	1	2	yes	—
77	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
78	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₈×D₁₃	#41	1	1	yes	characteristic
80	C₈×D₁₃	#41	1	1	yes	characteristic
81	#6092	—	2	4	no	—
82	C₁₃×C₁₆	—	2	2	no	—
83	#6090	—	2	2	no	—
84	C₂×#364	—	13	1	no	—
85	C₂×#364	—	13	1	no	—
86	C₄×M₃₂	—	13	2	no	—
87	C₄²×C₈	—	13	1	no	—
88	#1376	—	13	2	no	—
89	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×#261	—	2	4	no	—
91	C₂×#261	—	2	2	no	—
92	C₂×#261	—	2	4	no	—
93	C₂×#261	C₄⋊C₈	1	2	yes	—
94	C₂×#261	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₄×D₁₃	—	2	4	no	—
100	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₁₃	#117	1	1	yes	characteristic
102	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
104	#1057	—	2	2	no	—
105	#1057	C₄⋊C₈	1	2	yes	—
106	#1057	C₄⋊C₈	1	2	yes	—
107	C₈×C₁₃	C₄×C₈	1	2	yes	—
108	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
110	#1055	C₄×C₈	1	2	yes	—
111	#1055	C₄×D₄	1	1	yes	characteristic
112	#1055	C₄×Q₈	1	1	yes	characteristic
113	C₂×M₃₂	—	13	2	no	—
114	C₂×M₃₂	—	13	2	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₄³	—	13	1	no	—
122	#364	—	13	2	no	—
123	#364	—	13	2	no	—
124	#364	—	13	2	no	—
125	#364	—	13	1	no	—
126	#364	#261	1	1	yes	characteristic, direct product factor
127	#346	—	26	2	no	—
128	C₄×C₁₆	—	26	2	no	—
129	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₂×D₁₃	#431	1	1	yes	characteristic
133	#261	—	2	4	no	—
134	#261	—	2	2	no	—
135	#261	#364	1	4	yes	direct product factor
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
141	Dic₁₃	#431	1	1	yes	characteristic
142	C₂²×C₈	—	13	1	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂²×C₈	—	13	1	no	—
145	C₂²×C₈	—	13	1	no	—
146	C₂×C₄²	—	13	1	no	—
147	C₂×C₄²	—	13	1	no	—
148	C₂×C₄²	—	13	1	no	—
149	C₂×C₄²	—	26	2	no	—
150	M₃₂	—	13	4	no	—
151	M₃₂	—	13	2	no	—
152	M₃₂	—	26	2	no	—
153	M₃₂	—	26	2	no	—
154	M₃₂	C₂×#261	1	2	yes	—
155	C₂×C₁₆	—	26	2	no	—
156	C₂×C₁₆	—	26	2	no	—
157	C₄×C₈	—	13	4	no	—
158	C₄×C₈	—	13	4	no	—
159	C₄×C₈	—	13	1	no	—
160	C₄×C₈	—	13	1	no	—
161	C₄×C₈	—	13	1	no	—
162	C₄×C₈	—	13	1	no	—
163	C₄×C₈	—	13	1	no	—
164	C₄×C₈	—	13	1	no	—
165	C₄×C₈	—	13	1	no	—
166	C₄×C₈	—	26	1	no	—
167	C₄×C₈	—	26	1	no	—
168	C₄×C₈	—	26	2	no	—
169	C₄×C₈	C₂×#261	1	1	yes	characteristic
170	C₂×C₁₃	—	2	1	no	—
171	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
172	D₁₃	—	2	1	no	—
173	D₁₃	C₂×#364	1	2	yes	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	13	2	no	—
176	C₂²×C₄	—	26	1	no	—
177	C₂²×C₄	—	26	1	no	—
178	C₂×C₈	—	13	1	no	—
179	C₂×C₈	—	13	1	no	—
180	C₂×C₈	—	13	2	no	—
181	C₂×C₈	—	13	2	no	—
182	C₂×C₈	—	13	1	no	—
183	C₂×C₈	—	13	1	no	—
184	C₂×C₈	—	13	2	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	13	1	no	—
187	C₂×C₈	—	13	1	no	—
188	C₂×C₈	—	13	1	no	—
189	C₂×C₈	—	13	1	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	26	1	no	—
192	C₂×C₈	—	26	1	no	—
193	C₂×C₈	C₄×#261	1	1	yes	characteristic
194	C₂×C₈	C₂²×#261	1	1	yes	characteristic
195	C₄²	—	13	1	no	—
196	C₄²	—	13	1	no	—
197	C₄²	—	13	2	no	—
198	C₄²	—	13	2	no	—
199	C₄²	—	13	1	no	—
200	C₄²	—	26	1	no	—
201	C₄²	—	26	1	no	—
202	C₄²	—	26	8	no	—
203	C₄²	C₄×#261	1	1	yes	characteristic
204	C₁₆	—	2	2	no	—
205	C₁₆	—	26	4	no	—
206	C₁₆	—	26	2	no	—
207	C₁₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	13	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	13	2	no	—
211	C₂×C₄	—	13	1	no	—
212	C₂×C₄	—	13	1	no	—
213	C₂×C₄	—	13	1	no	—
214	C₂×C₄	—	13	1	no	—
215	C₂×C₄	—	13	1	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	26	4	no	—
218	C₂×C₄	—	26	4	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	4	no	—
221	C₂×C₄	—	26	1	no	—
222	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
223	C₈	—	13	2	no	—
224	C₈	—	13	2	no	—
225	C₈	—	13	1	no	—
226	C₈	—	13	1	no	—
227	C₈	—	13	2	no	—
228	C₈	—	26	2	no	—
229	C₈	C₈×#261	1	2	yes	—
230	C₈	D₄×#261	1	1	yes	characteristic
231	C₈	Q₈×#261	1	1	yes	characteristic
232	C₂²	—	13	1	no	—
233	C₂²	—	13	1	no	—
234	C₂²	—	26	2	no	—
235	C₂²	M₁₆×#261	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	13	4	no	—
238	C₄	—	13	1	no	—
239	C₄	—	13	1	no	—
240	C₄	—	26	2	no	—
241	C₄	—	26	2	no	—
242	C₄	—	26	4	no	—
243	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
244	C₄	C₂×C₈×#261	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	13	2	no	—
247	C₂	—	26	1	no	—
248	C₂	C₄⋊C₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#364	C₄
socle	C₂×C₁₃	C₄×C₄⋊C₈
prime radical	C₂²×D₁₃	C₂×M₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#364	#261
hypocenter	C₁₃	C₄×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
15. #11718261	C₄	semidirect
16. #11718233	C₄	semidirect
19. C₁₃×#364	C₄	semidirect
20. #11718103	C₄	semidirect
21. #11718055	C₄	semidirect
126. #364	#261	direct (N × H)
135. #261	#364	direct (N × H)
207. C₁₃	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#364	2
M₃₂×#261	2
C₄×C₈×#261	2
#13129879	2
D₁₃×#364	2
C₄×#364	13