Subgroups — #261×#347 #16789021

Order 3328.

418 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131053	C₂	1	1	yes	maximal, characteristic
2	#13131051	C₂	1	2	yes	maximal
3	M₃₂×#261	C₂	1	2	yes	maximal
4	#97×#261	C₂	1	1	yes	maximal, characteristic
5	D₁₃×#347	C₂	1	1	yes	maximal, characteristic
6	C₂×#88100	C₄	1	1	yes	characteristic
7	C₂×#88100	C₂²	1	1	yes	characteristic
8	C₂×C₈×#261	C₄	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	#11719047	C₄	1	1	yes	characteristic
13	D₁₃×M₃₂	C₂²	1	2	yes	—
14	D₁₃×#97	C₂²	1	1	yes	characteristic
15	#11718262	C₄	1	2	yes	complemented
16	#11718232	C₄	1	2	yes	complemented
17	#11718218	C₄	1	4	yes	complemented
18	#11718217	C₄	1	4	yes	complemented
19	#11718216	C₄	1	4	yes	complemented
20	C₁₆×#261	C₄	1	4	yes	complemented
21	C₁₃×#347	C₄	1	1	yes	characteristic, complemented
22	#11718104	C₄	1	2	yes	complemented
23	#11718058	C₄	1	1	yes	characteristic, complemented
24	C₂×C₄×#261	—	2	2	no	—
25	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
27	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
31	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
32	#88110	—	2	2	no	—
33	#88106	C₂×C₄	1	4	yes	—
34	#88100	—	2	2	no	—
35	#88100	C₂×C₄	1	2	yes	—
36	#88100	C₂×C₄	1	2	yes	—
37	C₈×#261	—	2	2	no	—
38	C₈×#261	C₂×C₄	1	2	yes	—
39	C₈×#261	C₂×C₄	1	2	yes	—
40	#88098	C₂×C₄	1	4	yes	—
41	D₁₃⋊C₁₆	C₂×C₄	1	4	yes	—
42	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
43	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
44	#88055	C₂×C₄	1	1	yes	characteristic
45	#88055	C₂×C₄	1	1	yes	characteristic
46	#88052	C₂×C₄	1	2	yes	—
47	#88043	C₂×C₄	1	1	yes	characteristic
48	#88038	C₂×C₄	1	4	yes	—
49	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
50	C₄×#347	—	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	1	no	—
58	C₄×#261	—	4	4	no	—
59	C₄×#261	C₄²	1	2	yes	—
60	C₄×#261	C₄²	1	1	yes	characteristic
61	C₄×#261	C₄²	1	1	yes	characteristic
62	C₄×#261	M₁₆	1	2	yes	—
63	D₁₃⋊C₈	—	2	1	no	—
64	D₁₃⋊C₈	C₄²	1	1	yes	characteristic
65	D₁₃⋊C₈	C₄²	1	1	yes	characteristic
66	D₁₃⋊C₈	C₄²	1	2	yes	—
67	D₁₃⋊C₈	M₁₆	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	—	2	1	no	—
76	C₈×D₁₃	—	2	1	no	—
77	C₈×D₁₃	C₂²×C₄	1	2	yes	—
78	C₈×D₁₃	C₂²×C₄	1	2	yes	—
79	#6092	C₄²	1	8	yes	complemented
80	C₁₃×C₁₆	C₄²	1	4	yes	complemented
81	#6090	C₄²	1	4	yes	complemented
82	C₂×#347	—	13	2	no	—
83	C₂×#347	—	13	1	no	—
84	C₂×#347	—	13	1	no	—
85	C₄×M₃₂	—	13	2	no	—
86	C₄×#97	—	13	1	no	—
87	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×#261	—	2	4	no	—
89	C₂×#261	—	2	2	no	—
90	C₂×#261	—	4	2	no	—
91	C₂×#261	#97	1	2	yes	—
92	C₂×#261	#97	1	2	yes	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
97	C₄×D₁₃	—	4	2	no	—
98	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
99	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
100	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
102	#1057	—	2	1	no	—
103	#1057	—	2	1	no	—
104	#1057	#97	1	4	yes	—
105	C₈×C₁₃	—	2	1	no	—
106	C₈×C₁₃	C₂×C₄²	1	2	yes	—
107	#1055	—	2	1	no	—
108	#1055	C₂×C₄²	1	2	yes	—
109	C₂×M₃₂	—	13	2	no	—
110	C₂×M₃₂	—	13	2	no	—
111	C₂×#97	—	13	1	no	—
112	C₂×#97	—	13	1	no	—
113	C₂×#97	—	13	1	no	—
114	C₂×#97	—	13	1	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₄³	—	13	1	no	—
118	#347	—	13	4	no	—
119	#347	—	13	4	no	—
120	#347	—	13	2	no	—
121	#347	—	13	2	no	—
122	#347	—	13	2	no	—
123	#347	—	13	1	no	—
124	#347	#261	1	1	yes	characteristic, direct product factor
125	#346	—	13	4	no	—
126	C₄×C₁₆	—	13	4	no	—
127	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
128	C₂×D₁₃	—	2	2	no	—
129	C₂×D₁₃	C₂×#97	1	1	yes	characteristic
130	C₂×D₁₃	C₄×M₁₆	1	1	yes	characteristic
131	#261	—	2	2	no	—
132	#261	—	4	2	no	—
133	#261	#347	1	4	yes	direct product factor
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	C₄³	1	1	yes	characteristic
136	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
137	Dic₁₃	—	4	1	no	—
138	Dic₁₃	C₂×#97	1	1	yes	characteristic
139	Dic₁₃	C₄×M₁₆	1	1	yes	characteristic
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	1	no	—
142	C₂²×C₈	—	13	1	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂×C₄²	—	13	1	no	—
145	C₂×C₄²	—	13	1	no	—
146	C₂×C₄²	—	13	1	no	—
147	C₂×C₄²	—	26	2	no	—
148	M₃₂	—	13	4	no	—
149	M₃₂	—	13	4	no	—
150	M₃₂	—	13	2	no	—
151	M₃₂	—	13	4	no	—
152	M₃₂	C₂×#261	1	2	yes	—
153	C₂×C₁₆	—	13	4	no	—
154	C₂×C₁₆	—	13	4	no	—
155	#97	—	13	2	no	—
156	#97	—	13	2	no	—
157	#97	—	13	1	no	—
158	#97	—	13	1	no	—
159	#97	—	13	1	no	—
160	#97	—	26	2	no	—
161	#97	—	26	2	no	—
162	#97	C₂×#261	1	1	yes	characteristic
163	C₄×C₈	—	13	2	no	—
164	C₄×C₈	—	13	2	no	—
165	C₄×C₈	—	26	2	no	—
166	C₂×C₁₃	—	2	1	no	—
167	C₂×C₁₃	C₄×#97	1	1	yes	characteristic
168	D₁₃	—	2	1	no	—
169	D₁₃	C₂×#347	1	2	yes	—
170	C₂²×C₄	—	13	1	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	26	1	no	—
173	C₂²×C₄	—	26	1	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	13	1	no	—
181	C₂×C₈	—	13	1	no	—
182	C₂×C₈	—	13	1	no	—
183	C₂×C₈	—	26	1	no	—
184	C₂×C₈	—	26	1	no	—
185	C₂×C₈	—	26	1	no	—
186	C₂×C₈	—	26	1	no	—
187	C₂×C₈	C₄×#261	1	1	yes	characteristic
188	C₂×C₈	C₂²×#261	1	1	yes	characteristic
189	C₄²	—	13	1	no	—
190	C₄²	—	13	1	no	—
191	C₄²	—	13	2	no	—
192	C₄²	—	13	2	no	—
193	C₄²	—	13	1	no	—
194	C₄²	—	26	1	no	—
195	C₄²	—	26	1	no	—
196	C₄²	—	52	4	no	—
197	C₄²	C₄×#261	1	1	yes	characteristic
198	C₁₆	—	13	8	no	—
199	C₁₆	—	13	4	no	—
200	C₁₆	C₄×#261	1	4	yes	complemented
201	C₁₃	C₄×#347	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	13	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	13	1	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	13	1	no	—
208	C₂×C₄	—	13	1	no	—
209	C₂×C₄	—	13	1	no	—
210	C₂×C₄	—	13	2	no	—
211	C₂×C₄	—	26	4	no	—
212	C₂×C₄	—	26	2	no	—
213	C₂×C₄	—	26	1	no	—
214	C₂×C₄	—	52	2	no	—
215	C₂×C₄	—	52	2	no	—
216	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	13	4	no	—
219	C₈	—	13	2	no	—
220	C₈	—	26	1	no	—
221	C₈	—	26	1	no	—
222	C₈	—	26	1	no	—
223	C₈	C₂×C₄×#261	1	2	yes	—
224	C₂²	—	13	1	no	—
225	C₂²	—	13	1	no	—
226	C₂²	—	26	2	no	—
227	C₂²	M₁₆×#261	1	1	yes	characteristic
228	C₄	—	4	1	no	—
229	C₄	—	13	4	no	—
230	C₄	—	13	1	no	—
231	C₄	—	13	1	no	—
232	C₄	—	26	2	no	—
233	C₄	—	52	1	no	—
234	C₄	—	52	2	no	—
235	C₄	C₄²×#261	1	1	yes	characteristic
236	C₄	M₁₆×#261	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	13	2	no	—
239	C₂	—	26	1	no	—
240	C₂	#97×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#261
commutator	C₄×C₁₃	C₄³
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#347	C₄
socle	C₂×C₁₃	C₄×#97
prime radical	C₂²×D₁₃	C₂×M₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#347	#261
hypocenter	C₁₃	C₄×#347
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #347

Complements

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

N	H ≅ G/N	type
15. #11718262	C₄	semidirect
16. #11718232	C₄	semidirect
17. #11718218	C₄	semidirect
18. #11718217	C₄	semidirect
19. #11718216	C₄	semidirect
20. C₁₆×#261	C₄	semidirect
21. C₁₃×#347	C₄	semidirect
22. #11718104	C₄	semidirect
23. #11718058	C₄	semidirect
79. #6092	C₄²	semidirect
80. C₁₃×C₁₆	C₄²	semidirect
81. #6090	C₄²	semidirect
124. #347	#261	direct (N × H)
133. #261	#347	direct (N × H)
200. C₁₆	C₄×#261	semidirect
201. C₁₃	C₄×#347	semidirect

Maximal subgroups

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

subgroup	index
#13131053	2
#13131051	2
M₃₂×#261	2
#97×#261	2
D₁₃×#347	2
C₄×#347	13