Subgroups — #16807068

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131202	C₂	1	4	yes	maximal, complemented
2	#13131194	C₂	1	4	yes	maximal, complemented
3	#13131173	C₂	1	2	yes	maximal, complemented
4	#13131066	C₂	1	2	yes	maximal
5	#13131042	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718257	C₂	1	1	yes	maximal, characteristic, complemented
7	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
9	#41×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×#88202	C₂²	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	2	yes	—
12	C₂²×#6092	C₂²	1	2	yes	—
13	#11718783	C₂²	1	1	yes	characteristic
14	#11718783	C₂²	1	2	yes	—
15	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
16	M₁₆×Dic₁₃	C₄	1	2	yes	—
17	#11718762	C₄	1	2	yes	—
18	#11718762	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	C₂²⋊#6092	C₂²	1	8	yes	—
22	#11718260	C₂²	1	8	yes	—
23	#11718258	C₂²	1	2	yes	—
24	#11718257	C₂²	1	2	yes	—
25	#11718257	C₂²	1	2	yes	—
26	#11718257	C₂²	1	2	yes	—
27	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
28	C₁₃×#131	—	2	4	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	#88202	—	2	4	no	—
34	Q₈×Dic₁₃	C₈	1	2	yes	—
35	D₄×Dic₁₃	C₈	1	4	yes	—
36	D₄×Dic₁₃	C₈	1	2	yes	—
37	#88180	C₂×C₄	1	2	yes	—
38	#88180	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#6099	C₂×C₄	1	1	yes	characteristic
42	C₂×#6099	C₂×C₄	1	2	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₂²×#1055	C₂³	1	1	yes	characteristic
45	#88106	—	2	4	no	—
46	C₂×#6092	—	2	4	no	—
47	C₂×#6092	C₂³	1	4	yes	—
48	C₂×#6092	C₂³	1	4	yes	—
49	#88055	C₂×C₄	1	2	yes	—
50	#88055	C₂×C₄	1	2	yes	—
51	#88055	C₂×C₄	1	2	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₈×Dic₁₃	C₂³	1	2	yes	—
59	#33420	—	13	1	no	maximal, Sylow
60	C₁₃×#41	—	2	2	no	—
61	C₁₃×#41	—	2	2	no	—
62	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
68	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
69	C₁₃×M₁₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
77	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
78	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
79	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
80	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
81	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
82	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
85	#6099	—	2	2	no	—
86	#6099	—	2	4	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	C₂²×C₄	1	2	yes	—
89	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
90	C₂×#1055	#41	1	4	yes	—
91	C₂×#1055	C₂⁴	1	1	yes	characteristic
92	#6092	—	2	4	no	—
93	#6092	—	2	4	no	—
94	C₄×#131	—	13	1	no	—
95	#2164	—	13	4	no	—
96	#2162	—	13	4	no	—
97	#2151	—	13	2	no	—
98	#2113	—	13	2	no	—
99	C₄×M₃₂	—	13	1	no	—
100	C₂×C₄×C₁₆	—	13	1	no	—
101	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	2	no	—
104	D₄×C₁₃	—	2	4	no	—
105	D₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	#117	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	2	8	no	—
115	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
116	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
117	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
120	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
121	#1055	—	2	2	no	—
122	#1055	#117	1	2	yes	—
123	#1055	C₂×#41	1	2	yes	—
124	C₂×#131	—	13	1	no	—
125	C₂×#131	#261	1	1	yes	characteristic
126	C₄×#41	—	13	1	no	—
127	C₂×M₃₂	—	13	2	no	—
128	C₂²×C₁₆	—	13	2	no	—
129	#405	—	13	1	no	—
130	#405	—	13	2	no	—
131	#405	—	13	2	no	—
132	#405	—	13	1	no	—
133	C₄×M₁₆	—	13	2	no	—
134	C₄×M₁₆	—	13	1	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	C₂×C₄×C₈	—	13	2	no	—
137	C₄⋊C₁₆	—	13	8	no	—
138	C₂²⋊C₁₆	—	13	8	no	—
139	#346	—	13	2	no	—
140	C₄×C₁₆	—	13	2	no	—
141	C₄×C₁₆	—	13	2	no	—
142	C₄×C₁₆	—	13	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	4	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	#431	1	2	yes	—
147	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	#431	1	2	yes	—
150	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
151	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	4	no	—
154	Dic₁₃	#504	1	4	yes	—
155	C₂×#41	C₂×#261	1	1	yes	characteristic
156	#131	—	2	4	no	—
157	#131	—	26	4	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	—	13	1	no	—
160	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
161	C₂×M₁₆	C₂×#261	1	2	yes	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	C₂×#261	1	2	yes	—
165	C₂²×C₈	C₂×#261	1	1	yes	characteristic
166	C₄×Q₈	—	13	2	no	—
167	C₄×D₄	—	13	4	no	—
168	C₄×D₄	—	13	2	no	—
169	#117	—	13	2	no	—
170	#117	—	13	1	no	—
171	C₂×C₄²	—	13	2	no	—
172	C₂×C₄²	—	13	1	no	—
173	M₃₂	—	26	4	no	—
174	C₂×C₁₆	—	13	4	no	—
175	C₂×C₁₆	—	13	4	no	—
176	C₂×C₁₆	—	26	4	no	—
177	#97	—	13	2	no	—
178	#97	—	13	2	no	—
179	#97	—	13	2	no	—
180	C₄×C₈	—	13	1	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	1	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	13	1	no	—
185	C₄×C₈	—	13	1	no	—
186	C₄×C₈	—	13	2	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#504	1	2	yes	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
194	C₂×D₄	D₁₃⋊C₈	1	2	yes	—
195	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
196	C₂²×C₄	—	13	4	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	C₂²×#261	1	2	yes	—
199	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	26	4	no	—
203	M₁₆	—	26	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	13	4	no	—
206	C₂×C₈	—	13	1	no	—
207	C₂×C₈	—	13	1	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	#6122	1	4	yes	—
211	C₂×C₈	C₂²×#261	1	2	yes	—
212	C₂×C₈	C₂²×#261	1	1	yes	characteristic
213	C₂×C₈	C₂²×#261	1	1	yes	characteristic
214	C₄⋊C₄	—	13	4	no	—
215	C₄⋊C₄	—	13	2	no	—
216	C₂²⋊C₄	—	13	4	no	—
217	C₂²⋊C₄	—	13	2	no	—
218	C₄²	—	13	4	no	—
219	C₄²	—	13	2	no	—
220	C₄²	—	13	1	no	—
221	C₄²	—	13	1	no	—
222	C₄²	—	13	2	no	—
223	C₁₆	—	26	4	no	—
224	C₁₆	—	26	4	no	—
225	C₁₃	#33420	1	1	yes	characteristic, complemented, Sylow
226	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
227	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	4	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	26	4	no	—
237	C₂×C₄	—	26	8	no	—
238	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
239	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
240	C₂×C₄	#88237	1	2	yes	—
241	C₂×C₄	C₂×#6122	1	2	yes	—
242	C₂×C₄	C₂³×#261	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	13	2	no	—
245	C₈	—	13	2	no	—
246	C₈	—	26	2	no	—
247	C₈	C₂×#6122	1	2	yes	—
248	C₈	C₂×#6122	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#11719125	1	2	yes	—
253	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	13	4	no	—
256	C₄	—	26	2	no	—
257	C₄	—	26	4	no	—
258	C₄	#11719125	1	2	yes	—
259	C₄	C₂×#88237	1	1	yes	characteristic
260	C₄	C₂²×#6122	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#11719125	1	1	yes	characteristic
264	C₂	D₁₃⋊#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#131	C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#33420
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #13131202	C₂	semidirect
2. #13131194	C₂	semidirect
3. #13131173	C₂	semidirect
5. #13131042	C₂	semidirect
6. C₂×#11718257	C₂	semidirect
225. C₁₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#13131202	2
#13131194	2
#13131173	2
#13131066	2
#13131042	2
C₂×#11718257	2
#131×Dic₁₃	2
#33420	13