Subgroups — #16807261

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	4	yes	maximal, complemented
2	#13131252	C₂	1	2	yes	maximal, complemented
3	#13131250	C₂	1	2	yes	maximal, complemented
4	#13131173	C₂	1	2	yes	maximal
5	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
6	#13131047	C₂	1	2	yes	maximal
7	C₄×#88106	C₂	1	1	yes	maximal, characteristic
8	#13123766	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₄	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	1	yes	characteristic
12	C₂²×#6092	C₂²	1	2	yes	—
13	C₁₃×#423	C₄	1	1	yes	characteristic
14	#11718783	C₂²	1	2	yes	—
15	#11718780	C₄	1	2	yes	—
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718492	C₄	1	2	yes	—
18	#11718492	C₄	1	2	yes	—
19	#11718491	C₂²	1	2	yes	—
20	#11718491	C₂²	1	2	yes	—
21	#11718260	C₂²	1	4	yes	—
22	#11718260	C₂²	1	4	yes	—
23	#11718258	C₂²	1	4	yes	—
24	#11718257	C₂²	1	4	yes	—
25	#11718231	C₂²	1	2	yes	—
26	C₄⋊#6092	C₂²	1	1	yes	characteristic
27	C₄⋊#6092	C₂²	1	1	yes	characteristic
28	C₄⋊#6092	C₂²	1	2	yes	—
29	C₄×#6092	C₂²	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	#88180	C₂×C₄	1	2	yes	—
33	#88178	C₈	1	2	yes	—
34	#88178	C₈	1	2	yes	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂³	1	1	yes	characteristic
37	#88124	C₈	1	2	yes	—
38	#88123	C₈	1	1	yes	characteristic
39	#88123	C₈	1	1	yes	characteristic
40	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
42	#88106	—	2	2	no	—
43	#88106	Q₈	1	4	yes	complemented
44	C₂×#6092	—	2	4	no	—
45	C₂×#6092	C₂³	1	2	yes	—
46	C₂×#6092	C₂³	1	2	yes	—
47	C₂×#6092	C₂³	1	2	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	#88055	C₂×C₄	1	4	yes	—
52	#88054	C₂×C₄	1	4	yes	—
53	#88054	C₂×C₄	1	4	yes	—
54	C₈×Dic₁₃	C₂³	1	4	yes	—
55	C₄×#1055	C₂×C₄	1	1	yes	characteristic
56	C₄×#1055	C₂×C₄	1	1	yes	characteristic
57	C₄×#1055	C₂³	1	1	yes	characteristic
58	C₄×#1055	C₂³	1	1	yes	characteristic
59	#33476	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
64	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
72	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
73	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
74	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
75	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
76	#6101	C₂×C₈	1	4	yes	—
77	#6101	C₂×C₈	1	4	yes	—
78	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
81	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
82	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
83	C₂×#1055	C₂×Q₈	1	2	yes	—
84	C₂×#1055	#41	1	2	yes	—
85	C₂×#1055	C₂⁴	1	1	yes	characteristic
86	#6092	—	2	2	no	—
87	#6092	—	2	2	no	—
88	#6092	—	2	2	no	—
89	#6092	C₂×Q₈	1	4	yes	—
90	#2960	—	13	1	no	—
91	#2183	—	13	2	no	—
92	#2183	—	13	4	no	—
93	Q₈×C₁₆	—	13	2	no	—
94	#2151	—	13	2	no	—
95	C₂×C₄⋊C₁₆	—	13	1	no	—
96	#2108	—	13	2	no	—
97	C₄×M₃₂	—	13	1	no	—
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	4	1	no	—
101	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#131	1	2	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
110	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
111	Dic₂₆	—	2	4	no	—
112	Dic₂₆	—	4	2	no	—
113	C₈×C₁₃	—	2	4	no	—
114	#1055	—	2	1	no	—
115	#1055	—	2	1	no	—
116	#1055	C₄×Q₈	1	2	yes	—
117	#1055	C₂²×Q₈	1	1	yes	characteristic
118	#1055	C₂×#41	1	1	yes	characteristic
119	#533	—	13	1	no	—
120	C₂×M₃₂	—	13	1	no	—
121	C₂×M₃₂	—	13	1	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	#446	—	13	2	no	—
124	#446	—	13	2	no	—
125	C₈×Q₈	—	13	2	no	—
126	C₈×Q₈	—	13	2	no	—
127	#424	—	13	2	no	—
128	#423	#261	1	1	yes	characteristic
129	#405	—	13	2	no	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₄⋊C₁₆	—	13	1	no	—
132	C₄⋊C₁₆	—	13	1	no	—
133	C₄⋊C₁₆	—	13	2	no	—
134	C₄⋊C₁₆	—	13	4	no	—
135	C₄⋊C₁₆	—	13	4	no	—
136	#346	—	13	2	no	—
137	#346	—	13	4	no	—
138	C₄×C₁₆	—	13	2	no	—
139	C₄×C₁₆	—	13	4	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	C₈×Q₈	1	2	yes	—
148	C₄×C₁₃	#504	1	2	yes	—
149	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	—	2	4	no	—
153	C₂×M₁₆	C₂×#261	1	2	yes	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₄⋊Q₈	—	13	1	no	—
157	C₄⋊Q₈	—	13	1	no	—
158	#125	—	13	2	no	—
159	C₂²⋊Q₈	—	13	2	no	—
160	C₂²⋊Q₈	—	13	2	no	—
161	C₄×Q₈	—	13	2	no	—
162	C₄×Q₈	—	13	2	no	—
163	#117	—	13	2	no	—
164	C₂×C₄²	C₂×#261	1	1	yes	characteristic
165	M₃₂	—	13	4	no	—
166	M₃₂	—	26	2	no	—
167	C₂×C₁₆	—	13	2	no	—
168	C₂×C₁₆	—	13	2	no	—
169	C₂×C₁₆	—	13	2	no	—
170	C₂×C₁₆	—	13	2	no	—
171	C₂×C₁₆	—	26	4	no	—
172	C₄⋊C₈	—	13	4	no	—
173	C₄⋊C₈	—	13	4	no	—
174	C₄⋊C₈	C₂×#261	1	2	yes	—
175	C₄⋊C₈	C₂×#261	1	2	yes	—
176	#97	—	13	4	no	—
177	C₄×C₈	—	13	4	no	—
178	C₄×C₈	—	13	1	no	—
179	C₄×C₈	—	13	1	no	—
180	C₄×C₈	—	13	1	no	—
181	C₄×C₈	—	13	1	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
185	C₂×C₁₃	#3407	1	1	yes	characteristic
186	C₂×Q₈	—	13	2	no	—
187	C₂×Q₈	—	13	2	no	—
188	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
189	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	1	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	C₂²×#261	1	4	yes	—
200	C₄⋊C₄	—	13	2	no	—
201	C₄⋊C₄	—	13	2	no	—
202	C₄⋊C₄	—	13	4	no	—
203	C₄⋊C₄	—	13	4	no	—
204	C₄⋊C₄	—	13	2	no	—
205	C₄⋊C₄	—	13	2	no	—
206	C₂²⋊C₄	—	13	4	no	—
207	C₄²	—	13	4	no	—
208	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
209	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
210	C₄²	C₂²×#261	1	1	yes	characteristic
211	C₄²	C₂²×#261	1	1	yes	characteristic
212	C₁₆	—	13	4	no	—
213	C₁₆	—	26	2	no	—
214	C₁₆	—	26	2	no	—
215	C₁₆	—	26	2	no	—
216	C₁₃	#33476	1	1	yes	characteristic, complemented, Sylow
217	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
218	Q₈	—	26	4	no	—
219	Q₈	—	52	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	13	4	no	—
223	C₂×C₄	—	13	4	no	—
224	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
229	C₂×C₄	#88238	1	2	yes	—
230	C₂×C₄	Q₈×#261	1	2	yes	—
231	C₂×C₄	C₂³×#261	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	13	1	no	—
234	C₈	—	13	1	no	—
235	C₈	—	13	2	no	—
236	C₈	—	26	1	no	—
237	C₈	—	26	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	#11719074	1	2	yes	—
241	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	26	4	no	—
246	C₄	—	26	4	no	—
247	C₄	#11719074	1	2	yes	—
248	C₄	#11719096	1	2	yes	—
249	C₄	C₂×#88238	1	1	yes	characteristic
250	C₄	C₂×Q₈×#261	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×#11719074	1	1	yes	characteristic
253	C₂	C₂×#11719096	1	1	yes	characteristic
254	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#423	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#423	#261
hypocenter	C₁₃	#33476
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131252	C₂	semidirect
3. #13131250	C₂	semidirect
43. #88106	Q₈	semidirect
216. C₁₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131252	2
#13131250	2
#13131173	2
C₂×C₄⋊#6092	2
#13131047	2
C₄×#88106	2
#13123766	2
#33476	13