Subgroups — #16780967

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130632	C₂	1	2	yes	maximal
2	C₂×#11718253	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718251	C₂	1	1	yes	maximal, characteristic
4	#13129672	C₂	1	2	yes	maximal
5	#107×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#88110	—	2	2	no	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
9	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
11	C₂×#88058	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
14	#11718290	—	2	2	no	—
15	#11718253	C₄	1	4	yes	complemented
16	#11718251	C₄	1	4	yes	complemented
17	#11718121	C₄	1	2	yes	—
18	C₂²×#1057	—	2	2	no	—
19	C₂²×#1057	—	2	1	no	—
20	C₂²×#1057	D₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₁₃	—	2	2	no	—
25	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	#88110	—	4	4	no	—
28	C₄⋊#1057	—	2	2	no	—
29	C₄⋊#1057	C₂×C₄	1	4	yes	—
30	C₄×#1057	—	2	1	no	—
31	C₄×#1057	C₂×C₄	1	2	yes	—
32	C₄×#1057	C₂×C₄	1	2	yes	—
33	C₄×#1057	Q₈	1	2	yes	—
34	C₁₃×#107	—	2	2	no	—
35	#88071	—	2	2	no	—
36	#88058	—	2	2	no	—
37	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
38	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
39	#8287	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	—	2	2	no	—
43	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	#41	1	1	yes	characteristic
45	C₂×#1057	—	2	2	no	—
46	C₂×#1057	—	2	2	no	—
47	C₂×#1057	—	2	2	no	—
48	C₂×#1057	—	2	4	no	—
49	C₂×#1057	—	4	1	no	—
50	C₂×#1057	—	4	1	no	—
51	C₂×#1057	—	4	2	no	—
52	C₂×#1057	—	4	2	no	—
53	C₂×#1057	C₄⋊C₄	1	4	yes	—
54	C₂×C₈×C₁₃	C₄²	1	2	yes	—
55	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
56	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄×Dic₁₃	—	4	2	no	—
62	C₄×Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₁₃	#41	1	1	yes	characteristic
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	C₄²	1	2	yes	—
70	C₄×#107	—	13	1	no	—
71	#1770	—	13	2	no	—
72	C₂×#334	—	13	1	no	—
73	C₂×#322	—	13	1	no	—
74	#1279	—	13	2	no	—
75	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	4	2	no	—
88	C₂×Dic₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₃	#117	1	1	yes	characteristic
92	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₁₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×Dic₁₃	#137	1	2	yes	—
96	#1057	—	2	4	no	—
97	#1057	—	4	2	no	—
98	#1057	—	4	4	no	—
99	C₈×C₁₃	#97	1	4	yes	complemented
100	#1055	—	2	2	no	—
101	C₂×#107	—	13	1	no	—
102	C₂×#107	#261	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	13	2	no	—
104	C₂×#97	—	26	2	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₄×C₄⋊C₄	—	13	2	no	—
109	#340	—	13	2	no	—
110	#336	—	26	2	no	—
111	#334	—	13	4	no	—
112	#322	—	13	4	no	—
113	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₃	#436	1	1	yes	characteristic
115	C₂²×C₁₃	#436	1	1	yes	characteristic
116	C₂²×C₁₃	#441	1	1	yes	characteristic
117	C₂²×C₁₃	#442	1	1	yes	characteristic
118	C₂²×C₁₃	#444	1	1	yes	characteristic
119	C₂²×C₁₃	#444	1	1	yes	characteristic
120	C₄×C₁₃	—	8	2	no	—
121	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
122	C₄×C₁₃	#405	1	1	yes	characteristic
123	C₄×C₁₃	#446	1	1	yes	characteristic
124	C₄×C₁₃	#446	1	1	yes	characteristic
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	—	8	2	no	—
128	Dic₁₃	#428	1	2	yes	—
129	Dic₁₃	#440	1	2	yes	—
130	C₂²×C₈	—	13	2	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	26	2	no	—
133	C₂²×C₈	—	26	1	no	—
134	C₂²×C₈	C₂×#261	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	13	2	no	—
136	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
137	C₂×C₄²	—	13	1	no	—
138	C₂×C₄²	—	26	2	no	—
139	#107	—	2	2	no	—
140	#107	—	26	2	no	—
141	C₄⋊C₈	—	13	4	no	—
142	C₄⋊C₈	—	26	2	no	—
143	#97	—	52	4	no	—
144	C₄×C₈	—	13	2	no	—
145	C₄×C₈	—	13	2	no	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	26	1	no	—
150	#95	—	26	2	no	—
151	C₂×C₁₃	#1593	1	1	yes	characteristic
152	C₂×C₁₃	#1593	1	1	yes	characteristic
153	C₂×C₁₃	#1594	1	1	yes	characteristic
154	C₂×C₁₃	#1594	1	1	yes	characteristic
155	C₂×C₁₃	#1770	1	1	yes	characteristic
156	C₂×C₁₃	#1773	1	1	yes	characteristic
157	C₂×C₁₃	#1776	1	1	yes	characteristic
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	26	2	no	—
162	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	26	1	no	—
168	C₂×C₈	—	26	1	no	—
169	C₂×C₈	—	26	2	no	—
170	C₂×C₈	—	26	4	no	—
171	C₂×C₈	—	52	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	—	52	1	no	—
174	C₂×C₈	—	52	1	no	—
175	C₂×C₈	#6118	1	2	yes	—
176	C₂×C₈	C₄×#261	1	2	yes	—
177	C₂×C₈	#6122	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	26	2	no	—
181	C₄⋊C₄	—	26	2	no	—
182	C₄²	—	13	1	no	—
183	C₄²	—	13	1	no	—
184	C₄²	—	13	1	no	—
185	C₄²	—	13	1	no	—
186	C₄²	—	52	2	no	—
187	C₄²	—	52	2	no	—
188	C₁₃	#8287	1	1	yes	characteristic, complemented, Sylow
189	C₂³	D₄×#261	1	1	yes	characteristic
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	1	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	—	52	2	no	—
203	C₂×C₄	—	52	2	no	—
204	C₂×C₄	C₂×#6118	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
206	C₂×C₄	#88238	1	1	yes	characteristic
207	C₂×C₄	#88240	1	1	yes	characteristic
208	C₂×C₄	Q₈×#261	1	1	yes	characteristic
209	C₂×C₄	C₂×#6122	1	1	yes	characteristic
210	C₈	—	26	4	no	—
211	C₈	—	26	2	no	—
212	C₈	—	52	4	no	—
213	C₈	—	52	2	no	—
214	C₈	#88110	1	4	yes	complemented
215	C₂²	#11719076	1	1	yes	characteristic
216	C₂²	C₄⋊C₄×#261	1	1	yes	characteristic
217	C₂²	#11719102	1	1	yes	characteristic
218	C₂²	#11719104	1	1	yes	characteristic
219	C₂²	#11719110	1	1	yes	characteristic
220	C₂²	#11719112	1	1	yes	characteristic
221	C₂²	#11719145	1	1	yes	characteristic
222	C₄	—	8	2	no	—
223	C₄	—	13	2	no	—
224	C₄	—	13	2	no	—
225	C₄	—	26	1	no	—
226	C₄	—	26	1	no	—
227	C₄	—	104	2	no	—
228	C₄	#11719072	1	1	yes	characteristic
229	C₄	#11719080	1	1	yes	characteristic
230	C₄	C₂×#88110	1	1	yes	characteristic
231	C₄	#11719158	1	1	yes	characteristic
232	C₂	#13130482	1	1	yes	characteristic
233	C₂	#13130483	1	1	yes	characteristic
234	C₂	#13130485	1	1	yes	characteristic
235	C₂	#13130486	1	1	yes	characteristic
236	C₂	#13130632	1	1	yes	characteristic
237	C₂	#13130643	1	1	yes	characteristic
238	C₂	#13130651	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#107	C₄
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#107	#261
hypocenter	C₁₃	#8287
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
15. #11718253	C₄	semidirect
16. #11718251	C₄	semidirect
99. C₈×C₁₃	#97	semidirect
188. C₁₃	#8287	semidirect
214. C₈	#88110	semidirect

Maximal subgroups

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

subgroup	index
#13130632	2
C₂×#11718253	2
C₂×#11718251	2
#13129672	2
#107×Dic₁₃	2
#8287	13