Subgroups — #16780946

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#11718256	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#1057	C₂	1	1	yes	maximal, characteristic
4	#13129672	C₂	1	2	yes	maximal
5	#13114728	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×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
12	C₂×#88047	C₄	1	2	yes	—
13	#11718290	—	2	2	no	—
14	#11718256	C₄	1	4	yes	complemented
15	C₈×#1057	C₄	1	4	yes	complemented
16	C₁₃×#340	C₄	1	1	yes	characteristic
17	#11718143	C₄	1	1	yes	characteristic
18	C₂²×#1057	—	2	1	no	—
19	C₂²×#1057	—	2	2	no	—
20	C₂²×#1057	D₄	1	1	yes	characteristic
21	C₂²×#1057	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₃	—	2	2	no	—
26	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	#88110	—	4	4	no	—
29	C₄⋊#1057	—	2	2	no	—
30	C₄⋊#1057	C₂×C₄	1	4	yes	—
31	C₄×#1057	—	2	1	no	—
32	C₄×#1057	C₂×C₄	1	2	yes	—
33	C₄×#1057	C₂×C₄	1	2	yes	—
34	C₄×#1057	Q₈	1	2	yes	—
35	#88071	—	2	2	no	—
36	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
37	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
38	#88047	—	2	4	no	—
39	#8282	—	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	4	no	—
46	C₂×#1057	—	2	2	no	—
47	C₂×#1057	—	2	1	no	—
48	C₂×#1057	—	2	1	no	—
49	C₂×#1057	—	2	4	no	—
50	C₂×#1057	—	4	2	no	—
51	C₂×#1057	—	4	2	no	—
52	C₂×#1057	C₄⋊C₄	1	2	yes	—
53	C₂×#1057	C₄⋊C₄	1	2	yes	—
54	C₂×#1057	D₈	1	2	yes	—
55	C₂×#1057	Q₁₆	1	2	yes	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	C₄²	1	2	yes	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄×Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₄×Dic₁₃	#41	1	1	yes	characteristic
68	C₂×#1055	C₄²	1	2	yes	—
69	C₂×#1055	M₁₆	1	4	yes	—
70	C₄×#107	—	13	1	no	—
71	#1770	—	13	2	no	—
72	C₂×#335	—	13	1	no	—
73	C₂×C₈²	—	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	2	yes	—
82	C₂×C₄×C₁₃	#131	1	2	yes	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	4	2	no	—
85	C₂×Dic₁₃	—	4	2	no	—
86	C₂×Dic₁₃	—	4	2	no	—
87	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₁₃	#117	1	1	yes	characteristic
89	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
90	C₂×Dic₁₃	C₄×Q₈	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×Dic₁₃	#135	1	1	yes	characteristic
94	C₂×Dic₁₃	#135	1	1	yes	characteristic
95	#1057	—	2	4	no	—
96	#1057	—	2	2	no	—
97	#1057	—	4	4	no	—
98	#1057	#107	1	4	yes	complemented
99	C₈×C₁₃	—	2	2	no	—
100	#1055	#97	1	4	yes	complemented
101	C₂×#107	—	13	2	no	—
102	C₂×C₄⋊C₈	—	13	2	no	—
103	C₂×#97	—	26	2	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₄×C₄⋊C₄	—	13	2	no	—
108	#340	—	13	1	no	—
109	#340	#261	1	1	yes	characteristic
110	#336	—	26	2	no	—
111	#335	—	13	4	no	—
112	C₈²	—	13	4	no	—
113	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₃	#436	1	2	yes	—
115	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
116	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
117	C₂²×C₁₃	#443	1	2	yes	—
118	C₄×C₁₃	—	8	2	no	—
119	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
120	C₄×C₁₃	#405	1	1	yes	characteristic
121	C₄×C₁₃	#446	1	2	yes	—
122	Dic₁₃	—	2	1	no	—
123	Dic₁₃	—	2	1	no	—
124	Dic₁₃	—	8	2	no	—
125	Dic₁₃	C₂×#107	1	1	yes	characteristic
126	Dic₁₃	#427	1	1	yes	characteristic
127	Dic₁₃	C₄×D₈	1	1	yes	characteristic
128	Dic₁₃	C₄×Q₁₆	1	1	yes	characteristic
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	26	1	no	—
133	C₂²×C₈	—	26	2	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	—	26	4	no	—
140	C₄⋊C₈	—	13	4	no	—
141	C₄⋊C₈	—	26	2	no	—
142	#97	—	52	4	no	—
143	C₄×C₈	—	13	2	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	—	26	1	no	—
149	#95	—	26	2	no	—
150	C₂×C₁₃	#1589	1	2	yes	—
151	C₂×C₁₃	#1591	1	2	yes	—
152	C₂×C₁₃	#1770	1	1	yes	characteristic
153	C₂×C₁₃	#1772	1	1	yes	characteristic
154	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	26	2	no	—
159	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	13	4	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	26	2	no	—
168	C₂×C₈	—	26	1	no	—
169	C₂×C₈	—	26	1	no	—
170	C₂×C₈	—	26	4	no	—
171	C₂×C₈	—	26	4	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	—	52	2	no	—
174	C₂×C₈	C₄×#261	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	26	2	no	—
178	C₄⋊C₄	—	26	2	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	1	no	—
183	C₄²	—	52	2	no	—
184	C₄²	—	52	2	no	—
185	C₁₃	#8282	1	1	yes	characteristic, complemented, Sylow
186	C₂³	D₄×#261	1	1	yes	characteristic
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	13	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	1	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	52	2	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
203	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
204	C₂×C₄	#88238	1	2	yes	—
205	C₂×C₄	Q₈×#261	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	13	4	no	—
208	C₈	—	13	4	no	—
209	C₈	—	26	2	no	—
210	C₈	—	26	4	no	—
211	C₈	—	52	4	no	—
212	C₂²	#11719066	1	2	yes	—
213	C₂²	C₄⋊C₄×#261	1	1	yes	characteristic
214	C₂²	D₈×#261	1	1	yes	characteristic
215	C₂²	#11719107	1	2	yes	—
216	C₂²	Q₁₆×#261	1	1	yes	characteristic
217	C₄	—	8	2	no	—
218	C₄	—	13	1	no	—
219	C₄	—	13	1	no	—
220	C₄	—	13	1	no	—
221	C₄	—	13	1	no	—
222	C₄	—	26	1	no	—
223	C₄	—	26	1	no	—
224	C₄	—	104	2	no	—
225	C₄	#11719047	1	1	yes	characteristic
226	C₄	#11719062	1	1	yes	characteristic
227	C₄	#11719078	1	2	yes	—
228	C₂	#13130467	1	2	yes	—
229	C₂	#13130474	1	2	yes	—
230	C₂	#13130631	1	1	yes	characteristic
231	C₂	#13130639	1	1	yes	characteristic
232	C₂	#107×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#340	C₄
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#340	#261
hypocenter	C₁₃	#8282
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
14. #11718256	C₄	semidirect
15. C₈×#1057	C₄	semidirect
98. #1057	#107	semidirect
100. #1055	#97	semidirect
185. C₁₃	#8282	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×#11718256	2
C₂×C₈×#1057	2
#13129672	2
#13114728	2
#8282	13