Subgroups — #15761965

Order 3328.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123231	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718077	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88128	—	2	1	no	—
6	C₂×#88128	—	2	1	no	—
7	C₂×#88128	—	2	2	no	—
8	C₂×#88128	C₄	1	4	yes	—
9	C₂×#88128	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	—	2	1	no	—
15	#11718116	—	2	1	no	—
16	#11718077	—	2	8	no	—
17	C₂²×D₅₂	—	2	4	no	—
18	C₂²×D₅₂	—	2	4	no	—
19	C₂²×D₅₂	—	2	2	no	—
20	C₂²×D₅₂	—	2	2	no	—
21	C₂²×D₅₂	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×#1055	—	2	2	no	—
29	#88128	—	2	4	no	—
30	#88128	—	2	4	no	—
31	#88128	—	4	8	no	—
32	#88128	—	4	4	no	—
33	#88128	—	4	2	no	—
34	#88128	—	4	4	no	—
35	#88128	—	4	2	no	—
36	#88128	—	4	4	no	—
37	C₁₃×C₄⋊C₈	—	2	4	no	—
38	C₁₃×C₄⋊C₈	—	4	4	no	—
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	D₄	1	4	yes	complemented
41	C₄×#1055	—	4	2	no	—
42	#9695	—	13	1	no	maximal, Sylow
43	C₂³×D₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×D₅₂	—	2	8	no	—
50	C₂×D₅₂	—	4	8	no	—
51	C₂×D₅₂	—	4	8	no	—
52	C₂×D₅₂	—	4	8	no	—
53	C₂×D₅₂	—	4	8	no	—
54	C₂×D₅₂	—	4	8	no	—
55	C₂×D₅₂	—	4	8	no	—
56	C₂×D₅₂	—	4	8	no	—
57	C₂×D₅₂	—	4	4	no	—
58	C₂×D₅₂	—	4	2	no	—
59	C₂×D₅₂	—	4	2	no	—
60	C₂×D₅₂	—	4	2	no	—
61	C₂×D₅₂	—	4	4	no	—
62	C₂×D₅₂	—	4	2	no	—
63	C₂×D₅₂	—	4	4	no	—
64	C₂×D₅₂	—	4	4	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	—	4	4	no	—
67	C₄²×C₁₃	—	2	4	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	D₈	1	2	yes	—
70	C₄²×C₁₃	D₈	1	4	yes	—
71	C₄²×C₁₃	SD₁₆	1	2	yes	—
72	C₄²×C₁₃	SD₁₆	1	4	yes	—
73	C₄²×C₁₃	C₂×D₄	1	2	yes	—
74	C₄²×C₁₃	#41	1	2	yes	—
75	C₂×#1055	—	2	4	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	2	no	—
78	#2867	—	13	1	no	—
79	#1842	D₁₃	1	1	yes	characteristic, complemented
80	C₄×C₄⋊C₈	—	13	1	no	—
81	C₂×#331	—	13	4	no	—
82	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×D₁₃	—	4	4	no	—
84	C₂²×D₁₃	—	4	8	no	—
85	C₂²×D₁₃	—	4	4	no	—
86	C₂²×D₁₃	—	4	8	no	—
87	C₂²×D₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	#100	1	4	yes	—
92	C₂×C₄×C₁₃	#102	1	4	yes	—
93	C₂×C₄×C₁₃	#102	1	4	yes	—
94	C₂×C₄×C₁₃	#102	1	4	yes	—
95	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	#121	1	2	yes	—
97	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
104	D₅₂	—	8	16	no	—
105	D₅₂	—	8	8	no	—
106	D₅₂	—	8	16	no	—
107	D₅₂	—	8	4	no	—
108	D₅₂	—	8	4	no	—
109	D₅₂	—	8	4	no	—
110	D₅₂	—	8	2	no	—
111	D₅₂	—	8	2	no	—
112	C₈×C₁₃	—	8	2	no	—
113	#1055	—	4	4	no	—
114	C₂×#127	—	13	4	no	—
115	C₂×#127	—	13	2	no	—
116	C₂×#127	—	26	1	no	—
117	C₂×#127	—	26	1	no	—
118	C₂×#127	—	26	2	no	—
119	C₂×C₄⋊C₈	—	2	2	no	—
120	C₂×C₄⋊C₈	—	13	2	no	—
121	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
122	C₂×C₄×C₈	—	26	1	no	—
123	C₄³	C₂×D₁₃	1	1	yes	characteristic
124	#336	—	26	1	no	—
125	#331	—	26	8	no	—
126	C₂²×C₁₃	#390	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×#100	1	2	yes	—
128	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×#102	1	2	yes	—
130	C₂²×C₁₃	#418	1	1	yes	characteristic
131	C₂×D₁₃	—	8	8	no	—
132	C₂×D₁₃	—	8	8	no	—
133	C₂×D₁₃	—	8	4	no	—
134	C₂×D₁₃	—	8	4	no	—
135	C₂×D₁₃	—	8	4	no	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	#331	1	8	yes	complemented
139	C₄×C₁₃	#459	1	2	yes	—
140	C₄×C₁₃	#460	1	2	yes	—
141	C₄×C₁₃	#486	1	2	yes	—
142	C₄×C₁₃	#492	1	1	yes	characteristic
143	C₄×C₁₃	#493	1	1	yes	characteristic
144	C₂²×D₄	—	13	4	no	—
145	C₂²×D₄	—	26	4	no	—
146	C₂²×D₄	—	26	4	no	—
147	C₂²×D₄	—	26	2	no	—
148	C₂²×D₄	—	26	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	26	2	no	—
151	#127	—	26	4	no	—
152	#127	—	26	4	no	—
153	#127	—	52	8	no	—
154	#127	—	52	4	no	—
155	#127	—	52	2	no	—
156	#127	—	52	2	no	—
157	#127	—	52	4	no	—
158	#127	—	52	4	no	—
159	C₂×C₄²	C₄×D₁₃	1	2	yes	—
160	C₂×C₄²	D₅₂	1	1	yes	characteristic
161	C₂×C₄²	D₅₂	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
163	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	4	4	no	—
166	C₄⋊C₈	—	13	4	no	—
167	C₄⋊C₈	—	26	2	no	—
168	C₄×C₈	—	52	2	no	—
169	C₂×C₁₃	C₂×#331	1	2	yes	—
170	C₂×C₁₃	#1545	1	2	yes	—
171	C₂×C₁₃	#1957	1	1	yes	characteristic
172	C₂×C₁₃	#1965	1	1	yes	characteristic
173	C₂×C₁₃	#1982	1	1	yes	characteristic
174	D₁₃	—	16	4	no	—
175	C₂⁴	—	26	4	no	—
176	C₂×D₄	—	26	8	no	—
177	C₂×D₄	—	52	8	no	—
178	C₂×D₄	—	52	8	no	—
179	C₂×D₄	—	52	8	no	—
180	C₂×D₄	—	52	4	no	—
181	C₂×D₄	—	52	2	no	—
182	C₂×D₄	—	52	2	no	—
183	C₂×D₄	—	52	4	no	—
184	C₂×D₄	—	52	4	no	—
185	C₂×D₄	—	52	2	no	—
186	C₂×D₄	—	52	4	no	—
187	C₂×D₄	—	52	2	no	—
188	C₂×D₄	—	52	8	no	—
189	C₂×D₄	—	52	8	no	—
190	C₂×D₄	—	52	8	no	—
191	C₂×D₄	—	52	8	no	—
192	C₂²×C₄	#6103	1	2	yes	—
193	C₂²×C₄	#6103	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	26	4	no	—
200	C₂×C₈	—	52	2	no	—
201	C₂×C₈	—	52	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₄²	#6095	1	2	yes	—
205	C₄²	D₁₀₄	1	2	yes	—
206	C₄²	#6104	1	4	yes	—
207	C₄²	#6106	1	4	yes	—
208	C₄²	D₄×D₁₃	1	2	yes	—
209	C₄²	#6131	1	2	yes	—
210	C₁₃	#9695	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	52	4	no	—
212	C₂³	—	52	8	no	—
213	C₂³	—	52	4	no	—
214	C₂³	—	52	8	no	—
215	C₂³	—	52	4	no	—
216	C₂³	C₂×#6103	1	1	yes	characteristic
217	D₄	—	104	8	no	—
218	D₄	—	104	16	no	—
219	D₄	—	104	2	no	—
220	D₄	—	104	2	no	—
221	D₄	—	104	4	no	—
222	D₄	—	104	4	no	—
223	D₄	—	104	4	no	—
224	D₄	—	104	16	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	#88049	1	4	yes	—
229	C₂×C₄	#88049	1	4	yes	—
230	C₂×C₄	#88061	1	4	yes	—
231	C₂×C₄	#88063	1	4	yes	—
232	C₂×C₄	#88147	1	2	yes	—
233	C₂×C₄	#88149	1	2	yes	—
234	C₂×C₄	C₂×#6095	1	1	yes	characteristic
235	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
236	C₂×C₄	#88162	1	2	yes	—
237	C₂×C₄	C₂×#6104	1	2	yes	—
238	C₂×C₄	#88194	1	1	yes	characteristic
239	C₂×C₄	C₂×#6106	1	2	yes	—
240	C₂×C₄	#88201	1	1	yes	characteristic
241	C₈	—	8	2	no	—
242	C₈	—	52	4	no	—
243	C₂²	—	104	4	no	—
244	C₂²	—	104	4	no	—
245	C₂²	—	104	8	no	—
246	C₂²	—	104	8	no	—
247	C₂²	—	104	4	no	—
248	C₂²	C₂×#88049	1	2	yes	—
249	C₂²	#11718649	1	1	yes	characteristic
250	C₂²	C₂×#88061	1	1	yes	characteristic
251	C₂²	C₂×#88063	1	2	yes	—
252	C₂²	#11718792	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	#11718077	1	8	yes	complemented
256	C₄	#11718504	1	2	yes	—
257	C₄	#11718509	1	2	yes	—
258	C₄	#11718734	1	1	yes	characteristic
259	C₄	#11718746	1	1	yes	characteristic
260	C₄	#11718749	1	2	yes	—
261	C₂	—	208	4	no	—
262	C₂	C₂×#11718077	1	2	yes	—
263	C₂	#13113505	1	2	yes	—
264	C₂	#13115641	1	1	yes	characteristic
265	C₂	#13115675	1	1	yes	characteristic
266	C₂	#13115762	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1842	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	#13123231	C₂
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#9695
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊#1055	C₂	semidirect
3. C₂×#11718077	C₂	semidirect
4. C₁₃×#1842	C₂	semidirect
40. C₄⋊#1055	D₄	semidirect
79. #1842	D₁₃	semidirect
138. C₄×C₁₃	#331	semidirect
210. C₁₃	#9695	semidirect
255. C₄	#11718077	semidirect

Maximal subgroups

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

subgroup	index
#13123231	2
C₄×C₄⋊#1055	2
C₂×#11718077	2
C₁₃×#1842	2
#9695	13