Subgroups — #15761513

Order 3328.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718071	C₂	1	2	yes	maximal
4	#13112670	C₂	1	2	yes	maximal
5	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#97	—	2	1	no	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1055	—	2	1	no	—
11	C₂×C₄⋊#1055	—	2	1	no	—
12	C₂×C₄⋊#1055	C₄	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	1	no	—
16	#11718116	C₄	1	1	yes	characteristic
17	#11718116	C₄	1	1	yes	characteristic
18	#11718071	—	2	4	no	—
19	C₂²×C₈×C₁₃	—	2	1	no	—
20	C₂²×C₈×C₁₃	—	2	1	no	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×#1055	—	2	2	no	—
29	C₂²×#1055	C₂×C₄	1	2	yes	—
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	C₁₃×C₄⋊C₈	—	2	1	no	—
32	C₁₃×C₄⋊C₈	—	2	1	no	—
33	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
34	C₁₃×C₄⋊C₈	Q₈	1	2	yes	—
35	C₁₃×#97	—	4	2	no	—
36	C₄⋊#1055	—	2	4	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄⋊#1055	—	2	2	no	—
39	C₄⋊#1055	—	4	2	no	—
40	C₄⋊#1055	—	4	2	no	—
41	C₄×#1055	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	#9604	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	4	1	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	—	4	1	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	—	2	1	no	—
61	C₄²×C₁₃	—	2	1	no	—
62	C₄²×C₁₃	—	4	1	no	—
63	C₄²×C₁₃	D₈	1	2	yes	—
64	C₄²×C₁₃	Q₁₆	1	2	yes	—
65	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₁₃	#41	1	1	yes	characteristic
68	C₄²×C₁₃	#41	1	1	yes	characteristic
69	C₂×#1055	—	2	2	no	—
70	C₂×#1055	—	2	2	no	—
71	C₂×#1055	—	2	2	no	—
72	C₂×#1055	—	2	2	no	—
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	—	4	2	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	M₁₆	1	4	yes	—
78	#1842	—	13	1	no	—
79	#1767	D₁₃	1	1	yes	characteristic
80	C₄×C₄⋊C₈	—	13	1	no	—
81	C₂×#335	—	13	2	no	—
82	#1277	—	13	2	no	—
83	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	#107	1	2	yes	—
96	C₂×C₄×C₁₃	#107	1	2	yes	—
97	C₂×C₄×C₁₃	#108	1	2	yes	—
98	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₁₃	#131	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
110	C₈×C₁₃	—	4	2	no	—
111	C₈×C₁₃	—	4	2	no	—
112	#1055	—	2	4	no	—
113	#1055	—	4	2	no	—
114	#1055	—	8	2	no	—
115	C₂×C₄⋊C₈	—	13	2	no	—
116	C₂×C₄⋊C₈	—	13	2	no	—
117	C₂×C₄⋊C₈	—	26	1	no	—
118	C₂×C₄⋊C₈	—	26	1	no	—
119	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
121	C₂×#97	—	2	1	no	—
122	C₂×C₄×C₈	—	13	2	no	—
123	C₄³	C₂×D₁₃	1	1	yes	characteristic
124	#336	—	2	1	no	—
125	#336	—	13	1	no	—
126	#336	—	13	1	no	—
127	#335	—	26	4	no	—
128	C₂²×C₁₃	#389	1	1	yes	characteristic
129	C₂²×C₁₃	#423	1	1	yes	characteristic
130	C₂²×C₁₃	#424	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
132	C₂²×C₁₃	#428	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
134	C₂²×C₁₃	#430	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	#435	1	2	yes	—
142	C₄×C₁₃	#446	1	2	yes	—
143	C₄×C₁₃	#459	1	1	yes	characteristic
144	C₄×C₁₃	#462	1	1	yes	characteristic
145	C₄×C₁₃	#474	1	1	yes	characteristic
146	C₄×C₁₃	#477	1	1	yes	characteristic
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	13	2	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	26	2	no	—
152	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
153	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
154	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
155	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	26	4	no	—
162	C₄⋊C₈	—	26	2	no	—
163	C₄⋊C₈	—	26	2	no	—
164	C₄⋊C₈	—	52	2	no	—
165	C₄⋊C₈	—	52	2	no	—
166	C₄⋊C₈	Dic₂₆	1	2	yes	—
167	C₄⋊C₈	D₅₂	1	2	yes	—
168	#97	—	4	2	no	—
169	C₄×C₈	—	26	2	no	—
170	C₄×C₈	—	26	2	no	—
171	C₂×C₁₃	#1567	1	1	yes	characteristic
172	C₂×C₁₃	#1569	1	1	yes	characteristic
173	C₂×C₁₃	#1571	1	1	yes	characteristic
174	C₂×C₁₃	#1573	1	1	yes	characteristic
175	C₂×C₁₃	#1940	1	1	yes	characteristic
176	C₂×C₁₃	#1947	1	1	yes	characteristic
177	C₂×C₁₃	#1950	1	1	yes	characteristic
178	C₂²×C₄	#6101	1	1	yes	characteristic
179	C₂²×C₄	#6101	1	1	yes	characteristic
180	C₂²×C₄	#6101	1	1	yes	characteristic
181	C₂²×C₄	#6101	1	1	yes	characteristic
182	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	13	4	no	—
192	C₂×C₈	—	26	2	no	—
193	C₂×C₈	—	26	2	no	—
194	C₂×C₈	—	26	2	no	—
195	C₂×C₈	—	26	2	no	—
196	C₂×C₈	—	26	2	no	—
197	C₂×C₈	—	52	2	no	—
198	C₂×C₈	—	52	2	no	—
199	C₂×C₈	—	52	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	4	1	no	—
206	C₄²	#6104	1	2	yes	—
207	C₄²	#6107	1	2	yes	—
208	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
209	C₄²	C₂×D₅₂	1	1	yes	characteristic
210	C₄²	#6127	1	1	yes	characteristic
211	C₄²	#6127	1	1	yes	characteristic
212	C₁₃	#9604	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×#6101	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	#88047	1	2	yes	—
226	C₂×C₄	#88047	1	2	yes	—
227	C₂×C₄	#88062	1	2	yes	—
228	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
229	C₂×C₄	#88123	1	1	yes	characteristic
230	C₂×C₄	#88124	1	1	yes	characteristic
231	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
232	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
233	C₂×C₄	#88161	1	2	yes	—
234	C₂×C₄	#88178	1	1	yes	characteristic
235	C₂×C₄	#88184	1	1	yes	characteristic
236	C₂×C₄	C₂×#6104	1	1	yes	characteristic
237	C₂×C₄	C₂×#6107	1	1	yes	characteristic
238	C₂×C₄	#88203	1	1	yes	characteristic
239	C₂×C₄	#88205	1	1	yes	characteristic
240	C₈	—	4	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	26	4	no	—
243	C₈	—	52	2	no	—
244	C₈	—	104	2	no	—
245	C₂²	#11718589	1	1	yes	characteristic
246	C₂²	C₂×#88047	1	1	yes	characteristic
247	C₂²	#11718656	1	1	yes	characteristic
248	C₂²	#11718779	1	1	yes	characteristic
249	C₂²	#11718780	1	1	yes	characteristic
250	C₂²	C₂×#88062	1	1	yes	characteristic
251	C₂²	#11718785	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#11718362	1	2	yes	—
259	C₄	#11718367	1	2	yes	—
260	C₄	#11718668	1	1	yes	characteristic
261	C₄	#11718676	1	1	yes	characteristic
262	C₄	#11718682	1	1	yes	characteristic
263	C₄	#11718692	1	1	yes	characteristic
264	C₂	#13113594	1	1	yes	characteristic
265	C₂	#13113604	1	1	yes	characteristic
266	C₂	#13113612	1	1	yes	characteristic
267	C₂	#13113622	1	1	yes	characteristic
268	C₂	#13115559	1	1	yes	characteristic
269	C₂	#13115594	1	1	yes	characteristic
270	C₂	#13115611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1767	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1767	D₁₃
hypocenter	C₁₃	#9604
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
212. C₁₃	#9604	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
C₄×C₄⋊#1055	2
C₂×#11718071	2
#13112670	2
C₁₃×#1767	2
#9604	13