Subgroups — #15881246

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718377	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13117756	C₂	1	2	yes	maximal
4	#13115574	C₂	1	2	yes	maximal
5	#13115093	C₂	1	4	yes	maximal
6	#13115000	C₂	1	1	yes	maximal, characteristic
7	#13114852	C₂	1	2	yes	maximal
8	C₁₃×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₂²	1	4	yes	—
11	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88057	C₂²	1	2	yes	—
19	C₂×#88056	C₂²	1	4	yes	—
20	C₂×#88056	C₂²	1	2	yes	—
21	#11718633	—	2	4	no	—
22	#11718597	C₂²	1	2	yes	—
23	#11718589	C₂²	1	4	yes	—
24	#11718377	—	2	8	no	—
25	C₄×Dic₅₂	C₄	1	8	yes	complemented
26	#11718311	—	2	2	no	—
27	#11718310	—	2	2	no	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	#11718143	C₂²	1	4	yes	—
30	C₂²×Dic₂₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
39	C₂×#6101	—	2	2	no	—
40	C₂×#6101	—	2	4	no	—
41	C₂×#6101	—	2	4	no	—
42	C₂×C₄×Dic₁₃	—	2	2	no	—
43	C₂×C₄×Dic₁₃	—	2	4	no	—
44	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
45	C₂×Dic₅₂	D₄	1	4	yes	—
46	C₂×Dic₅₂	Q₈	1	4	yes	—
47	C₄×Dic₂₆	—	2	4	no	—
48	C₄×Dic₂₆	—	2	8	no	—
49	C₄×Dic₂₆	—	4	4	no	—
50	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₁₃×#97	—	2	4	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
54	C₁₃×#95	D₄	1	2	yes	—
55	#88071	—	2	2	no	—
56	#88071	—	2	4	no	—
57	#88071	—	2	2	no	—
58	#88058	—	4	2	no	—
59	#88058	C₂×C₄	1	4	yes	—
60	#88057	—	2	4	no	—
61	#88056	—	2	8	no	—
62	#88056	C₂×C₄	1	8	yes	—
63	#22675	—	13	1	no	maximal, Sylow
64	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×Dic₂₆	—	2	2	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
75	C₂×Dic₂₆	C₂×D₄	1	4	yes	—
76	C₂×Dic₂₆	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
86	C₁₃×C₄⋊C₄	—	4	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₃	#41	1	2	yes	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	2	4	no	—
94	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
95	#6101	—	2	8	no	—
96	#6101	—	4	8	no	—
97	#6101	—	4	8	no	—
98	C₄×Dic₁₃	—	2	8	no	—
99	C₄×Dic₁₃	—	4	8	no	—
100	Dic₅₂	C₄⋊C₄	1	16	yes	complemented
101	C₂×#441	—	13	2	no	—
102	C₂×C₄×Q₁₆	—	13	1	no	—
103	#2351	—	13	2	no	—
104	#1943	—	13	2	no	—
105	#1863	—	13	4	no	—
106	#1846	—	13	1	no	—
107	#1805	—	13	2	no	—
108	#1770	D₁₃	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	—	2	4	no	—
125	C₂×Dic₁₃	—	2	4	no	—
126	C₂×Dic₁₃	—	2	4	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	Dic₂₆	—	2	4	no	—
132	Dic₂₆	—	2	4	no	—
133	Dic₂₆	—	2	4	no	—
134	Dic₂₆	C₂×C₄⋊C₄	1	8	yes	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	4	2	no	—
137	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₂²×Q₁₆	—	13	1	no	—
139	C₂×C₄×Q₈	—	13	2	no	—
140	C₂×C₄×Q₈	—	13	4	no	—
141	#441	—	26	8	no	—
142	C₄×Q₁₆	—	13	8	no	—
143	C₂×#107	—	13	1	no	—
144	C₂×#107	—	13	2	no	—
145	C₂×#106	—	13	2	no	—
146	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₂×#103	—	13	2	no	—
148	C₂×#103	—	13	4	no	—
149	C₂×#97	C₂×D₁₃	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	#389	—	13	4	no	—
152	#389	—	26	4	no	—
153	#382	—	13	2	no	—
154	#382	—	26	2	no	—
155	C₄×C₄⋊C₄	—	26	2	no	—
156	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
157	#340	—	13	4	no	—
158	#336	C₂×D₁₃	1	2	yes	—
159	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₁₃	D₄²	1	1	yes	characteristic
161	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
162	C₂²×C₁₃	#575	1	2	yes	—
163	C₂²×C₁₃	#578	1	2	yes	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	4	no	—
167	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₁₃	#520	1	1	yes	characteristic
169	C₄×C₁₃	#547	1	1	yes	characteristic
170	C₄×C₁₃	#554	1	1	yes	characteristic
171	Dic₁₃	—	2	8	no	—
172	Dic₁₃	—	4	4	no	—
173	Dic₁₃	—	8	4	no	—
174	C₂²×Q₈	—	13	2	no	—
175	C₂×Q₁₆	—	13	4	no	—
176	C₂×Q₁₆	—	13	4	no	—
177	C₂×Q₁₆	—	13	4	no	—
178	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
179	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₄×Q₈	—	13	8	no	—
182	C₄×Q₈	—	26	8	no	—
183	C₄×Q₈	—	26	4	no	—
184	C₄×Q₈	—	52	4	no	—
185	C₂×C₄⋊C₄	—	13	2	no	—
186	C₂×C₄⋊C₄	—	13	4	no	—
187	C₂×C₄⋊C₄	—	26	2	no	—
188	C₂×C₄⋊C₄	—	26	4	no	—
189	C₂×C₄⋊C₄	—	26	4	no	—
190	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
191	C₂×C₄²	—	26	2	no	—
192	C₂×C₄²	—	26	4	no	—
193	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	#107	—	13	4	no	—
196	#107	—	52	2	no	—
197	#106	—	26	4	no	—
198	C₄⋊C₈	—	2	2	no	—
199	#103	—	13	8	no	—
200	#103	—	26	8	no	—
201	#97	—	2	4	no	—
202	C₄×C₈	C₄×D₁₃	1	4	yes	—
203	#95	—	26	2	no	—
204	#95	—	26	2	no	—
205	#95	—	26	4	no	—
206	#95	D₅₂	1	2	yes	—
207	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₁₃	#2943	1	1	yes	characteristic
209	C₂×C₁₃	#2950	1	1	yes	characteristic
210	C₂×C₁₃	#3277	1	2	yes	—
211	C₂×C₁₃	#3390	1	2	yes	—
212	C₂×Q₈	—	13	4	no	—
213	C₂×Q₈	—	13	4	no	—
214	C₂×Q₈	—	13	4	no	—
215	C₂×Q₈	—	26	2	no	—
216	C₂×Q₈	—	26	2	no	—
217	C₂×Q₈	—	26	2	no	—
218	C₂²×C₄	—	26	2	no	—
219	C₂²×C₄	—	26	2	no	—
220	C₂²×C₄	—	26	4	no	—
221	C₂²×C₄	C₂×D₅₂	1	4	yes	—
222	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	Q₁₆	—	13	16	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
233	C₂×C₈	D₄×D₁₃	1	2	yes	—
234	C₂×C₈	Q₈×D₁₃	1	2	yes	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	13	4	no	—
237	C₄⋊C₄	—	26	8	no	—
238	C₄⋊C₄	—	26	2	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	—	26	4	no	—
241	C₄⋊C₄	—	52	8	no	—
242	C₄⋊C₄	—	52	8	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	26	8	no	—
246	C₄²	—	52	8	no	—
247	C₄²	C₂×C₄×D₁₃	1	2	yes	—
248	C₄²	#6127	1	2	yes	—
249	C₁₃	#22675	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂²×D₅₂	1	1	yes	characteristic
251	Q₈	—	13	8	no	—
252	Q₈	—	26	4	no	—
253	Q₈	—	26	4	no	—
254	Q₈	—	26	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	26	4	no	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	26	4	no	—
263	C₂×C₄	—	26	4	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	52	4	no	—
266	C₂×C₄	—	52	4	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	C₄×D₅₂	1	2	yes	—
269	C₂×C₄	C₄×D₅₂	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#6127	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	#88256	1	1	yes	characteristic
275	C₂×C₄	#88257	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	2	no	—
278	C₈	C₄⋊C₄×D₁₃	1	4	yes	—
279	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
280	C₂²	D₄×D₅₂	1	1	yes	characteristic
281	C₂²	Q₈×D₅₂	1	1	yes	characteristic
282	C₂²	#11719453	1	2	yes	—
283	C₂²	#11719458	1	2	yes	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	26	8	no	—
288	C₄	—	52	4	no	—
289	C₄	—	104	4	no	—
290	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
291	C₄	#11719228	1	1	yes	characteristic
292	C₄	#11719256	1	1	yes	characteristic
293	C₄	#11719284	1	1	yes	characteristic
294	C₂	C₄⋊C₄×D₅₂	1	1	yes	characteristic
295	C₂	#13123638	1	1	yes	characteristic
296	C₂	#13123705	1	1	yes	characteristic
297	C₂	#13126494	1	2	yes	—
298	C₂	#13127805	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1770	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1770	D₁₃
hypocenter	C₁₃	#22675
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
25. C₄×Dic₅₂	C₄	semidirect
100. Dic₅₂	C₄⋊C₄	semidirect
249. C₁₃	#22675	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718377	2
C₂×C₄×Dic₅₂	2
#13117756	2
#13115574	2
#13115093	2
#13115000	2
#13114852	2
C₁₃×#1770	2
#22675	13