Subgroups — #15881360

Order 3328.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13117756	C₂	1	2	yes	maximal
4	#13115569	C₂	1	2	yes	maximal
5	#13115107	C₂	1	4	yes	maximal
6	#13115000	C₂	1	1	yes	maximal, characteristic
7	#13114852	C₂	1	2	yes	maximal
8	C₈×C₁₃×C₄⋊C₄	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×C₄×C₈×C₁₃	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	C₄×Dic₅₂	—	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	C₄×C₈×C₁₃	—	2	4	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
54	C₁₃×#95	D₄	1	1	yes	characteristic
55	C₁₃×#95	D₄	1	1	yes	characteristic
56	#88071	—	2	2	no	—
57	#88071	—	2	4	no	—
58	#88071	—	2	2	no	—
59	#88058	—	2	4	no	—
60	#88058	C₂×C₄	1	4	yes	—
61	#88057	—	4	2	no	—
62	#88056	—	2	8	no	—
63	#88056	C₂×C₄	1	8	yes	—
64	#22685	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	—	2	2	no	—
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	2	no	—
75	C₂×Dic₂₆	—	2	2	no	—
76	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
77	C₂×Dic₂₆	C₂×D₄	1	4	yes	—
78	C₂×Dic₂₆	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃	#41	1	2	yes	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	2	4	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
97	#6101	—	2	8	no	—
98	#6101	—	4	8	no	—
99	#6101	—	4	8	no	—
100	C₄×Dic₁₃	—	2	8	no	—
101	C₄×Dic₁₃	—	4	8	no	—
102	Dic₅₂	C₄⋊C₄	1	16	yes	complemented
103	C₂×C₄×Q₁₆	—	13	1	no	—
104	C₂×C₄×Q₁₆	—	13	2	no	—
105	#2351	—	13	2	no	—
106	#1942	—	13	2	no	—
107	#1865	—	13	4	no	—
108	#1846	—	13	1	no	—
109	#1805	—	13	2	no	—
110	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	#135	1	4	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	4	no	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	2	4	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₂×Dic₁₃	—	4	4	no	—
136	Dic₂₆	—	2	4	no	—
137	Dic₂₆	—	2	4	no	—
138	Dic₂₆	—	2	4	no	—
139	Dic₂₆	C₂×C₄⋊C₄	1	8	yes	—
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	—	4	2	no	—
142	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
143	C₂²×Q₁₆	—	13	1	no	—
144	C₂×C₄×Q₈	—	13	2	no	—
145	C₂×C₄×Q₈	—	13	4	no	—
146	C₄×Q₁₆	—	13	8	no	—
147	C₄×Q₁₆	—	26	8	no	—
148	C₂×#107	—	13	1	no	—
149	C₂×#107	—	13	2	no	—
150	C₂×#106	—	13	2	no	—
151	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂×#103	—	13	2	no	—
153	C₂×#103	—	13	4	no	—
154	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	#389	—	13	4	no	—
157	#389	—	26	4	no	—
158	#382	—	13	2	no	—
159	#382	—	26	2	no	—
160	C₄×C₄⋊C₄	—	26	2	no	—
161	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
162	#340	—	13	4	no	—
163	#336	C₂×D₁₃	1	2	yes	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₁₃	D₄²	1	1	yes	characteristic
166	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#135	1	2	yes	—
168	C₂²×C₁₃	#578	1	2	yes	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	4	2	no	—
173	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₁₃	#520	1	1	yes	characteristic
175	C₄×C₁₃	#547	1	1	yes	characteristic
176	C₄×C₁₃	#554	1	1	yes	characteristic
177	Dic₁₃	—	2	8	no	—
178	Dic₁₃	—	4	4	no	—
179	Dic₁₃	—	8	4	no	—
180	C₂²×Q₈	—	13	2	no	—
181	C₂×Q₁₆	—	13	4	no	—
182	C₂×Q₁₆	—	13	4	no	—
183	C₂×Q₁₆	—	13	4	no	—
184	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₄×Q₈	—	13	8	no	—
188	C₄×Q₈	—	26	8	no	—
189	C₄×Q₈	—	26	4	no	—
190	C₄×Q₈	—	52	4	no	—
191	C₂×C₄⋊C₄	—	13	2	no	—
192	C₂×C₄⋊C₄	—	13	4	no	—
193	C₂×C₄⋊C₄	—	26	2	no	—
194	C₂×C₄⋊C₄	—	26	4	no	—
195	C₂×C₄⋊C₄	—	26	4	no	—
196	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
197	C₂×C₄²	—	26	2	no	—
198	C₂×C₄²	—	26	4	no	—
199	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
200	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
201	#107	—	13	4	no	—
202	#107	—	26	4	no	—
203	#106	—	52	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	13	8	no	—
206	#103	—	26	8	no	—
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₄×D₁₃	1	4	yes	—
209	#95	—	26	2	no	—
210	#95	—	26	2	no	—
211	#95	—	26	4	no	—
212	#95	D₅₂	1	1	yes	characteristic
213	#95	D₅₂	1	1	yes	characteristic
214	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
216	C₂×C₁₃	#2950	1	1	yes	characteristic
217	C₂×C₁₃	#3285	1	2	yes	—
218	C₂×C₁₃	#3383	1	2	yes	—
219	C₂×Q₈	—	13	4	no	—
220	C₂×Q₈	—	13	4	no	—
221	C₂×Q₈	—	13	4	no	—
222	C₂×Q₈	—	26	2	no	—
223	C₂×Q₈	—	26	2	no	—
224	C₂×Q₈	—	26	2	no	—
225	C₂²×C₄	—	26	2	no	—
226	C₂²×C₄	—	26	2	no	—
227	C₂²×C₄	—	26	4	no	—
228	C₂²×C₄	C₂×D₅₂	1	2	yes	—
229	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₅₂	1	2	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
233	Q₁₆	—	13	16	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
241	C₂×C₈	D₄×D₁₃	1	2	yes	—
242	C₂×C₈	Q₈×D₁₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	13	4	no	—
245	C₄⋊C₄	—	26	8	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₄⋊C₄	—	26	4	no	—
248	C₄⋊C₄	—	26	4	no	—
249	C₄⋊C₄	—	52	8	no	—
250	C₄⋊C₄	—	52	8	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	26	8	no	—
254	C₄²	—	52	8	no	—
255	C₄²	C₂×C₄×D₁₃	1	2	yes	—
256	C₄²	#6127	1	2	yes	—
257	C₁₃	#22685	1	1	yes	characteristic, complemented, Sylow
258	C₂³	C₂²×D₅₂	1	1	yes	characteristic
259	Q₈	—	13	8	no	—
260	Q₈	—	26	4	no	—
261	Q₈	—	26	4	no	—
262	Q₈	—	26	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	52	4	no	—
275	C₂×C₄	—	52	4	no	—
276	C₂×C₄	—	52	4	no	—
277	C₂×C₄	—	52	4	no	—
278	C₂×C₄	C₄×D₅₂	1	2	yes	—
279	C₂×C₄	C₄×D₅₂	1	2	yes	—
280	C₂×C₄	#88158	1	4	yes	—
281	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#6127	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
285	C₂×C₄	#88256	1	1	yes	characteristic
286	C₂×C₄	#88257	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	C₄⋊C₄×D₁₃	1	4	yes	—
290	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
291	C₂²	D₄×D₅₂	1	1	yes	characteristic
292	C₂²	Q₈×D₅₂	1	1	yes	characteristic
293	C₂²	C₂×#88158	1	2	yes	—
294	C₂²	#11719458	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	2	no	—
299	C₄	—	26	8	no	—
300	C₄	—	52	4	no	—
301	C₄	—	104	4	no	—
302	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
303	C₄	#11719228	1	1	yes	characteristic
304	C₄	#11719256	1	1	yes	characteristic
305	C₄	#11719284	1	1	yes	characteristic
306	C₂	C₄⋊C₄×D₅₂	1	1	yes	characteristic
307	C₂	C₄×#88158	1	1	yes	characteristic
308	C₂	#13123705	1	1	yes	characteristic
309	C₂	#13126579	1	2	yes	—
310	C₂	#13127728	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#22685
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
25. C₄×Dic₅₂	C₄	semidirect
102. Dic₅₂	C₄⋊C₄	semidirect
257. C₁₃	#22685	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13117756	2
#13115569	2
#13115107	2
#13115000	2
#13114852	2
C₈×C₁₃×C₄⋊C₄	2
#22685	13