Subgroups — #15882679

Order 3328.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 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	1	yes	maximal, characteristic
2	#13117799	C₂	1	2	yes	maximal
3	#13115697	C₂	1	2	yes	maximal
4	#13115471	C₂	1	4	yes	maximal
5	#13115000	C₂	1	1	yes	maximal, characteristic
6	#13114852	C₂	1	2	yes	maximal
7	C₄×#88058	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	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×#88058	C₂²	1	2	yes	—
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	C₂×#88057	C₂²	1	2	yes	—
18	C₂×#88056	C₂²	1	4	yes	—
19	C₂×#88056	C₂²	1	2	yes	—
20	#11718639	—	2	4	no	—
21	#11718597	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
23	#11718590	C₂²	1	4	yes	—
24	C₄×Dic₅₂	C₄	1	8	yes	complemented
25	#11718312	—	2	2	no	—
26	#11718309	—	2	2	no	—
27	C₁₃×#336	C₂²	1	2	yes	—
28	#11718143	C₂²	1	4	yes	—
29	C₂²×Dic₂₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
38	C₂×#6101	—	2	2	no	—
39	C₂×C₄×Dic₁₃	—	2	2	no	—
40	C₂×C₄×Dic₁₃	—	2	4	no	—
41	C₂×Dic₅₂	—	2	2	no	—
42	C₂×Dic₅₂	—	2	2	no	—
43	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
44	C₄×Dic₂₆	—	2	4	no	—
45	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
46	C₁₃×C₄⋊C₈	—	2	2	no	—
47	C₄×C₈×C₁₃	—	2	4	no	—
48	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
49	C₁₃×#95	D₄	1	1	yes	characteristic
50	C₁₃×#95	D₄	1	1	yes	characteristic
51	#88071	—	2	2	no	—
52	#88071	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88071	—	2	4	no	—
55	#88071	—	2	2	no	—
56	#88058	—	2	4	no	—
57	#88058	C₂×C₄	1	4	yes	—
58	#88057	—	4	2	no	—
59	#88056	—	4	4	no	—
60	#88056	C₂×C₄	1	8	yes	—
61	#22809	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	2	2	no	—
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂×Dic₂₆	—	2	2	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×Dic₂₆	—	2	2	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	2	no	—
75	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	#41	1	2	yes	—
84	C₂×C₈×C₁₃	#41	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₃	#41	1	2	yes	—
90	C₄⋊Dic₁₃	—	2	2	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
94	#6101	—	2	8	no	—
95	C₄×Dic₁₃	—	2	8	no	—
96	C₄×Dic₁₃	—	4	8	no	—
97	Dic₅₂	—	2	8	no	—
98	C₂×C₄×Q₁₆	—	13	1	no	—
99	#2356	—	13	2	no	—
100	#1969	—	13	2	no	—
101	#1927	—	13	4	no	—
102	#1846	—	13	1	no	—
103	#1805	—	13	2	no	—
104	C₄×#107	—	13	2	no	—
105	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	#135	1	4	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	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	—	4	2	no	—
126	C₂×Dic₁₃	—	4	2	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	—	4	2	no	—
134	Dic₂₆	—	4	2	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	4	2	no	—
137	C₈×C₁₃	#117	1	4	yes	—
138	C₂²×Q₁₆	—	13	1	no	—
139	C₂×C₄×Q₈	—	13	2	no	—
140	C₄×Q₁₆	—	13	8	no	—
141	C₂×#107	—	13	1	no	—
142	C₂×#107	—	13	2	no	—
143	C₂×#106	—	13	2	no	—
144	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
145	C₂×#103	—	13	2	no	—
146	C₂×#103	—	13	4	no	—
147	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	#391	—	13	4	no	—
150	#387	—	26	4	no	—
151	#383	—	26	2	no	—
152	#382	—	13	2	no	—
153	C₄×C₄⋊C₄	—	13	4	no	—
154	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
155	#376	—	26	2	no	—
156	#340	—	13	4	no	—
157	#336	C₂×D₁₃	1	2	yes	—
158	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₁₃	#547	1	1	yes	characteristic
160	C₂²×C₁₃	#550	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#135	1	2	yes	—
162	C₂²×C₁₃	#578	1	2	yes	—
163	C₄×C₁₃	—	2	4	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
168	C₄×C₁₃	#520	1	1	yes	characteristic
169	C₄×C₁₃	#555	1	1	yes	characteristic
170	C₄×C₁₃	#556	1	1	yes	characteristic
171	Dic₁₃	—	4	4	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	—	26	2	no	—
177	C₂×Q₁₆	—	26	2	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	4	no	—
183	C₂×C₄⋊C₄	—	13	2	no	—
184	C₂×C₄⋊C₄	—	13	4	no	—
185	C₂×C₄⋊C₄	—	26	2	no	—
186	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
187	C₂×C₄²	—	26	2	no	—
188	C₂×C₄²	—	26	4	no	—
189	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
190	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
191	#107	—	13	4	no	—
192	#107	—	26	4	no	—
193	#106	—	52	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	#103	—	13	8	no	—
196	#103	—	52	4	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	C₄×D₁₃	1	4	yes	—
199	#95	—	26	2	no	—
200	#95	—	26	4	no	—
201	#95	—	26	4	no	—
202	#95	—	26	2	no	—
203	#95	—	26	4	no	—
204	#95	D₅₂	1	1	yes	characteristic
205	#95	D₅₂	1	1	yes	characteristic
206	C₂×C₁₃	#2354	1	1	yes	characteristic
207	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
208	C₂×C₁₃	#2950	1	1	yes	characteristic
209	C₂×C₁₃	#3336	1	2	yes	—
210	C₂×C₁₃	#3393	1	2	yes	—
211	C₂×Q₈	—	13	4	no	—
212	C₂×Q₈	—	26	2	no	—
213	C₂×Q₈	—	26	2	no	—
214	C₂×Q₈	—	26	2	no	—
215	C₂×Q₈	—	26	2	no	—
216	C₂×Q₈	—	26	2	no	—
217	C₂²×C₄	—	26	2	no	—
218	C₂²×C₄	—	26	2	no	—
219	C₂²×C₄	—	26	4	no	—
220	C₂²×C₄	C₂×D₅₂	1	2	yes	—
221	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₅₂	1	2	yes	—
224	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	Q₁₆	—	26	8	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	#6129	1	2	yes	—
234	C₂×C₈	#6131	1	2	yes	—
235	C₄⋊C₄	—	2	4	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	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	26	8	no	—
244	C₄²	—	52	8	no	—
245	C₄²	C₂×C₄×D₁₃	1	2	yes	—
246	C₄²	#6127	1	2	yes	—
247	C₁₃	#22809	1	1	yes	characteristic, complemented, Sylow
248	C₂³	C₂²×D₅₂	1	1	yes	characteristic
249	Q₈	—	26	4	no	—
250	Q₈	—	26	4	no	—
251	Q₈	—	52	2	no	—
252	Q₈	—	52	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	26	4	no	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	52	4	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	2	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	#88158	1	4	yes	—
271	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#6127	1	1	yes	characteristic
273	C₂×C₄	C₂×#6129	1	1	yes	characteristic
274	C₂×C₄	C₂×#6131	1	1	yes	characteristic
275	C₂×C₄	#88256	1	1	yes	characteristic
276	C₂×C₄	#88257	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	#88146	1	4	yes	—
280	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
281	C₂²	#11719260	1	1	yes	characteristic
282	C₂²	#11719282	1	1	yes	characteristic
283	C₂²	C₂×#88158	1	2	yes	—
284	C₂²	#11719458	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	52	4	no	—
290	C₄	—	52	4	no	—
291	C₄	—	104	4	no	—
292	C₄	C₂×#88146	1	1	yes	characteristic
293	C₄	#11719228	1	1	yes	characteristic
294	C₄	#11719268	1	1	yes	characteristic
295	C₄	#11719289	1	1	yes	characteristic
296	C₂	#13117783	1	1	yes	characteristic
297	C₂	C₄×#88158	1	1	yes	characteristic
298	C₂	#13123705	1	1	yes	characteristic
299	C₂	#13127205	1	2	yes	—
300	C₂	#13127825	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	#22809
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
24. C₄×Dic₅₂	C₄	semidirect
247. C₁₃	#22809	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
#13117799	2
#13115697	2
#13115471	2
#13115000	2
#13114852	2
C₄×#88058	2
C₈×C₁₃×C₄⋊C₄	2
#22809	13