Subgroups — #15882671

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 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	#13117791	C₂	1	2	yes	maximal
3	#13115697	C₂	1	2	yes	maximal
4	#13115478	C₂	1	4	yes	maximal
5	#13114996	C₂	1	1	yes	maximal, characteristic
6	#13114842	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	2	yes	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88056	C₂²	1	4	yes	—
19	C₂×#88056	C₂²	1	2	yes	—
20	#11718638	—	2	2	no	—
21	C₄⋊C₄×Dic₁₃	—	2	2	no	—
22	#11718596	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
24	#11718590	C₂²	1	4	yes	—
25	C₄×Dic₅₂	C₄	1	8	yes	complemented
26	#11718316	—	2	4	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	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6101	—	2	2	no	—
42	C₂×C₄×Dic₁₃	—	2	2	no	—
43	C₂×C₄×Dic₁₃	—	2	4	no	—
44	C₂×Dic₅₂	—	2	2	no	—
45	C₂×Dic₅₂	—	2	2	no	—
46	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
47	C₄×Dic₂₆	—	2	4	no	—
48	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
49	C₁₃×C₄⋊C₈	—	2	2	no	—
50	C₄×C₈×C₁₃	—	2	4	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
52	C₁₃×#95	D₄	1	2	yes	—
53	#88071	—	2	4	no	—
54	#88071	—	2	4	no	—
55	#88071	—	2	4	no	—
56	#88058	—	2	4	no	—
57	#88058	—	4	2	no	—
58	#88058	C₂×C₄	1	4	yes	—
59	#88056	—	4	4	no	—
60	#88056	C₂×C₄	1	8	yes	—
61	#22808	—	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	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
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	2	no	—
86	C₁₃×C₄⋊C₄	Q₁₆	1	4	yes	complemented
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	—	4	4	no	—
95	C₄⋊Dic₁₃	—	8	2	no	—
96	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
97	#6101	—	2	8	no	—
98	C₄×Dic₁₃	—	2	8	no	—
99	C₄×Dic₁₃	—	8	4	no	—
100	Dic₅₂	—	2	8	no	—
101	C₂×C₄×Q₁₆	—	13	1	no	—
102	#2355	—	13	2	no	—
103	#1969	—	13	2	no	—
104	#1928	—	13	4	no	—
105	#1845	—	13	1	no	—
106	#1803	—	13	2	no	—
107	C₄×#107	—	13	2	no	—
108	C₈×C₄⋊C₄	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	1	no	—
113	C₂×C₄×C₁₃	—	2	1	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	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
120	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₁₃	#135	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	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	—	4	2	no	—
131	C₂×Dic₁₃	—	4	2	no	—
132	C₂×Dic₁₃	—	4	2	no	—
133	C₂×Dic₁₃	—	4	2	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₂×Dic₁₃	—	4	4	no	—
136	C₂×Dic₁₃	—	4	4	no	—
137	Dic₂₆	—	2	4	no	—
138	Dic₂₆	—	2	4	no	—
139	Dic₂₆	—	4	2	no	—
140	Dic₂₆	—	4	2	no	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	#117	1	4	yes	—
144	C₂²×Q₁₆	—	13	1	no	—
145	C₂×C₄×Q₈	—	13	2	no	—
146	C₄×Q₁₆	—	13	8	no	—
147	C₂×#107	—	13	1	no	—
148	C₂×#107	—	13	2	no	—
149	C₂×#107	—	13	2	no	—
150	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×#103	—	13	2	no	—
152	C₂×#103	—	13	4	no	—
153	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
154	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
155	#391	—	13	4	no	—
156	#387	—	26	4	no	—
157	#384	—	13	2	no	—
158	#382	—	26	2	no	—
159	C₄×C₄⋊C₄	—	13	4	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	#547	1	1	yes	characteristic
166	C₂²×C₁₃	#550	1	1	yes	characteristic
167	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
169	C₂²×C₁₃	#576	1	1	yes	characteristic
170	C₂²×C₁₃	#578	1	1	yes	characteristic
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	4	2	no	—
174	C₄×C₁₃	C₄×Q₁₆	1	4	yes	—
175	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
176	C₄×C₁₃	#520	1	1	yes	characteristic
177	C₄×C₁₃	#555	1	1	yes	characteristic
178	C₄×C₁₃	#556	1	1	yes	characteristic
179	Dic₁₃	—	4	4	no	—
180	Dic₁₃	—	4	4	no	—
181	Dic₁₃	—	8	4	no	—
182	C₂²×Q₈	—	13	2	no	—
183	C₂×Q₁₆	—	13	4	no	—
184	C₂×Q₁₆	—	26	2	no	—
185	C₂×Q₁₆	—	26	2	no	—
186	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×Q₈	—	13	8	no	—
190	C₄×Q₈	—	26	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	2	no	—
195	C₂×C₄⋊C₄	—	26	2	no	—
196	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
197	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
198	C₂×C₄²	—	26	2	no	—
199	C₂×C₄²	—	26	4	no	—
200	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
201	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
202	#107	—	13	4	no	—
203	#107	—	26	4	no	—
204	#107	—	52	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	#103	—	13	8	no	—
207	#103	—	52	4	no	—
208	C₄×C₈	—	2	4	no	—
209	C₄×C₈	C₄×D₁₃	1	4	yes	—
210	#95	—	26	4	no	—
211	#95	—	26	4	no	—
212	#95	—	26	4	no	—
213	#95	D₅₂	1	2	yes	—
214	C₂×C₁₃	#2354	1	1	yes	characteristic
215	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
216	C₂×C₁₃	#2951	1	1	yes	characteristic
217	C₂×C₁₃	#3338	1	1	yes	characteristic
218	C₂×C₁₃	#3342	1	1	yes	characteristic
219	C₂×C₁₃	#3395	1	1	yes	characteristic
220	C₂×C₁₃	#3396	1	1	yes	characteristic
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	C₂×Q₈	—	26	2	no	—
226	C₂×Q₈	—	26	2	no	—
227	C₂²×C₄	—	26	2	no	—
228	C₂²×C₄	—	26	2	no	—
229	C₂²×C₄	—	26	4	no	—
230	C₂²×C₄	C₂×D₅₂	1	2	yes	—
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₅₂	1	2	yes	—
233	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
234	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
235	Q₁₆	—	26	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
243	C₂×C₈	#6129	1	2	yes	—
244	C₂×C₈	#6131	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	13	4	no	—
247	C₄⋊C₄	—	26	8	no	—
248	C₄⋊C₄	—	26	2	no	—
249	C₄⋊C₄	—	26	4	no	—
250	C₄⋊C₄	—	26	4	no	—
251	C₄⋊C₄	—	52	4	no	—
252	C₄⋊C₄	—	104	2	no	—
253	C₄⋊C₄	Dic₅₂	1	4	yes	complemented
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	26	8	no	—
257	C₄²	—	104	4	no	—
258	C₄²	C₂×C₄×D₁₃	1	2	yes	—
259	C₄²	#6127	1	2	yes	—
260	C₁₃	#22808	1	1	yes	characteristic, complemented, Sylow
261	C₂³	C₂²×D₅₂	1	1	yes	characteristic
262	Q₈	—	26	4	no	—
263	Q₈	—	26	4	no	—
264	Q₈	—	52	2	no	—
265	Q₈	—	52	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	—	52	2	no	—
276	C₂×C₄	—	52	2	no	—
277	C₂×C₄	—	52	2	no	—
278	C₂×C₄	—	52	2	no	—
279	C₂×C₄	—	52	4	no	—
280	C₂×C₄	—	52	4	no	—
281	C₂×C₄	—	52	4	no	—
282	C₂×C₄	C₄×D₅₂	1	2	yes	—
283	C₂×C₄	C₄×D₅₂	1	2	yes	—
284	C₂×C₄	#88158	1	2	yes	—
285	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
286	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#6127	1	1	yes	characteristic
289	C₂×C₄	C₂×#6129	1	1	yes	characteristic
290	C₂×C₄	C₂×#6131	1	1	yes	characteristic
291	C₂×C₄	#88256	1	1	yes	characteristic
292	C₂×C₄	#88257	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#88146	1	4	yes	—
296	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
297	C₂²	#11719260	1	1	yes	characteristic
298	C₂²	#11719282	1	1	yes	characteristic
299	C₂²	C₂×#88158	1	1	yes	characteristic
300	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
301	C₂²	#11719457	1	1	yes	characteristic
302	C₂²	#11719458	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	52	4	no	—
307	C₄	—	52	4	no	—
308	C₄	—	104	4	no	—
309	C₄	C₄×Dic₅₂	1	4	yes	—
310	C₄	C₂×#88146	1	1	yes	characteristic
311	C₄	#11719228	1	1	yes	characteristic
312	C₄	#11719268	1	1	yes	characteristic
313	C₄	#11719289	1	1	yes	characteristic
314	C₂	#13117783	1	1	yes	characteristic
315	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
316	C₂	#13123713	1	1	yes	characteristic
317	C₂	#13127229	1	1	yes	characteristic
318	C₂	#13127267	1	1	yes	characteristic
319	C₂	#13127842	1	1	yes	characteristic
320	C₂	#13127849	1	1	yes	characteristic

	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	#22808
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
86. C₁₃×C₄⋊C₄	Q₁₆	semidirect
253. C₄⋊C₄	Dic₅₂	semidirect
260. C₁₃	#22808	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
#13117791	2
#13115697	2
#13115478	2
#13114996	2
#13114842	2
C₄×#88058	2
C₈×C₁₃×C₄⋊C₄	2
#22808	13