Subgroups — #15846127

Order 3328.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125779	C₂	1	2	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
3	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114321	C₂	1	4	yes	maximal, complemented
5	#13114197	C₂	1	4	yes	maximal, complemented
6	#13113503	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₀₄	—	2	1	no	—
9	C₂²×D₁₀₄	—	2	1	no	—
10	C₂×#88128	C₂²	1	2	yes	—
11	#11719175	C₂²	1	1	yes	characteristic
12	C₁₃×#431	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718763	—	2	1	no	—
15	#11718762	—	2	1	no	—
16	#11718585	C₂²	1	1	yes	characteristic
17	#11718509	C₂²	1	8	yes	complemented
18	#11718392	—	2	4	no	—
19	#11718356	—	2	1	no	—
20	#11718356	—	2	1	no	—
21	#11718356	C₂²	1	2	yes	—
22	#11718356	C₂²	1	2	yes	—
23	#11718355	—	2	2	no	—
24	#11718354	—	2	2	no	—
25	#11718340	—	2	1	no	—
26	#11718340	C₂²	1	2	yes	complemented
27	C₈×Dic₂₆	—	2	1	no	—
28	C₈×Dic₂₆	C₂²	1	2	yes	complemented
29	C₁₃×#335	C₂²	1	2	yes	complemented
30	C₁₃×#334	C₂²	1	2	yes	complemented
31	#11718077	C₂²	1	8	yes	complemented
32	C₂²×D₅₂	—	2	2	no	—
33	C₂²×D₅₂	—	2	2	no	—
34	C₂²×D₅₂	—	2	2	no	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	#88180	—	2	1	no	—
39	#88178	—	2	1	no	—
40	#88178	D₄	1	2	yes	—
41	C₂×#6099	—	2	1	no	—
42	C₂×D₁₀₄	—	2	2	no	—
43	C₂×D₁₀₄	—	2	2	no	—
44	C₂×D₁₀₄	—	2	2	no	—
45	C₂×D₁₀₄	—	2	2	no	—
46	C₂×D₁₀₄	—	4	4	no	—
47	C₂×D₁₀₄	—	4	4	no	—
48	C₂×#6095	—	2	8	no	—
49	#88128	—	2	2	no	—
50	#88128	—	2	2	no	—
51	#88128	—	2	2	no	—
52	#88128	C₂³	1	4	yes	—
53	#88124	—	2	1	no	—
54	#88123	D₄	1	1	yes	characteristic
55	#88123	D₄	1	1	yes	characteristic
56	C₄×Dic₂₆	—	2	1	no	—
57	C₄×Dic₂₆	C₂³	1	2	yes	—
58	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
59	C₁₃×C₂²⋊C₈	—	2	2	no	—
60	C₄×C₈×C₁₃	—	2	2	no	—
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#88061	—	2	8	no	—
66	#88055	—	2	2	no	—
67	#88054	—	2	2	no	—
68	#88054	—	2	2	no	—
69	C₈×Dic₁₃	—	2	2	no	—
70	C₄⋊#1055	—	2	1	no	—
71	C₄⋊#1055	C₂³	1	2	yes	—
72	#19203	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂×D₅₂	—	2	4	no	—
78	C₂×D₅₂	—	2	4	no	—
79	C₂×D₅₂	—	2	4	no	—
80	C₂×D₅₂	—	4	4	no	—
81	C₂×D₅₂	—	4	4	no	—
82	C₂×D₅₂	—	4	2	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	8	4	no	—
86	C₂×D₅₂	—	8	4	no	—
87	C₂×Dic₂₆	—	2	1	no	—
88	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	#41	1	1	yes	characteristic
99	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	1	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	2	1	no	—
104	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
105	#6101	—	2	2	no	—
106	#6101	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	#6099	—	4	2	no	—
109	C₂×#1055	—	2	2	no	—
110	D₁₀₄	—	4	4	no	—
111	D₁₀₄	—	4	4	no	—
112	D₁₀₄	—	4	4	no	—
113	D₁₀₄	—	4	4	no	—
114	#6095	—	8	8	no	—
115	#3208	—	13	2	no	—
116	C₂×#493	—	13	1	no	—
117	#2960	—	13	1	no	—
118	#1688	—	13	4	no	—
119	#1668	—	13	4	no	—
120	#1564	D₁₃	1	1	yes	characteristic, complemented
121	#1545	—	13	2	no	—
122	C₂³×C₁₃	#121	1	1	yes	characteristic
123	C₂²×D₁₃	—	4	4	no	—
124	C₂²×D₁₃	—	8	4	no	—
125	C₂²×D₁₃	—	8	4	no	—
126	C₂²×D₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	D₅₂	—	4	4	no	—
140	D₅₂	—	4	4	no	—
141	D₅₂	—	4	8	no	—
142	D₅₂	—	8	4	no	—
143	D₅₂	—	8	4	no	—
144	D₅₂	—	8	4	no	—
145	D₅₂	—	8	4	no	—
146	Dic₂₆	—	2	4	no	—
147	Dic₂₆	—	4	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	4	4	no	—
151	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
152	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
153	#1055	—	4	2	no	—
154	C₂²×D₈	—	26	1	no	—
155	C₂²×D₈	—	26	1	no	—
156	#533	—	13	1	no	—
157	C₂×#127	—	13	2	no	—
158	#493	—	13	2	no	—
159	#493	—	13	2	no	—
160	#493	—	26	1	no	—
161	#493	—	26	1	no	—
162	#492	—	26	2	no	—
163	#486	—	26	2	no	—
164	#460	—	13	8	no	—
165	#450	—	26	4	no	—
166	#446	—	13	2	no	—
167	#446	—	26	1	no	—
168	C₈×Q₈	—	13	2	no	—
169	C₈×Q₈	—	26	1	no	—
170	#431	C₂×D₁₃	1	2	yes	—
171	#424	—	26	1	no	—
172	#423	—	13	1	no	—
173	#405	—	26	1	no	—
174	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
175	#335	C₂×D₁₃	1	2	yes	complemented
176	#334	C₂×D₁₃	1	2	yes	complemented
177	#331	—	13	8	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	#460	1	2	yes	—
181	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
182	C₂×D₁₃	—	8	4	no	—
183	C₂×D₁₃	—	8	4	no	—
184	C₂×D₁₃	—	8	4	no	—
185	C₂×D₁₃	—	16	8	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	#460	1	2	yes	—
189	C₄×C₁₃	#472	1	2	yes	—
190	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
191	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₄×C₁₃	#576	1	2	yes	—
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	—	4	2	no	—
195	C₂²×D₄	—	26	2	no	—
196	C₂²×D₄	—	26	2	no	—
197	C₂²×D₄	—	26	2	no	—
198	C₂×SD₁₆	—	26	8	no	—
199	C₂×D₈	—	26	2	no	—
200	C₂×D₈	—	26	2	no	—
201	C₂×D₈	—	26	2	no	—
202	C₂×D₈	—	26	2	no	—
203	C₂×D₈	—	52	4	no	—
204	C₂×D₈	—	52	4	no	—
205	C₂×M₁₆	—	26	1	no	—
206	C₂²×C₈	D₅₂	1	1	yes	characteristic
207	C₂²×C₈	D₅₂	1	1	yes	characteristic
208	C₄⋊Q₈	—	13	1	no	—
209	C₄⋊Q₈	—	13	1	no	—
210	#127	—	13	4	no	—
211	#127	—	26	2	no	—
212	#127	—	26	2	no	—
213	#127	—	26	2	no	—
214	#125	—	26	1	no	—
215	C₂²⋊Q₈	—	13	2	no	—
216	C₂²⋊Q₈	—	26	1	no	—
217	C₄×Q₈	—	13	2	no	—
218	C₄×Q₈	—	26	1	no	—
219	#117	—	26	1	no	—
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₄⋊C₈	—	13	2	no	—
222	C₄⋊C₈	—	26	2	no	—
223	C₄⋊C₈	—	26	2	no	—
224	C₄⋊C₈	—	26	1	no	—
225	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
226	#102	—	26	8	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	#97	—	26	2	no	—
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	—	26	2	no	—
231	C₄×C₈	D₅₂	1	1	yes	characteristic
232	C₄×C₈	D₅₂	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₂×C₁₃	—	2	2	no	—
236	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
237	C₂×C₁₃	#3058	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
239	D₁₃	—	16	4	no	—
240	C₂⁴	—	52	2	no	—
241	C₂×Q₈	—	13	2	no	—
242	C₂×Q₈	—	26	1	no	—
243	C₂×D₄	—	26	4	no	—
244	C₂×D₄	—	26	4	no	—
245	C₂×D₄	—	26	4	no	—
246	C₂×D₄	—	52	2	no	—
247	C₂×D₄	—	52	2	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	4	no	—
250	C₂×D₄	—	52	4	no	—
251	C₂×D₄	—	104	4	no	—
252	C₂×D₄	—	104	4	no	—
253	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	#6131	1	1	yes	characteristic
256	SD₁₆	—	104	8	no	—
257	D₈	—	52	4	no	—
258	D₈	—	52	4	no	—
259	D₈	—	52	4	no	—
260	D₈	—	52	4	no	—
261	M₁₆	—	52	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	#6095	1	4	yes	complemented
266	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
267	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
268	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
269	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
270	C₄⋊C₄	—	13	2	no	—
271	C₄⋊C₄	—	26	1	no	—
272	C₄⋊C₄	—	26	1	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	1	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₂²⋊C₄	—	26	2	no	—
277	C₄²	—	26	2	no	—
278	C₄²	C₂×D₅₂	1	1	yes	characteristic
279	C₄²	D₄×D₁₃	1	1	yes	characteristic
280	C₄²	#6131	1	1	yes	characteristic
281	C₄²	C₂³×D₁₃	1	1	yes	characteristic
282	C₁₃	#19203	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	52	4	no	—
284	C₂³	—	104	4	no	—
285	C₂³	—	104	4	no	—
286	C₂³	—	104	4	no	—
287	C₂³	#88149	1	1	yes	characteristic
288	Q₈	—	26	4	no	—
289	Q₈	—	52	2	no	—
290	D₄	—	52	4	no	—
291	D₄	—	52	4	no	—
292	D₄	—	52	8	no	—
293	D₄	—	104	4	no	—
294	D₄	—	104	4	no	—
295	D₄	—	104	4	no	—
296	D₄	—	104	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	#88149	1	1	yes	characteristic
302	C₂×C₄	#88149	1	1	yes	characteristic
303	C₂×C₄	#88149	1	1	yes	characteristic
304	C₂×C₄	C₂×#6095	1	2	yes	—
305	C₂×C₄	#88162	1	2	yes	—
306	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#6131	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₈	—	52	2	no	—
313	C₈	C₂×#6095	1	2	yes	—
314	C₈	C₂×#6095	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	104	4	no	—
318	C₂²	—	104	4	no	—
319	C₂²	—	104	4	no	—
320	C₂²	—	208	8	no	—
321	C₂²	#11718509	1	2	yes	—
322	C₂²	C₂×#88149	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	52	2	no	—
326	C₄	—	52	2	no	—
327	C₄	#11718509	1	2	yes	—
328	C₄	#11718558	1	2	yes	—
329	C₄	C₂²×#6095	1	1	yes	characteristic
330	C₄	C₂×#88162	1	1	yes	characteristic
331	C₄	#11719474	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	208	4	no	—
334	C₂	C₂×#11718509	1	1	yes	characteristic
335	C₂	#13124485	1	1	yes	characteristic
336	C₂	C₂×#11718558	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂×#11718356	C₂
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#19203
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #13125779	C₂	semidirect
3. #13123770	C₂	semidirect
4. #13114321	C₂	semidirect
5. #13114197	C₂	semidirect
6. #13113503	C₂	semidirect
7. C₁₃×#1564	C₂	semidirect
17. #11718509	C₂²	semidirect
26. #11718340	C₂²	semidirect
28. C₈×Dic₂₆	C₂²	semidirect
29. C₁₃×#335	C₂²	semidirect
30. C₁₃×#334	C₂²	semidirect
31. #11718077	C₂²	semidirect
91. C₂×C₈×C₁₃	SD₁₆	semidirect
120. #1564	D₁₃	semidirect
175. #335	C₂×D₁₃	semidirect
176. #334	C₂×D₁₃	semidirect
265. C₂×C₈	#6095	semidirect
282. C₁₃	#19203	semidirect

Maximal subgroups

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

subgroup	index
#13125779	2
C₂×#11718356	2
#13123770	2
#13114321	2
#13114197	2
#13113503	2
C₁₃×#1564	2
#19203	13