Subgroups — #15845560

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic
2	#13124875	C₂	1	2	yes	maximal
3	#13114364	C₂	1	2	yes	maximal, complemented
4	#13114354	C₂	1	2	yes	maximal, complemented
5	#13114342	C₂	1	4	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic
7	#13113494	C₂	1	2	yes	maximal
8	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11719175	C₂²	1	2	yes	—
11	C₁₃×#446	—	2	1	no	—
12	C₁₃×#446	—	2	1	no	—
13	C₁₃×#446	C₂²	1	2	yes	—
14	C₁₃×#446	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718889	—	2	2	no	—
20	#11718767	—	2	1	no	—
21	#11718767	—	2	1	no	—
22	#11718692	C₂²	1	4	yes	—
23	C₄×#6107	—	2	4	no	—
24	#11718682	C₂²	1	4	yes	—
25	#11718631	—	2	2	no	—
26	#11718586	C₂²	1	2	yes	—
27	#11718344	—	2	1	no	—
28	#11718344	—	2	1	no	—
29	#11718343	C₂²	1	1	yes	characteristic
30	#11718343	C₂²	1	1	yes	characteristic
31	#11718341	C₂²	1	2	yes	—
32	#11718078	C₂²	1	4	yes	—
33	#11718073	C₂²	1	4	yes	—
34	#11718050	C₂²	1	2	yes	—
35	#11718049	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	—	2	1	no	—
41	C₄×Q₈×C₁₃	—	2	1	no	—
42	C₄×Q₈×C₁₃	D₄	1	2	yes	—
43	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	—	2	1	no	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×#6107	—	2	4	no	—
49	#88180	—	2	2	no	—
50	#88178	—	2	2	no	—
51	#88178	—	2	2	no	—
52	#88124	—	2	2	no	—
53	#88123	C₂³	1	2	yes	—
54	#88123	C₂³	1	2	yes	—
55	C₄×Dic₂₆	—	2	2	no	—
56	C₄×Dic₂₆	—	2	2	no	—
57	C₁₃×C₄⋊C₈	—	2	2	no	—
58	C₁₃×C₄⋊C₈	—	2	2	no	—
59	C₁₃×C₄⋊C₈	—	2	1	no	—
60	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
61	C₁₃×#97	—	2	2	no	—
62	C₁₃×#97	—	2	2	no	—
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	#88075	—	2	4	no	—
68	C₂²⋊#1055	—	2	2	no	—
69	#88058	—	2	1	no	—
70	#88058	—	2	1	no	—
71	#88058	—	2	1	no	—
72	#88058	—	2	1	no	—
73	#88057	—	2	2	no	—
74	#88057	—	2	2	no	—
75	#88050	—	2	4	no	—
76	#88047	—	2	4	no	—
77	C₄⋊#1055	C₂³	1	4	yes	—
78	C₄×#1055	—	2	2	no	—
79	#19150	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	2	no	—
82	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	—	2	1	no	—
84	C₂²×C₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₂×Dic₂₆	—	2	2	no	—
89	C₂×Dic₂₆	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	4	2	no	—
93	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	#41	1	1	yes	characteristic
107	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₁₃	—	4	2	no	—
109	#6107	—	4	8	no	—
110	C₄⋊Dic₁₃	—	2	2	no	—
111	C₄⋊Dic₁₃	—	2	2	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	#6101	—	4	2	no	—
115	#6101	—	4	2	no	—
116	C₄×Dic₁₃	—	4	2	no	—
117	C₂×#1055	—	2	4	no	—
118	#3160	—	13	1	no	—
119	#3104	—	13	2	no	—
120	C₂×#446	D₁₃	1	1	yes	characteristic
121	#1696	—	13	2	no	—
122	#1694	—	13	2	no	—
123	#1692	—	13	4	no	—
124	#1564	—	13	1	no	—
125	#1542	—	13	2	no	—
126	C₂³×C₁₃	#121	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	4	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	#137	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	Dic₂₆	—	4	4	no	—
146	Dic₂₆	—	8	2	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	4	2	no	—
150	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
151	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
152	#1055	—	4	4	no	—
153	#533	—	13	2	no	—
154	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
155	#500	—	13	1	no	—
156	#500	—	13	1	no	—
157	#499	—	26	1	no	—
158	#499	—	26	1	no	—
159	#498	—	13	2	no	—
160	#484	—	26	2	no	—
161	#477	—	13	4	no	—
162	#462	—	13	4	no	—
163	#451	—	26	2	no	—
164	#446	—	2	1	no	—
165	#446	—	2	1	no	—
166	#446	C₂×D₁₃	1	2	yes	—
167	#446	C₂×D₁₃	1	2	yes	—
168	C₄×Q₁₆	—	26	4	no	—
169	#431	—	13	2	no	—
170	#427	—	26	1	no	—
171	#427	—	26	1	no	—
172	C₂×C₄⋊C₈	—	2	1	no	—
173	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
174	C₂×#97	—	2	1	no	—
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#335	—	13	2	no	—
177	#334	—	13	2	no	—
178	#333	—	13	4	no	—
179	#332	—	13	4	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	#462	1	2	yes	—
183	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	#462	1	2	yes	—
190	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
192	C₄×C₁₃	#575	1	2	yes	—
193	Dic₁₃	—	8	2	no	—
194	Dic₁₃	—	8	2	no	—
195	C₂²×Q₈	D₅₂	1	1	yes	characteristic
196	C₂×Q₁₆	—	26	4	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	13	2	no	—
201	C₄⋊Q₈	—	13	2	no	—
202	#125	—	26	2	no	—
203	C₂²⋊Q₈	—	26	2	no	—
204	C₂²⋊Q₈	—	26	2	no	—
205	C₄×Q₈	—	2	1	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	26	2	no	—
208	C₄×Q₈	—	26	2	no	—
209	C₄×Q₈	D₅₂	1	2	yes	—
210	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
211	#117	—	26	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	#107	—	26	1	no	—
217	#107	—	26	1	no	—
218	#107	—	26	1	no	—
219	#107	—	26	1	no	—
220	#107	—	26	4	no	—
221	#106	—	26	2	no	—
222	#106	—	26	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	13	4	no	—
227	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
228	#103	—	26	4	no	—
229	#103	—	26	4	no	—
230	C₂²⋊C₈	—	26	2	no	—
231	#97	—	2	2	no	—
232	#97	—	2	2	no	—
233	C₄×C₈	—	26	2	no	—
234	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₁₃	—	2	2	no	—
239	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
240	C₂×C₁₃	#3054	1	1	yes	characteristic
241	C₂×C₁₃	#3067	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	26	2	no	—
245	C₂×Q₈	—	26	2	no	—
246	C₂×Q₈	C₂×D₅₂	1	2	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
250	C₂²×C₄	#6127	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
252	Q₁₆	—	52	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	26	4	no	—
257	C₂×C₈	#6107	1	4	yes	complemented
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	52	2	no	—
270	C₄⋊C₄	—	52	2	no	—
271	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
272	C₂²⋊C₄	—	52	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	52	2	no	—
276	C₄²	C₂×D₅₂	1	1	yes	characteristic
277	C₄²	#6127	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
279	C₄²	C₂³×D₁₃	1	1	yes	characteristic
280	C₁₃	#19150	1	1	yes	characteristic, complemented, Sylow
281	C₂³	#88184	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	52	4	no	—
285	Q₈	—	104	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	#88184	1	1	yes	characteristic
295	C₂×C₄	#88184	1	1	yes	characteristic
296	C₂×C₄	#88184	1	1	yes	characteristic
297	C₂×C₄	C₂×#6107	1	2	yes	—
298	C₂×C₄	#88205	1	2	yes	—
299	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
300	C₂×C₄	C₂×#6127	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	4	2	no	—
305	C₈	—	52	4	no	—
306	C₈	C₂×#6107	1	2	yes	—
307	C₈	C₂×#6107	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#11718692	1	2	yes	—
311	C₂²	C₂×#88184	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	104	2	no	—
318	C₄	—	104	2	no	—
319	C₄	#11718692	1	2	yes	—
320	C₄	#11719453	1	2	yes	—
321	C₄	C₂²×#6107	1	1	yes	characteristic
322	C₄	C₂×#88205	1	1	yes	characteristic
323	C₂	—	2	2	no	—
324	C₂	C₂×#11718692	1	1	yes	characteristic
325	C₂	#13124445	1	1	yes	characteristic
326	C₂	#13124556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#19150
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #13114364	C₂	semidirect
4. #13114354	C₂	semidirect
5. #13114342	C₂	semidirect
93. C₂×C₈×C₁₃	Q₁₆	semidirect
257. C₂×C₈	#6107	semidirect
280. C₁₃	#19150	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13124875	2
#13114364	2
#13114354	2
#13114342	2
#13113584	2
#13113494	2
C₂×C₁₃×#446	2
#19150	13