Subgroups — #15846793

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124962	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13114263	C₂	1	4	yes	maximal, complemented
4	#13114227	C₂	1	2	yes	maximal, complemented
5	#13114203	C₂	1	2	yes	maximal, complemented
6	#13113599	C₂	1	1	yes	maximal, characteristic
7	#13113536	C₂	1	2	yes	maximal
8	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₂²	1	1	yes	characteristic
12	C₁₃×#494	C₂²	1	1	yes	characteristic
13	C₁₃×#487	C₂²	1	2	yes	—
14	C₁₃×#467	—	2	1	no	—
15	C₁₃×#467	—	2	1	no	—
16	C₄×C₁₃×Q₁₆	—	2	1	no	—
17	C₄×C₁₃×Q₁₆	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88054	—	2	1	no	—
20	C₂×C₈×Dic₁₃	—	2	1	no	—
21	#11718755	C₂²	1	4	yes	—
22	#11718751	C₂²	1	4	yes	—
23	#11718711	—	2	2	no	—
24	#11718704	—	2	2	no	—
25	#11718691	—	2	4	no	—
26	#11718587	C₂²	1	2	yes	—
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₈×Dic₂₆	—	2	1	no	—
29	C₈×Dic₂₆	—	2	1	no	—
30	C₈×Dic₂₆	C₂²	1	2	yes	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	#11718139	C₂²	1	4	yes	—
33	#11718139	C₂²	1	4	yes	—
34	#11718050	C₂²	1	2	yes	—
35	#11718049	C₂²	1	2	yes	—
36	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
37	C₂×C₁₃×Q₁₆	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	—	2	1	no	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×C₂²⋊Q₈	—	2	2	no	—
45	C₁₃×C₂²⋊Q₈	—	2	2	no	—
46	C₄×Q₈×C₁₃	—	2	2	no	—
47	C₄×Q₈×C₁₃	—	2	2	no	—
48	C₁₃×#117	—	2	2	no	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×#6107	—	2	4	no	—
51	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
52	C₂×#6101	—	2	1	no	—
53	C₂×C₄×Dic₁₃	—	2	1	no	—
54	C₂²×#1055	—	2	1	no	—
55	C₄×Dic₂₆	—	2	1	no	—
56	C₄×Dic₂₆	—	2	1	no	—
57	C₄×Dic₂₆	D₄	1	2	yes	—
58	C₄×Dic₂₆	C₂³	1	2	yes	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#103	—	2	2	no	—
62	C₁₃×#103	—	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	#88054	—	2	2	no	—
70	#88054	—	2	2	no	—
71	C₈×Dic₁₃	—	2	2	no	—
72	C₈×Dic₁₃	—	2	2	no	—
73	#88050	—	2	4	no	—
74	#88048	—	2	4	no	—
75	C₄⋊#1055	—	2	1	no	—
76	C₄⋊#1055	C₂³	1	4	yes	—
77	C₄⋊#1055	C₂³	1	2	yes	—
78	#88043	—	2	2	no	—
79	#19265	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
85	C₂²×Dic₁₃	—	2	1	no	—
86	C₂²×Dic₁₃	—	2	1	no	—
87	C₂×Dic₂₆	—	2	1	no	—
88	C₂×Dic₂₆	—	2	4	no	—
89	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
90	C₁₃×Q₁₆	—	4	2	no	—
91	C₁₃×Q₁₆	—	4	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
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	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₄²×C₁₃	—	4	2	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6107	—	4	8	no	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
113	#6101	—	2	2	no	—
114	#6101	—	2	2	no	—
115	C₄×Dic₁₃	—	2	2	no	—
116	C₄×Dic₁₃	—	2	2	no	—
117	C₂×#1055	—	2	4	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	4	2	no	—
120	#3170	D₁₃	1	1	yes	characteristic
121	#3113	—	13	2	no	—
122	C₂×C₈×Q₈	—	13	1	no	—
123	#1680	—	13	4	no	—
124	#1674	—	13	2	no	—
125	#1670	—	13	2	no	—
126	#1568	—	13	1	no	—
127	#1552	—	13	2	no	—
128	C₂³×C₁₃	#121	1	1	yes	characteristic
129	Q₈×C₁₃	—	4	4	no	—
130	Q₈×C₁₃	—	4	2	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₁₃	—	2	4	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	Dic₂₆	—	2	4	no	—
149	Dic₂₆	—	2	2	no	—
150	Dic₂₆	#135	1	4	yes	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	#137	1	2	yes	—
155	#1055	—	4	4	no	—
156	#1055	—	4	2	no	—
157	#533	C₂×D₁₃	1	2	yes	—
158	C₂×C₄×Q₈	—	13	1	no	—
159	#500	C₂×D₁₃	1	1	yes	characteristic
160	#494	C₂×D₁₃	1	1	yes	characteristic
161	#487	C₂×D₁₃	1	2	yes	—
162	#483	—	26	2	no	—
163	#475	—	13	4	no	—
164	#467	—	2	1	no	—
165	#467	—	2	1	no	—
166	#462	—	13	4	no	—
167	#451	—	26	2	no	—
168	C₈×Q₈	—	13	2	no	—
169	C₈×Q₈	—	13	2	no	—
170	C₈×Q₈	—	26	1	no	—
171	C₈×Q₈	—	26	1	no	—
172	#441	—	26	4	no	—
173	C₄×Q₁₆	—	2	1	no	—
174	C₄×Q₁₆	—	2	1	no	—
175	#432	—	13	2	no	—
176	C₂×C₄⋊C₈	—	13	1	no	—
177	C₂×C₄⋊C₈	—	26	1	no	—
178	C₂×C₄×C₈	—	26	1	no	—
179	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
180	#335	—	13	2	no	—
181	#334	—	13	2	no	—
182	#332	—	13	4	no	—
183	#332	—	13	4	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	#473	1	2	yes	—
187	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	C₂×#135	1	2	yes	—
195	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
196	C₄×C₁₃	#575	1	1	yes	characteristic
197	Dic₁₃	—	2	4	no	—
198	Dic₁₃	—	4	2	no	—
199	C₂²×Q₈	—	13	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	26	4	no	—
203	C₂²×C₈	—	26	1	no	—
204	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
206	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
207	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
208	#125	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	13	2	no	—
214	C₄×Q₈	—	13	2	no	—
215	C₄×Q₈	—	26	1	no	—
216	C₄×Q₈	—	26	1	no	—
217	#117	—	2	2	no	—
218	C₂×C₄⋊C₄	—	13	1	no	—
219	C₂×C₄⋊C₄	—	26	1	no	—
220	C₂×C₄²	—	26	1	no	—
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#106	—	26	4	no	—
225	C₄⋊C₈	—	13	2	no	—
226	C₄⋊C₈	—	13	4	no	—
227	C₄⋊C₈	—	26	2	no	—
228	C₄⋊C₈	—	26	2	no	—
229	C₄⋊C₈	—	26	1	no	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#103	—	26	4	no	—
233	#103	—	26	4	no	—
234	C₂²⋊C₈	—	26	2	no	—
235	#97	—	26	2	no	—
236	C₄×C₈	—	26	2	no	—
237	C₄×C₈	—	26	2	no	—
238	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×C₁₃	—	2	2	no	—
243	C₂×C₁₃	#3037	1	1	yes	characteristic
244	C₂×C₁₃	#3050	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	13	2	no	—
249	C₂×Q₈	—	26	4	no	—
250	C₂×Q₈	—	26	1	no	—
251	C₂²×C₄	—	26	1	no	—
252	C₂²×C₄	—	26	1	no	—
253	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	#6129	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	Q₁₆	—	4	2	no	—
257	Q₁₆	—	4	2	no	—
258	Q₁₆	—	52	8	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	26	4	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	52	2	no	—
264	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	13	2	no	—
275	C₄⋊C₄	—	26	1	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄⋊C₄	—	26	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	26	2	no	—
281	C₄²	—	26	2	no	—
282	C₄²	D₄×D₁₃	1	1	yes	characteristic
283	C₄²	#6129	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
285	C₄²	C₂³×D₁₃	1	1	yes	characteristic
286	C₁₃	#19265	1	1	yes	characteristic, complemented, Sylow
287	C₂³	#88192	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	13	4	no	—
292	Q₈	—	26	4	no	—
293	Q₈	—	26	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	26	4	no	—
301	C₂×C₄	—	26	2	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	#88192	1	1	yes	characteristic
305	C₂×C₄	#88192	1	1	yes	characteristic
306	C₂×C₄	#88192	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#6129	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	52	4	no	—
314	C₈	—	52	2	no	—
315	C₈	#88205	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#11718853	1	2	yes	—
319	C₂²	C₂×#88192	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	26	4	no	—
327	C₄	—	52	2	no	—
328	C₄	D₁₃×#135	1	2	yes	—
329	C₄	#11719527	1	1	yes	characteristic
330	C₄	C₂×#88205	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	#13124304	1	1	yes	characteristic
333	C₂	#13124413	1	1	yes	characteristic
334	C₂	C₂×#11718853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#19265
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #13114263	C₂	semidirect
4. #13114227	C₂	semidirect
5. #13114203	C₂	semidirect
286. C₁₃	#19265	semidirect

Maximal subgroups

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

subgroup	index
#13124962	2
C₂×C₈×Dic₂₆	2
#13114263	2
#13114227	2
#13114203	2
#13113599	2
#13113536	2
C₁₃×#3170	2
#19265	13