Subgroups — #15863371

Order 3328.

1070 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718656	C₂	1	2	yes	maximal
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
4	#13117456	C₂	1	2	yes	maximal
5	#13115001	C₂	1	4	yes	maximal, complemented
6	#13114999	C₂	1	2	yes	maximal, complemented
7	#13114994	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	2	yes	—
11	C₂×#88180	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂²×#6099	C₂²	1	1	yes	characteristic
14	C₁₃×#427	—	2	4	no	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#106	C₂²	1	2	yes	—
18	C₂×#88058	—	2	4	no	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	2	yes	—
21	#11718762	C₄	1	8	yes	complemented
22	C₂×#88055	C₂²	1	2	yes	—
23	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
24	#11718656	—	2	8	no	—
25	#11718638	C₂²	1	4	yes	—
26	#11718637	C₂²	1	4	yes	—
27	C₂×#88048	C₂²	1	4	yes	—
28	C₂×#88047	C₂²	1	4	yes	—
29	#11718607	—	2	4	no	—
30	#11718589	—	2	4	no	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂³×Dic₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	D₄	1	4	yes	—
35	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	—	2	4	no	—
39	C₁₃×#117	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₂²⋊Dic₁₃	D₄	1	2	yes	—
45	#88180	C₂×C₄	1	4	yes	—
46	#88180	Q₈	1	4	yes	—
47	C₂×C₄⋊Dic₁₃	—	2	4	no	—
48	C₂×C₄⋊Dic₁₃	—	2	4	no	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
52	C₂×#6101	—	2	8	no	—
53	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
54	C₂×#6099	C₂×C₄	1	4	yes	—
55	C₂×#6099	D₄	1	4	yes	—
56	C₂×#6099	Q₈	1	4	yes	—
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₁₃×#107	—	2	2	no	—
59	C₁₃×#107	—	2	2	no	—
60	C₁₃×#106	—	2	4	no	—
61	#88071	—	2	8	no	—
62	#88058	—	2	4	no	—
63	#88058	—	2	4	no	—
64	#88055	C₂×C₄	1	8	yes	complemented
65	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
66	#88048	—	2	8	no	—
67	#88047	—	2	8	no	—
68	#20803	—	13	1	no	maximal, Sylow
69	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₁₃	—	2	4	no	—
81	C₂²×Dic₁₃	—	4	4	no	—
82	C₂²×Dic₁₃	—	8	4	no	—
83	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₄²×C₁₃	—	4	4	no	—
97	C₂²⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
98	C₄⋊Dic₁₃	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	4	4	no	—
101	C₄⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
102	#6101	—	8	8	no	—
103	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
104	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
105	#6099	C₄⋊C₄	1	16	yes	complemented
106	C₂×#1055	C₂²×C₄	1	4	yes	—
107	C₂×#1055	C₂×D₄	1	4	yes	—
108	C₂×#1055	C₂×Q₈	1	4	yes	—
109	C₂×#428	—	13	2	no	—
110	C₂×#427	D₁₃	1	1	yes	characteristic
111	C₂²×#107	—	13	1	no	—
112	C₂×#405	—	13	1	no	—
113	#2317	—	13	2	no	—
114	#1847	—	13	4	no	—
115	#1846	—	13	2	no	—
116	#1845	—	13	2	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂³×C₁₃	#127	1	2	yes	—
121	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₃	#127	1	2	yes	—
131	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
132	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
133	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×Dic₁₃	—	2	4	no	—
141	C₂×Dic₁₃	—	2	4	no	—
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	4	8	no	—
144	C₂×Dic₁₃	—	8	8	no	—
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
146	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
147	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
148	#1055	C₂×C₄⋊C₄	1	8	yes	—
149	C₂²×M₁₆	—	13	1	no	—
150	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×#117	—	13	1	no	—
152	C₂×#117	C₂×D₁₃	1	2	yes	—
153	C₂²×C₄⋊C₄	—	13	2	no	—
154	#428	—	26	8	no	—
155	#427	—	2	4	no	—
156	C₂×#107	—	13	2	no	—
157	C₂×#107	—	13	2	no	—
158	C₂×#107	—	13	4	no	—
159	C₂×#107	—	26	4	no	—
160	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
161	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
162	C₂×#106	—	13	4	no	—
163	C₂×#106	C₂×D₁₃	1	2	yes	—
164	#405	—	13	8	no	—
165	C₂×#97	—	13	2	no	—
166	C₂×C₄×C₈	—	13	2	no	—
167	#389	—	26	4	no	—
168	#385	—	26	4	no	—
169	#384	—	13	4	no	—
170	#382	—	13	4	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	4	no	—
174	C₂²×C₁₃	—	2	4	no	—
175	C₂²×C₁₃	#384	1	4	yes	—
176	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
178	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₂²×C₁₃	#576	1	2	yes	—
180	C₂²×C₁₃	#578	1	2	yes	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	#384	1	4	yes	—
184	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
188	Dic₁₃	—	2	8	no	—
189	Dic₁₃	—	8	4	no	—
190	C₂³×C₄	—	26	2	no	—
191	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×M₁₆	—	13	4	no	—
193	C₂×M₁₆	—	13	4	no	—
194	C₂×M₁₆	—	13	4	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	Dic₂₆	1	4	yes	—
197	C₂²×C₈	C₄×D₁₃	1	4	yes	—
198	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
199	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	#117	—	2	4	no	—
202	#117	—	2	4	no	—
203	#117	—	13	4	no	—
204	#117	—	13	4	no	—
205	C₂×C₄⋊C₄	—	13	2	no	—
206	C₂×C₄⋊C₄	—	13	4	no	—
207	C₂×C₄⋊C₄	—	26	4	no	—
208	C₂×C₄⋊C₄	—	26	4	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄⋊C₄	—	26	8	no	—
211	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	13	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	13	2	no	—
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	26	8	no	—
220	#107	—	26	4	no	—
221	#107	—	26	4	no	—
222	#106	—	2	4	no	—
223	#106	—	26	8	no	—
224	#97	—	13	8	no	—
225	C₄×C₈	—	13	8	no	—
226	#95	—	26	8	no	—
227	C₂×C₁₃	—	2	4	no	—
228	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
229	C₂×C₁₃	#2913	1	2	yes	—
230	C₂×C₁₃	#3155	1	1	yes	characteristic
231	C₂×C₁₃	#3156	1	1	yes	characteristic
232	C₂×C₁₃	#3164	1	2	yes	—
233	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	13	4	no	—
237	C₂²×C₄	—	26	4	no	—
238	C₂²×C₄	—	52	4	no	—
239	C₂²×C₄	—	104	4	no	—
240	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
242	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₁₃	1	2	yes	—
244	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
245	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
247	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
248	M₁₆	—	13	16	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	—	13	4	no	—
251	C₂×C₈	—	13	4	no	—
252	C₂×C₈	#6101	1	16	yes	complemented
253	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
254	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
257	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	13	8	no	—
264	C₄⋊C₄	—	26	4	no	—
265	C₄⋊C₄	—	26	4	no	—
266	C₄⋊C₄	—	52	4	no	—
267	C₄⋊C₄	—	104	8	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₂²⋊C₄	—	13	8	no	—
270	C₄²	—	4	4	no	—
271	C₄²	—	13	4	no	—
272	C₄²	—	13	4	no	—
273	C₁₃	#20803	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	#88142	1	2	yes	—
277	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
278	C₂³	#88194	1	2	yes	—
279	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	13	8	no	—
285	C₂×C₄	—	26	4	no	—
286	C₂×C₄	—	26	4	no	—
287	C₂×C₄	—	52	8	no	—
288	C₂×C₄	—	52	4	no	—
289	C₂×C₄	—	104	8	no	—
290	C₂×C₄	#88142	1	4	yes	—
291	C₂×C₄	#88142	1	2	yes	—
292	C₂×C₄	C₄⋊C₄×D₁₃	1	4	yes	—
293	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
294	C₂×C₄	C₂×#6101	1	4	yes	—
295	C₂×C₄	#88194	1	2	yes	—
296	C₂×C₄	#88198	1	4	yes	—
297	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
303	C₈	—	13	8	no	—
304	C₈	C₂×#6101	1	4	yes	—
305	C₈	C₂×#6101	1	4	yes	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#11718637	1	4	yes	—
311	C₂²	C₂×#88142	1	1	yes	characteristic
312	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
313	C₂²	#11719474	1	2	yes	—
314	C₂²	#11719476	1	2	yes	—
315	C₂²	C₂×#88194	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	26	8	no	—
319	C₄	—	104	4	no	—
320	C₄	#11718637	1	4	yes	—
321	C₄	C₂×#88142	1	1	yes	characteristic
322	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
323	C₄	C₂²×#6101	1	1	yes	characteristic
324	C₄	C₂×#88198	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₂×#11718637	1	1	yes	characteristic
327	C₂	#13123433	1	2	yes	—
328	C₂	#13125323	1	1	yes	characteristic
329	C₂	#13125331	1	1	yes	characteristic
330	C₂	#13125376	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#20803
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
5. #13115001	C₂	semidirect
6. #13114999	C₂	semidirect
7. #13114994	C₂	semidirect
21. #11718762	C₄	semidirect
64. #88055	C₂×C₄	semidirect
65. C₈×Dic₁₃	C₂×C₄	semidirect
84. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
105. #6099	C₄⋊C₄	semidirect
252. C₂×C₈	#6101	semidirect
273. C₁₃	#20803	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718656	2
C₂²×#88058	2
C₂×#11718762	2
#13117456	2
#13115001	2
#13114999	2
#13114994	2
C₂×C₁₃×#427	2
#20803	13