Subgroups — #15866207

Order 3328.

1010 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718763	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13117243	C₂	1	1	yes	maximal, characteristic
4	#13115557	C₂	1	4	yes	maximal, complemented
5	#13115552	C₂	1	4	yes	maximal, complemented
6	#13114532	C₂	1	2	yes	maximal
7	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	C₂²×#6099	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	—	2	4	no	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	#11718873	C₄	1	2	yes	—
17	#11718873	C₄	1	2	yes	—
18	#11718763	—	2	8	no	—
19	C₂×#88054	C₂²	1	8	yes	—
20	#11718597	C₄	1	2	yes	—
21	#11718596	C₄	1	2	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
24	C₄×#6099	—	2	4	no	—
25	C₂×#88043	C₂²	1	2	yes	—
26	C₂×C₄×#1055	C₂²	1	2	yes	—
27	#11718142	C₂²	1	8	yes	—
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	—	2	4	no	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	4	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
42	#88180	—	2	4	no	—
43	#88180	—	2	4	no	—
44	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	D₄	1	4	yes	—
53	C₂×#6099	Q₈	1	4	yes	—
54	C₂²×#1055	C₂³	1	2	yes	—
55	C₂²×#1055	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₈	—	2	4	no	—
57	C₁₃×C₄⋊C₈	—	2	4	no	—
58	#88071	C₂×C₄	1	4	yes	—
59	#88071	C₂×C₄	1	4	yes	—
60	#88054	—	2	16	no	—
61	#88043	—	2	4	no	—
62	C₄×#1055	—	2	4	no	—
63	#21145	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	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	#41	1	2	yes	—
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	4	8	no	—
86	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
87	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
88	C₄²×C₁₃	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃	C₂×D₄	1	2	yes	—
90	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
92	C₂²⋊Dic₁₃	—	2	8	no	—
93	C₄⋊Dic₁₃	—	2	8	no	—
94	C₄⋊Dic₁₃	—	2	8	no	—
95	C₄⋊Dic₁₃	—	4	4	no	—
96	C₄×Dic₁₃	—	2	4	no	—
97	C₄×Dic₁₃	—	2	4	no	—
98	#6099	—	2	8	no	—
99	#6099	—	2	8	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	C₂×D₄	1	4	yes	—
105	C₂×#1055	C₂×Q₈	1	4	yes	—
106	C₂×#424	—	13	2	no	—
107	C₂²×C₄⋊C₈	D₁₃	1	1	yes	characteristic
108	C₂×C₄×M₁₆	—	13	1	no	—
109	#2285	—	13	1	no	—
110	#1940	—	13	4	no	—
111	#1939	—	13	4	no	—
112	#1729	—	13	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂³×C₁₃	C₄×D₄	1	2	yes	—
117	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
118	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	#121	1	4	yes	—
132	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
133	C₂×C₄×C₁₃	#125	1	4	yes	—
134	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
135	C₂×C₄×C₁₃	#131	1	4	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	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×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₁₃	—	2	8	no	—
143	C₂×Dic₁₃	—	2	8	no	—
144	C₂×Dic₁₃	—	2	8	no	—
145	C₈×C₁₃	—	4	4	no	—
146	#1055	—	2	4	no	—
147	#1055	—	2	4	no	—
148	C₂²×M₁₆	—	13	1	no	—
149	C₂²×M₁₆	—	13	1	no	—
150	C₂³×C₈	C₂×D₁₃	1	2	yes	—
151	C₂×#117	—	13	2	no	—
152	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
153	#424	—	26	8	no	—
154	C₂×C₄⋊C₈	—	2	4	no	—
155	C₂×C₄⋊C₈	—	13	8	no	—
156	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
157	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
158	C₄×M₁₆	—	26	4	no	—
159	C₂×#97	—	13	2	no	—
160	C₂×C₄×C₈	—	13	2	no	—
161	#384	—	13	2	no	—
162	#382	—	13	2	no	—
163	#380	—	13	2	no	—
164	#380	—	13	2	no	—
165	C₄×C₄⋊C₄	—	13	2	no	—
166	C₄×C₄⋊C₄	—	13	2	no	—
167	#336	—	13	8	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	2	4	no	—
171	C₂²×C₁₃	—	2	4	no	—
172	C₂²×C₁₃	#389	1	4	yes	—
173	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#131	1	2	yes	—
177	C₂²×C₁₃	#568	1	2	yes	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	#389	1	4	yes	—
182	C₄×C₁₃	#424	1	4	yes	complemented
183	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
184	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
185	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
186	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	Dic₁₃	—	4	8	no	—
188	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
189	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
190	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
191	C₂×M₁₆	—	13	4	no	—
192	C₂×M₁₆	—	13	4	no	—
193	C₂×M₁₆	—	26	2	no	—
194	C₂×M₁₆	—	26	2	no	—
195	C₂×M₁₆	—	26	2	no	—
196	C₂×M₁₆	—	26	2	no	—
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
203	#117	—	26	4	no	—
204	#117	—	26	4	no	—
205	C₂×C₄⋊C₄	—	13	4	no	—
206	C₂×C₄⋊C₄	—	13	4	no	—
207	C₂×C₄⋊C₄	—	13	4	no	—
208	C₂×C₂²⋊C₄	—	13	4	no	—
209	C₂×C₄²	—	13	4	no	—
210	C₂×C₄²	Dic₂₆	1	4	yes	—
211	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
214	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	26	16	no	—
219	#97	—	26	4	no	—
220	C₄×C₈	—	26	4	no	—
221	#95	—	13	4	no	—
222	#95	—	13	4	no	—
223	C₂×C₁₃	—	2	4	no	—
224	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
225	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
226	C₂×C₁₃	#2905	1	1	yes	characteristic
227	C₂×C₁₃	#2935	1	2	yes	—
228	C₂×C₁₃	#2962	1	2	yes	—
229	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	13	8	no	—
233	C₂²×C₄	#6101	1	4	yes	—
234	C₂²×C₄	#6101	1	4	yes	—
235	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₁₃	1	2	yes	—
242	C₂²×C₄	#6129	1	2	yes	—
243	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
244	C₂²×C₄	#6131	1	2	yes	—
245	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
246	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
247	M₁₆	—	26	8	no	—
248	M₁₆	—	26	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	8	no	—
253	C₂×C₈	—	13	4	no	—
254	C₂×C₈	—	13	4	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	2	no	—
259	C₄⋊C₄	—	26	8	no	—
260	C₄⋊C₄	—	26	8	no	—
261	C₄⋊C₄	—	52	4	no	—
262	C₂²⋊C₄	—	26	8	no	—
263	C₄²	—	26	4	no	—
264	C₄²	—	26	4	no	—
265	C₄²	#6101	1	4	yes	—
266	C₄²	#6101	1	4	yes	—
267	C₄²	C₂×Dic₂₆	1	2	yes	—
268	C₄²	C₂×Dic₂₆	1	2	yes	—
269	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
270	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
271	C₁₃	#21145	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	#88141	1	2	yes	—
275	C₂³	#88147	1	2	yes	—
276	C₂³	C₂×#6101	1	2	yes	—
277	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	26	8	no	—
284	C₂×C₄	—	26	8	no	—
285	C₂×C₄	—	26	8	no	—
286	C₂×C₄	#88141	1	2	yes	—
287	C₂×C₄	#88142	1	4	yes	—
288	C₂×C₄	#88144	1	4	yes	—
289	C₂×C₄	#88147	1	2	yes	—
290	C₂×C₄	#88155	1	4	yes	—
291	C₂×C₄	C₂×#6101	1	2	yes	—
292	C₂×C₄	C₂×#6101	1	2	yes	—
293	C₂×C₄	C₂×#6101	1	2	yes	—
294	C₂×C₄	C₂×#6101	1	2	yes	—
295	C₂×C₄	C₂×#6101	1	2	yes	—
296	C₂×C₄	#88192	1	4	yes	—
297	C₂×C₄	#88200	1	4	yes	—
298	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×#6129	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂×#6131	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
304	C₈	—	4	4	no	—
305	C₈	—	26	4	no	—
306	C₈	—	26	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	#11718634	1	4	yes	—
312	C₂²	C₂×#88141	1	1	yes	characteristic
313	C₂²	C₂×#88147	1	1	yes	characteristic
314	C₂²	C₂×#88155	1	2	yes	—
315	C₂²	#11719450	1	2	yes	—
316	C₂²	C₂²×#6101	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	52	8	no	—
321	C₄	#11718634	1	4	yes	—
322	C₄	#11718763	1	4	yes	complemented
323	C₄	C₂×#88142	1	1	yes	characteristic
324	C₄	C₂×#88144	1	1	yes	characteristic
325	C₄	C₂×#88192	1	1	yes	characteristic
326	C₄	C₂×#88200	1	1	yes	characteristic
327	C₂	—	2	4	no	—
328	C₂	C₂×#11718634	1	1	yes	characteristic
329	C₂	C₂×#11718763	1	1	yes	characteristic
330	C₂	#13123399	1	1	yes	characteristic
331	C₂	#13123579	1	2	yes	—
332	C₂	#13123779	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₂²×C₁₃×C₄⋊C₈	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#21145
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #13115557	C₂	semidirect
5. #13115552	C₂	semidirect
182. C₄×C₁₃	#424	semidirect
271. C₁₃	#21145	semidirect
322. C₄	#11718763	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718763	2
C₂×C₄×#6099	2
#13117243	2
#13115557	2
#13115552	2
#13114532	2
C₂²×C₁₃×C₄⋊C₈	2
#21145	13