Subgroups — #15768792

Order 3328.

810 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	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718125	C₂	1	4	yes	maximal
4	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6099	C₂²	1	2	yes	—
8	C₂²×#6099	C₂²	1	2	yes	—
9	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
11	C₁₃×#376	—	2	1	no	—
12	C₄³×C₁₃	—	2	1	no	—
13	#11718585	—	2	2	no	—
14	#11718585	—	2	2	no	—
15	#11718585	—	2	1	no	—
16	#11718585	—	2	1	no	—
17	#11718585	C₄	1	2	yes	complemented
18	#11718585	C₄	1	2	yes	complemented
19	C₂×C₄⋊#1055	—	2	2	no	—
20	C₂×C₄⋊#1055	C₄	1	4	yes	complemented
21	#11718125	—	2	8	no	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×#117	—	2	2	no	—
25	C₁₃×#117	—	2	2	no	—
26	C₁₃×#117	D₄	1	4	yes	—
27	C₁₃×#117	Q₈	1	4	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	—	2	4	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	4	no	—
45	C₂×#6099	—	2	4	no	—
46	C₂×#6099	—	2	4	no	—
47	C₂×#6099	—	2	4	no	—
48	C₂×#6099	C₂×C₄	1	4	yes	—
49	C₂²×#1055	—	2	2	no	—
50	C₂²×#1055	C₂×C₄	1	4	yes	—
51	C₁₃×#95	—	2	4	no	—
52	C₄⋊#1055	—	2	4	no	—
53	C₄⋊#1055	—	2	4	no	—
54	C₄⋊#1055	—	2	4	no	—
55	C₄⋊#1055	—	2	4	no	—
56	#10856	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃	#41	1	2	yes	—
75	C₂²×C₄×C₁₃	#41	1	2	yes	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₂²⋊C₄	—	2	4	no	—
79	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	—	4	8	no	—
83	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
93	#6099	—	2	8	no	—
94	#6099	—	4	4	no	—
95	#6099	—	4	4	no	—
96	#6099	—	4	2	no	—
97	#6099	—	4	2	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	2	8	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂×#1055	—	4	2	no	—
103	C₂×#423	—	13	1	no	—
104	C₂×#423	—	13	1	no	—
105	C₄×#117	D₁₃	1	1	yes	characteristic
106	C₂×#344	—	13	4	no	—
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂³×C₁₃	#117	1	2	yes	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	8	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	8	no	—
121	C₂×C₄×C₁₃	#95	1	2	yes	—
122	C₂×C₄×C₁₃	#95	1	2	yes	—
123	C₂×C₄×C₁₃	#95	1	2	yes	—
124	C₂×C₄×C₁₃	#95	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
133	C₂×C₄×C₁₃	#121	1	2	yes	—
134	C₂×C₄×C₁₃	#121	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#125	1	2	yes	—
140	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
143	#1055	—	4	4	no	—
144	#1055	—	4	4	no	—
145	C₂²×M₁₆	—	13	2	no	—
146	C₂²×M₁₆	—	13	2	no	—
147	C₂×#117	C₂×D₁₃	1	2	yes	—
148	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
149	#423	—	13	2	no	—
150	#423	—	13	2	no	—
151	#423	—	26	2	no	—
152	#423	—	26	1	no	—
153	#423	—	26	1	no	—
154	#423	—	26	2	no	—
155	C₂×C₄⋊C₈	—	13	4	no	—
156	C₂×C₄⋊C₈	—	26	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₂²⋊C₄	Dic₁₃	1	4	yes	—
159	#376	—	2	1	no	—
160	C₄³	—	2	1	no	—
161	#344	—	26	8	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	4	2	no	—
166	C₂²×C₁₃	—	4	2	no	—
167	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
168	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
169	C₂²×C₁₃	#380	1	2	yes	—
170	C₂²×C₁₃	#382	1	1	yes	characteristic
171	C₂²×C₁₃	#384	1	1	yes	characteristic
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	8	no	—
176	C₄×C₁₃	#344	1	4	yes	complemented
177	C₄×C₁₃	#389	1	2	yes	—
178	C₄×C₁₃	#389	1	2	yes	—
179	C₄×C₁₃	#390	1	1	yes	characteristic
180	C₄×C₁₃	#390	1	1	yes	characteristic
181	C₄×C₁₃	#391	1	1	yes	characteristic
182	C₄×C₁₃	#391	1	1	yes	characteristic
183	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
184	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₂×M₁₆	—	13	4	no	—
186	C₂×M₁₆	—	26	4	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	—	26	4	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	—	26	2	no	—
191	C₂×M₁₆	—	26	4	no	—
192	C₂×M₁₆	—	26	4	no	—
193	C₂²×C₈	—	13	4	no	—
194	C₂²×C₈	—	26	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	Dic₂₆	1	4	yes	—
198	#117	D₅₂	1	4	yes	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
204	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
205	C₂×C₄²	C₄×D₁₃	1	2	yes	—
206	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
208	C₂×C₄²	D₅₂	1	2	yes	—
209	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
212	C₄⋊C₈	—	26	4	no	—
213	C₄⋊C₈	—	26	4	no	—
214	C₄⋊C₈	—	26	4	no	—
215	C₄⋊C₈	—	26	4	no	—
216	#95	—	2	4	no	—
217	C₂×C₁₃	—	2	2	no	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	—	2	1	no	—
220	C₂×C₁₃	—	2	1	no	—
221	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#344	1	2	yes	—
223	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
227	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
230	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
232	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
233	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₅₂	1	2	yes	—
237	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
238	C₂²×C₄	#6127	1	2	yes	—
239	C₂²×C₄	#6127	1	2	yes	—
240	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	M₁₆	—	26	8	no	—
243	M₁₆	—	52	2	no	—
244	M₁₆	—	52	2	no	—
245	M₁₆	—	52	4	no	—
246	M₁₆	—	52	4	no	—
247	C₂×C₈	—	26	8	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	52	2	no	—
251	C₂×C₈	—	52	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	8	no	—
259	C₄²	#6101	1	2	yes	—
260	C₄²	#6101	1	1	yes	characteristic
261	C₄²	#6101	1	1	yes	characteristic
262	C₄²	#6101	1	1	yes	characteristic
263	C₄²	#6101	1	2	yes	—
264	C₄²	#6101	1	1	yes	characteristic
265	C₄²	#6103	1	2	yes	—
266	C₄²	#6103	1	2	yes	—
267	C₄²	#6103	1	2	yes	—
268	C₄²	#6103	1	2	yes	—
269	C₁₃	#10856	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
272	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
273	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
274	C₂³	#88180	1	2	yes	—
275	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	#88071	1	2	yes	—
285	C₂×C₄	#88071	1	2	yes	—
286	C₂×C₄	#88071	1	2	yes	—
287	C₂×C₄	#88071	1	2	yes	—
288	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
289	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
290	C₂×C₄	#88123	1	2	yes	—
291	C₂×C₄	#88123	1	1	yes	characteristic
292	C₂×C₄	#88124	1	2	yes	—
293	C₂×C₄	C₄×D₅₂	1	2	yes	—
294	C₂×C₄	C₄×D₅₂	1	2	yes	—
295	C₂×C₄	#88128	1	1	yes	characteristic
296	C₂×C₄	#88129	1	1	yes	characteristic
297	C₂×C₄	#88129	1	1	yes	characteristic
298	C₂×C₄	C₂×#6101	1	1	yes	characteristic
299	C₂×C₄	C₂×#6101	1	1	yes	characteristic
300	C₂×C₄	#88178	1	2	yes	—
301	C₂×C₄	#88178	1	2	yes	—
302	C₂×C₄	C₂×#6103	1	1	yes	characteristic
303	C₂×C₄	C₂×#6103	1	1	yes	characteristic
304	C₂×C₄	#88184	1	2	yes	—
305	C₂×C₄	#88184	1	2	yes	—
306	C₈	—	52	4	no	—
307	C₈	—	52	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
314	C₂²	#11718596	1	1	yes	characteristic
315	C₂²	#11718597	1	1	yes	characteristic
316	C₂²	C₂×#88071	1	1	yes	characteristic
317	C₂²	#11718873	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	8	no	—
322	C₄	#11718125	1	4	yes	complemented
323	C₄	#11718589	1	2	yes	—
324	C₄	#11718589	1	2	yes	—
325	C₄	#11718590	1	1	yes	characteristic
326	C₄	#11718590	1	1	yes	characteristic
327	C₄	#11718600	1	1	yes	characteristic
328	C₄	#11718600	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	#13113066	1	1	yes	characteristic
334	C₂	C₂×#11718125	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6103
commutator	C₂³×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₄×C₁₃×#117	C₂
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#10856
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
17. #11718585	C₄	semidirect
18. #11718585	C₄	semidirect
20. C₂×C₄⋊#1055	C₄	semidirect
176. C₄×C₁₃	#344	semidirect
269. C₁₃	#10856	semidirect
322. C₄	#11718125	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂×#11718125	2
C₄×C₁₃×#117	2
#10856	13