Subgroups — #15882665

Order 3328.

934 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117785	C₂	1	2	yes	maximal, complemented
2	#13115703	C₂	1	2	yes	maximal
3	#13115458	C₂	1	4	yes	maximal, complemented
4	#13114996	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114836	C₂	1	2	yes	maximal, complemented
6	#107×Dic₁₃	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂×#88180	—	2	2	no	—
12	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#102	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718882	C₂²	1	4	yes	—
17	C₂×#88072	C₂²	1	4	yes	—
18	#11718873	—	2	2	no	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₁₃	C₂²	1	4	yes	—
23	#11718612	—	2	4	no	—
24	#11718596	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
27	#11718142	C₂²	1	2	yes	—
28	#11718121	C₂²	1	4	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₃	—	2	4	no	—
37	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
42	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
43	#88180	—	4	8	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
49	C₂×C₄×Dic₁₃	—	2	4	no	—
50	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₁₃×#107	C₂×C₄	1	4	yes	—
53	C₁₃×#102	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
55	#88072	—	4	4	no	—
56	#88071	—	2	4	no	—
57	#88071	D₄	1	2	yes	—
58	#88058	—	2	4	no	—
59	#88058	—	4	2	no	—
60	C₈×Dic₁₃	—	2	4	no	—
61	C₄⋊#1055	—	2	2	no	—
62	#22807	—	13	1	no	maximal, Sylow
63	C₂⁴×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	8	no	—
72	C₂²×C₄×C₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₃	—	4	4	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	2	4	no	—
78	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
79	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
80	C₁₃×D₈	—	2	8	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
89	C₁₃×C₂²⋊C₄	—	2	8	no	—
90	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₁₃	#41	1	2	yes	—
92	C₂²⋊Dic₁₃	—	4	8	no	—
93	C₂²⋊Dic₁₃	—	8	4	no	—
94	C₄⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	4	no	—
98	C₄⋊Dic₁₃	—	8	2	no	—
99	C₄⋊Dic₁₃	D₈	1	4	yes	complemented
100	C₄×Dic₁₃	—	2	2	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	4	8	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
107	#2354	—	13	2	no	—
108	#1970	—	13	2	no	—
109	#1925	—	13	4	no	—
110	#1845	—	13	1	no	—
111	#1802	—	13	2	no	—
112	C₄×#107	—	13	2	no	—
113	C₈×C₄⋊C₄	—	13	1	no	—
114	C₂³×C₁₃	—	2	4	no	—
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	—	4	2	no	—
118	C₂³×C₁₃	—	4	2	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	D₄×C₁₃	—	2	4	no	—
123	D₄×C₁₃	—	2	4	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	16	no	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	#135	1	2	yes	—
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	4	4	no	—
148	C₂×Dic₁₃	—	4	4	no	—
149	C₂×Dic₁₃	—	4	4	no	—
150	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
151	C₈×C₁₃	—	2	2	no	—
152	C₈×C₁₃	#117	1	4	yes	complemented
153	#1055	—	4	2	no	—
154	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
155	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
156	C₂×#117	—	26	2	no	—
157	C₄×D₈	Dic₁₃	1	8	yes	complemented
158	C₂×#107	—	13	2	no	—
159	C₂×#107	—	13	2	no	—
160	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	13	1	no	—
162	C₂×#102	—	13	4	no	—
163	C₂×#102	C₂×D₁₃	1	2	yes	—
164	C₂×C₄×C₈	—	13	2	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#390	—	13	4	no	—
167	#388	—	26	4	no	—
168	#384	—	13	2	no	—
169	#380	—	26	2	no	—
170	C₄×C₄⋊C₄	—	13	1	no	—
171	C₄×C₄⋊C₄	—	13	4	no	—
172	#340	—	13	4	no	—
173	#336	—	13	2	no	—
174	C₂²×C₁₃	—	4	4	no	—
175	C₂²×C₁₃	—	4	4	no	—
176	C₂²×C₁₃	—	4	4	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂²×C₁₃	#547	1	1	yes	characteristic
183	C₂²×C₁₃	#550	1	1	yes	characteristic
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
186	C₂²×C₁₃	#576	1	1	yes	characteristic
187	C₂²×C₁₃	#578	1	1	yes	characteristic
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	4	4	no	—
190	C₄×C₁₃	C₄×D₈	1	4	yes	—
191	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
192	C₄×C₁₃	#520	1	1	yes	characteristic
193	C₄×C₁₃	#555	1	1	yes	characteristic
194	C₄×C₁₃	#556	1	1	yes	characteristic
195	Dic₁₃	—	2	4	no	—
196	Dic₁₃	—	4	2	no	—
197	Dic₁₃	—	8	4	no	—
198	C₂²×D₄	C₂²×D₁₃	1	2	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂×D₈	—	2	2	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
206	C₄×D₄	—	2	4	no	—
207	C₄×D₄	C₂×Dic₁₃	1	8	yes	—
208	#117	—	52	8	no	—
209	C₂×C₄⋊C₄	—	13	1	no	—
210	C₂×C₄⋊C₄	—	13	1	no	—
211	C₂×C₄⋊C₄	—	13	4	no	—
212	C₂×C₄⋊C₄	—	26	2	no	—
213	C₂×C₄⋊C₄	—	26	2	no	—
214	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	26	4	no	—
217	C₂×C₂²⋊C₄	—	26	4	no	—
218	C₂×C₄²	—	13	2	no	—
219	C₂×C₄²	—	26	4	no	—
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	#107	—	26	4	no	—
222	#107	—	52	2	no	—
223	#107	C₂×Dic₁₃	1	4	yes	—
224	C₄⋊C₈	—	26	2	no	—
225	#102	—	52	4	no	—
226	#102	C₂×Dic₁₃	1	8	yes	—
227	C₄×C₈	—	26	4	no	—
228	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
229	#95	—	13	2	no	—
230	#95	—	26	4	no	—
231	C₂×C₁₃	—	8	4	no	—
232	C₂×C₁₃	#2354	1	1	yes	characteristic
233	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
234	C₂×C₁₃	#2951	1	1	yes	characteristic
235	C₂×C₁₃	#3334	1	1	yes	characteristic
236	C₂×C₁₃	#3341	1	1	yes	characteristic
237	C₂×C₁₃	#3391	1	1	yes	characteristic
238	C₂×C₁₃	#3393	1	1	yes	characteristic
239	C₂⁴	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	13	2	no	—
251	C₂²×C₄	—	13	2	no	—
252	C₂²×C₄	—	26	4	no	—
253	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
256	D₈	—	2	8	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	#6127	1	2	yes	—
263	C₂×C₈	#6127	1	2	yes	—
264	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
265	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	13	4	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	4	no	—
270	C₄⋊C₄	—	26	4	no	—
271	C₄⋊C₄	—	52	4	no	—
272	C₄⋊C₄	—	104	2	no	—
273	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
274	C₂²⋊C₄	—	2	8	no	—
275	C₂²⋊C₄	—	52	8	no	—
276	C₂²⋊C₄	—	104	4	no	—
277	C₄²	—	26	2	no	—
278	C₄²	—	26	2	no	—
279	C₄²	—	52	8	no	—
280	C₄²	#6129	1	2	yes	—
281	C₄²	C₂²×Dic₁₃	1	2	yes	—
282	C₁₃	#22807	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	2	4	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
291	D₄	—	2	4	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	16	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	52	4	no	—
308	C₂×C₄	—	52	4	no	—
309	C₂×C₄	—	52	4	no	—
310	C₂×C₄	D₈×D₁₃	1	2	yes	—
311	C₂×C₄	#88166	1	2	yes	—
312	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
313	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
314	C₂×C₄	C₂×#6127	1	1	yes	characteristic
315	C₂×C₄	C₂×#6127	1	1	yes	characteristic
316	C₂×C₄	C₂×#6129	1	1	yes	characteristic
317	C₂×C₄	#88256	1	1	yes	characteristic
318	C₂×C₄	#88257	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	52	2	no	—
322	C₈	#88180	1	4	yes	complemented
323	C₂²	—	4	4	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	#11719256	1	1	yes	characteristic
331	C₂²	#11719258	1	1	yes	characteristic
332	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
333	C₂²	C₂×#88166	1	1	yes	characteristic
334	C₂²	#11719474	1	1	yes	characteristic
335	C₂²	#11719476	1	1	yes	characteristic
336	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	4	4	no	—
339	C₄	—	26	4	no	—
340	C₄	—	52	2	no	—
341	C₄	—	104	4	no	—
342	C₄	D₈×Dic₁₃	1	4	yes	—
343	C₄	#11719247	1	1	yes	characteristic
344	C₄	#11719268	1	1	yes	characteristic
345	C₄	C₂×#88180	1	1	yes	characteristic
346	C₄	#11719532	1	1	yes	characteristic
347	C₂	—	8	4	no	—
348	C₂	#13117785	1	1	yes	characteristic
349	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic
350	C₂	#13123712	1	1	yes	characteristic
351	C₂	#13127185	1	1	yes	characteristic
352	C₂	#13127260	1	1	yes	characteristic
353	C₂	#13127810	1	1	yes	characteristic
354	C₂	#13127822	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×C₁₃×D₈	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	#22807
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #13117785	C₂	semidirect
3. #13115458	C₂	semidirect
4. #13114996	C₂	semidirect
5. #13114836	C₂	semidirect
6. #107×Dic₁₃	C₂	semidirect
7. C₈×C₄⋊Dic₁₃	C₂	semidirect
12. C₄×C₁₃×D₈	C₄	semidirect
99. C₄⋊Dic₁₃	D₈	semidirect
152. C₈×C₁₃	#117	semidirect
157. C₄×D₈	Dic₁₃	semidirect
282. C₁₃	#22807	semidirect
322. C₈	#88180	semidirect

Maximal subgroups

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

subgroup	index
#13117785	2
#13115703	2
#13115458	2
#13114996	2
#13114836	2
#107×Dic₁₃	2
C₈×C₄⋊Dic₁₃	2
C₂×C₄×C₁₃×D₈	2
#22807	13