Subgroups — #15834340

Order 3328.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13123571	C₂	1	2	yes	maximal, complemented
3	#13113720	C₂	1	2	yes	maximal, complemented
4	#13113664	C₂	1	4	yes	maximal, complemented
5	#13113630	C₂	1	2	yes	maximal, complemented
6	#13113260	C₂	1	2	yes	maximal, complemented
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×M₁₆×D₁₃	—	2	2	no	—
10	C₂²×D₁₀₄	C₄	1	1	yes	characteristic
11	C₂×C₄×D₅₂	C₂²	1	2	yes	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718787	—	2	2	no	—
15	C₂×#88061	C₄	1	2	yes	—
16	C₂×#88058	C₄	1	1	yes	characteristic
17	#11718586	C₂²	1	1	yes	characteristic
18	C₄×#6099	C₂²	1	1	yes	characteristic
19	#11718499	C₂²	1	4	yes	complemented
20	#11718497	—	2	4	no	—
21	#11718495	C₂²	1	4	yes	complemented
22	C₄×D₁₀₄	C₄	1	2	yes	—
23	C₄×D₁₀₄	C₄	1	1	yes	characteristic
24	C₄×D₁₀₄	C₄	1	1	yes	characteristic
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	C₁₃×#335	C₂²	1	2	yes	complemented
29	C₁₃×#335	C₂²	1	2	yes	complemented
30	#11718075	C₂²	1	4	yes	complemented
31	#11718075	C₂²	1	4	yes	complemented
32	#11718046	C₂²	1	2	yes	complemented
33	C₈×#1055	C₂²	1	2	yes	complemented
34	C₂²×D₅₂	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₁₃	—	2	2	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6103	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
42	C₂×#6099	D₄	1	1	yes	characteristic
43	C₂×#6099	D₄	1	1	yes	characteristic
44	M₁₆×D₁₃	—	4	8	no	—
45	C₂×D₁₀₄	—	2	4	no	—
46	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
47	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
48	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
49	C₂×#6094	—	2	4	no	—
50	C₂×C₈×D₁₃	—	2	4	no	—
51	C₄×D₅₂	—	2	2	no	—
52	C₄×D₅₂	—	2	2	no	—
53	C₄×D₅₂	C₂×C₄	1	2	yes	—
54	C₄×D₅₂	C₂×C₄	1	2	yes	—
55	C₄×D₅₂	C₂³	1	2	yes	—
56	C₄×D₅₂	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₈	—	2	2	no	—
58	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊#1055	—	2	1	no	—
64	#88061	C₂×C₄	1	2	yes	—
65	#88061	C₂×C₄	1	2	yes	—
66	#88061	C₂×C₄	1	2	yes	—
67	#88061	C₂×C₄	1	2	yes	—
68	#88060	—	2	4	no	—
69	#88060	—	2	4	no	—
70	#88058	C₂×C₄	1	2	yes	—
71	#88058	C₂×C₄	1	1	yes	characteristic
72	#88058	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#1055	—	2	1	no	—
74	#88043	D₄	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	C₄×#1055	C₂³	1	2	yes	—
77	M₁₆×D₈	—	13	1	no	maximal, Sylow
78	C₂³×D₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₂×D₅₂	—	2	4	no	—
84	C₂×D₅₂	—	2	2	no	—
85	C₂×D₅₂	—	4	2	no	—
86	C₂×D₅₂	C₂²×C₄	1	2	yes	—
87	C₂×D₅₂	C₂²×C₄	1	2	yes	—
88	C₂×C₄×D₁₃	—	2	2	no	—
89	C₂×C₄×D₁₃	—	2	2	no	—
90	C₂×C₄×D₁₃	—	2	2	no	—
91	C₂×C₄×D₁₃	—	2	2	no	—
92	C₂×C₄×D₁₃	—	4	4	no	—
93	C₂×C₄×D₁₃	—	4	4	no	—
94	C₁₃×M₁₆	—	4	4	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	#41	1	1	yes	characteristic
105	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
106	#6103	—	2	2	no	—
107	#6103	—	2	2	no	—
108	#6103	—	2	2	no	—
109	#6103	—	2	2	no	—
110	C₄⋊Dic₁₃	—	2	2	no	—
111	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
113	#6099	—	2	2	no	—
114	#6099	D₈	1	4	yes	complemented
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	C₂×#1055	C₂×D₄	1	2	yes	—
118	D₁₀₄	—	2	4	no	—
119	D₁₀₄	—	2	2	no	—
120	D₁₀₄	M₁₆	1	4	yes	complemented
121	#6094	—	4	8	no	—
122	C₈×D₁₃	—	4	8	no	—
123	C₂×C₄×D₈	—	13	1	no	—
124	D₄×M₁₆	—	13	2	no	—
125	#1589	—	13	2	no	—
126	#1581	—	13	4	no	—
127	C₈×D₈	—	13	2	no	—
128	#1567	D₁₃	1	1	yes	characteristic, complemented
129	#1486	—	13	2	no	—
130	C₈×M₁₆	—	13	1	no	—
131	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂²×D₁₃	—	2	2	no	—
133	C₂²×D₁₃	—	2	2	no	—
134	C₂²×D₁₃	—	4	4	no	—
135	C₂²×D₁₃	—	4	4	no	—
136	C₂²×D₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₁₃	#135	1	2	yes	—
145	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	4	4	no	—
151	D₅₂	—	2	2	no	—
152	D₅₂	—	2	2	no	—
153	D₅₂	—	2	2	no	—
154	D₅₂	—	4	4	no	—
155	D₅₂	C₂×M₁₆	1	4	yes	complemented
156	C₄×D₁₃	—	4	2	no	—
157	C₄×D₁₃	—	4	2	no	—
158	C₄×D₁₃	—	4	2	no	—
159	C₄×D₁₃	—	4	2	no	—
160	C₄×D₁₃	—	4	4	no	—
161	C₄×D₁₃	—	4	4	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	2	1	no	—
165	C₈×C₁₃	—	4	4	no	—
166	C₈×C₁₃	C₂×M₁₆	1	2	yes	complemented
167	#1055	—	2	2	no	—
168	#1055	—	4	2	no	—
169	#1055	C₂×D₈	1	4	yes	complemented
170	C₂²×D₈	—	13	1	no	—
171	C₂²×M₁₆	—	26	2	no	—
172	C₂×C₄×D₄	—	13	2	no	—
173	C₄×D₈	—	13	1	no	—
174	C₄×D₈	—	13	2	no	—
175	C₄×D₈	—	13	1	no	—
176	C₄×D₈	—	13	1	no	—
177	C₄×D₈	—	13	2	no	—
178	C₄×D₈	—	13	1	no	—
179	#436	—	13	4	no	—
180	#435	—	26	4	no	—
181	C₈×D₄	—	13	4	no	—
182	#431	—	13	1	no	—
183	C₂×#107	—	13	1	no	—
184	#423	C₂×D₁₃	1	2	yes	—
185	C₂×#102	—	13	2	no	—
186	#407	—	26	2	no	—
187	C₄×M₁₆	—	13	1	no	—
188	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
189	#335	C₂×D₁₃	1	2	yes	complemented
190	#335	C₂×D₁₃	1	2	yes	complemented
191	D₄⋊C₈	—	13	4	no	—
192	D₄⋊C₈	—	13	4	no	—
193	#322	—	13	2	no	—
194	C₈²	—	13	2	no	—
195	C₂²×C₁₃	—	2	1	no	—
196	C₂²×C₁₃	—	2	1	no	—
197	C₂²×C₁₃	C₄×D₈	1	2	yes	—
198	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂×D₁₃	—	2	4	no	—
200	C₂×D₁₃	—	4	2	no	—
201	C₂×D₁₃	—	4	2	no	—
202	C₂×D₁₃	—	4	2	no	—
203	C₂×D₁₃	—	4	2	no	—
204	C₂×D₁₃	—	4	2	no	—
205	C₂×D₁₃	—	8	8	no	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	C₄×D₈	1	2	yes	—
210	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
211	C₄×C₁₃	#568	1	1	yes	characteristic
212	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
213	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
214	Dic₁₃	—	4	2	no	—
215	Dic₁₃	—	4	2	no	—
216	C₂²×D₄	—	13	2	no	—
217	C₂³×C₄	—	26	2	no	—
218	C₂×D₈	—	13	1	no	—
219	C₂×D₈	—	13	2	no	—
220	C₂×D₈	—	13	1	no	—
221	C₂×D₈	—	26	4	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	13	1	no	—
224	C₂×M₁₆	—	13	1	no	—
225	C₂×M₁₆	—	26	4	no	—
226	C₂×M₁₆	—	52	8	no	—
227	C₂²×C₈	—	26	4	no	—
228	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
230	C₄×D₄	—	13	2	no	—
231	C₄×D₄	—	13	2	no	—
232	C₄×D₄	—	13	2	no	—
233	C₄×D₄	—	13	2	no	—
234	C₄×D₄	—	26	2	no	—
235	C₄×D₄	—	26	2	no	—
236	C₂×C₄⋊C₄	—	13	2	no	—
237	C₂×C₂²⋊C₄	—	26	2	no	—
238	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	#107	—	13	2	no	—
240	#107	—	13	1	no	—
241	#107	—	13	1	no	—
242	C₄⋊C₈	—	2	2	no	—
243	C₄⋊C₈	—	26	1	no	—
244	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
245	#102	—	13	2	no	—
246	#102	—	13	2	no	—
247	#102	—	13	2	no	—
248	#102	—	13	2	no	—
249	C₂²⋊C₈	—	26	4	no	—
250	C₂²⋊C₈	—	26	4	no	—
251	C₂²⋊C₈	—	26	1	no	—
252	#97	—	13	2	no	—
253	C₄×C₈	—	13	2	no	—
254	C₄×C₈	—	13	2	no	—
255	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
256	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
259	C₂×C₁₃	—	2	2	no	—
260	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
261	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
262	C₂×C₁₃	#2957	1	1	yes	characteristic
263	D₁₃	—	4	4	no	—
264	D₁₃	—	8	2	no	—
265	C₂⁴	—	26	2	no	—
266	C₂×D₄	—	13	2	no	—
267	C₂×D₄	—	13	2	no	—
268	C₂×D₄	—	26	4	no	—
269	C₂×D₄	—	26	2	no	—
270	C₂×D₄	—	52	2	no	—
271	C₂²×C₄	—	26	2	no	—
272	C₂²×C₄	—	26	2	no	—
273	C₂²×C₄	—	26	2	no	—
274	C₂²×C₄	—	26	2	no	—
275	C₂²×C₄	—	26	2	no	—
276	C₂²×C₄	—	52	4	no	—
277	C₂²×C₄	—	52	4	no	—
278	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
280	C₂²×C₄	#6131	1	1	yes	characteristic
281	D₈	—	13	4	no	—
282	D₈	—	26	4	no	—
283	D₈	—	26	2	no	—
284	M₁₆	—	4	4	no	—
285	M₁₆	—	13	4	no	—
286	M₁₆	—	26	2	no	—
287	M₁₆	—	52	8	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	4	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	13	2	no	—
292	C₂×C₈	—	13	2	no	—
293	C₂×C₈	—	26	2	no	—
294	C₂×C₈	—	52	8	no	—
295	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
299	C₄⋊C₄	—	13	2	no	—
300	C₄⋊C₄	—	13	2	no	—
301	C₄⋊C₄	—	26	2	no	—
302	C₂²⋊C₄	—	26	2	no	—
303	C₂²⋊C₄	—	26	2	no	—
304	C₂²⋊C₄	—	26	2	no	—
305	C₂²⋊C₄	—	26	2	no	—
306	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
307	C₄²	D₄×D₁₃	1	1	yes	characteristic
308	C₄²	#6131	1	1	yes	characteristic
309	C₄²	C₂³×D₁₃	1	1	yes	characteristic
310	C₁₃	M₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	26	2	no	—
312	C₂³	—	26	2	no	—
313	C₂³	—	52	4	no	—
314	C₂³	—	52	4	no	—
315	C₂³	—	104	2	no	—
316	C₂³	#88147	1	1	yes	characteristic
317	D₄	—	13	4	no	—
318	D₄	—	26	2	no	—
319	D₄	—	26	2	no	—
320	D₄	—	26	2	no	—
321	D₄	—	52	4	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	52	4	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	—	52	2	no	—
332	C₂×C₄	—	52	4	no	—
333	C₂×C₄	—	52	4	no	—
334	C₂×C₄	#88147	1	1	yes	characteristic
335	C₂×C₄	#88147	1	1	yes	characteristic
336	C₂×C₄	#88147	1	1	yes	characteristic
337	C₂×C₄	D₈×D₁₃	1	2	yes	—
338	C₂×C₄	#88173	1	2	yes	—
339	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
341	C₂×C₄	C₂×#6131	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	4	no	—
346	C₈	—	13	4	no	—
347	C₈	—	26	2	no	—
348	C₈	—	52	2	no	—
349	C₈	M₁₆×D₁₃	1	2	yes	complemented
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	26	4	no	—
353	C₂²	—	52	2	no	—
354	C₂²	—	52	2	no	—
355	C₂²	—	52	2	no	—
356	C₂²	—	52	2	no	—
357	C₂²	—	52	2	no	—
358	C₂²	—	104	8	no	—
359	C₂²	#11718533	1	2	yes	—
360	C₂²	C₂×#88147	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	52	2	no	—
365	C₄	—	52	2	no	—
366	C₄	#11718533	1	2	yes	—
367	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
368	C₄	#11719450	1	1	yes	characteristic
369	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
370	C₄	C₂×#88173	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	52	4	no	—
373	C₂	—	104	2	no	—
374	C₂	#13123571	1	1	yes	characteristic
375	C₂	C₂×#11718533	1	1	yes	characteristic
376	C₂	#13123752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1567	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₀₄	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1567	D₁₃
hypocenter	C₁₃	M₁₆×D₈
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
2. #13123571	C₂	semidirect
3. #13113720	C₂	semidirect
4. #13113664	C₂	semidirect
5. #13113630	C₂	semidirect
6. #13113260	C₂	semidirect
7. C₈×#6099	C₂	semidirect
8. C₁₃×#1567	C₂	semidirect
19. #11718499	C₂²	semidirect
21. #11718495	C₂²	semidirect
28. C₁₃×#335	C₂²	semidirect
29. C₁₃×#335	C₂²	semidirect
30. #11718075	C₂²	semidirect
31. #11718075	C₂²	semidirect
32. #11718046	C₂²	semidirect
33. C₈×#1055	C₂²	semidirect
114. #6099	D₈	semidirect
120. D₁₀₄	M₁₆	semidirect
128. #1567	D₁₃	semidirect
155. D₅₂	C₂×M₁₆	semidirect
166. C₈×C₁₃	C₂×M₁₆	semidirect
169. #1055	C₂×D₈	semidirect
189. #335	C₂×D₁₃	semidirect
190. #335	C₂×D₁₃	semidirect
310. C₁₃	M₁₆×D₈	semidirect
349. C₈	M₁₆×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
#13123571	2
#13113720	2
#13113664	2
#13113630	2
#13113260	2
C₈×#6099	2
C₁₃×#1567	2
M₁₆×D₈	13