Subgroups — #16807359

Order 3328.

564 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131222	C₂	1	1	yes	maximal, characteristic, complemented
2	#13131220	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131218	C₂	1	2	yes	maximal, complemented
4	#13131217	C₂	1	2	yes	maximal, complemented
5	#13131215	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131200	C₂	1	1	yes	maximal, characteristic, complemented
8	#13131195	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131068	C₂	1	2	yes	maximal, complemented
10	#13131067	C₂	1	2	yes	maximal, complemented
11	#13123545	C₂	1	1	yes	maximal, characteristic
12	C₂×#88161	C₂²	1	2	yes	—
13	#11719253	C₄	1	1	yes	characteristic
14	C₂×#88106	C₂²	1	2	yes	—
15	C₂²×#6092	C₂²	1	2	yes	—
16	C₂×#88098	C₂²	1	2	yes	—
17	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
18	C₁₃×#436	C₄	1	1	yes	characteristic
19	#11718789	C₄	1	2	yes	—
20	D₄×#1055	C₂²	1	1	yes	characteristic
21	#11718498	C₄	1	1	yes	characteristic
22	#11718495	C₂²	1	1	yes	characteristic
23	#11718388	C₄	1	2	yes	—
24	#11718386	C₂²	1	2	yes	—
25	#11718349	C₄	1	1	yes	characteristic
26	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
27	C₂²⋊#6092	C₂²	1	2	yes	—
28	C₂²⋊#6092	C₂²	1	2	yes	—
29	#11718260	C₂²	1	1	yes	characteristic
30	#11718260	C₂²	1	1	yes	characteristic
31	#11718259	C₂²	1	1	yes	characteristic
32	#11718259	C₂²	1	2	yes	—
33	#11718259	C₂²	1	1	yes	characteristic
34	#11718259	C₂²	1	2	yes	—
35	#11718259	C₂²	1	2	yes	—
36	#11718258	C₂²	1	2	yes	—
37	C₄⋊#6092	C₂²	1	1	yes	characteristic
38	C₄⋊#6092	C₂²	1	1	yes	characteristic
39	C₄×#6092	C₂²	1	1	yes	characteristic
40	C₄×#6092	C₂²	1	1	yes	characteristic
41	C₂×#6127	C₂×C₄	1	2	yes	—
42	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
43	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	#88194	C₈	1	1	yes	characteristic
45	#88192	C₈	1	1	yes	characteristic
46	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	#88161	—	4	2	no	—
49	#88161	—	4	2	no	—
50	C₂×#6094	C₂×C₄	1	2	yes	—
51	C₂×#6094	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	2	yes	—
55	#88149	C₈	1	1	yes	characteristic
56	#88142	C₈	1	1	yes	characteristic
57	#88139	C₈	1	2	yes	—
58	#88138	C₈	1	2	yes	—
59	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
61	#88106	—	4	2	no	—
62	C₂×#6092	—	4	2	no	—
63	C₂×#6092	C₂³	1	1	yes	characteristic
64	C₂×#6092	C₂³	1	1	yes	characteristic
65	C₂×#6092	C₂³	1	1	yes	characteristic
66	C₂×#6092	C₂³	1	1	yes	characteristic
67	C₂×#6092	C₂³	1	2	yes	—
68	C₂×#6092	C₂³	1	2	yes	—
69	#88098	—	4	2	no	—
70	D₁₃⋊C₁₆	—	4	2	no	—
71	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊#1055	C₂×C₄	1	2	yes	—
75	#88060	C₂×C₄	1	2	yes	—
76	#88060	C₂×C₄	1	2	yes	—
77	#88054	C₂×C₄	1	2	yes	—
78	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
80	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
81	C₄×#1055	C₂³	1	1	yes	characteristic
82	#33495	—	13	1	no	maximal, Sylow
83	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
86	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
87	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
88	#6127	—	4	2	no	—
89	#6127	—	4	2	no	—
90	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
91	C₂×D₅₂	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
96	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
97	C₁₃×M₁₆	—	4	2	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	2	yes	—
101	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
102	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
105	#6103	C₂×C₈	1	2	yes	—
106	#6103	C₂×C₈	1	2	yes	—
107	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
108	#6101	C₂×C₈	1	2	yes	—
109	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	C₂×#1055	C₂²×C₄	1	2	yes	—
116	C₂×#1055	C₂⁴	1	1	yes	characteristic
117	#6094	—	4	2	no	—
118	#6094	—	4	2	no	—
119	C₈×D₁₃	—	2	2	no	—
120	C₈×D₁₃	—	2	2	no	—
121	C₈×D₁₃	—	2	1	no	—
122	C₈×D₁₃	—	2	1	no	—
123	C₈×D₁₃	—	4	1	no	—
124	#6092	—	2	1	no	—
125	#6092	—	2	2	no	—
126	#6092	—	2	1	no	—
127	#6092	—	2	2	no	—
128	#6092	—	2	1	no	—
129	#6092	—	2	1	no	—
130	#2931	—	13	1	no	—
131	#2169	—	13	1	no	—
132	#2169	—	13	1	no	—
133	#2168	—	13	2	no	—
134	#2168	—	13	2	no	—
135	D₄×C₁₆	—	13	1	no	—
136	D₄×C₁₆	—	13	1	no	—
137	#2163	—	13	1	no	—
138	#2162	—	13	1	no	—
139	#2113	—	13	2	no	—
140	#2113	—	13	2	no	—
141	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
142	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
143	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
144	D₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₂²⋊D₁₃	—	4	2	no	—
152	C₂²⋊D₁₃	—	4	2	no	—
153	C₂²⋊D₁₃	—	4	2	no	—
154	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
155	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
156	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
157	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
158	D₅₂	—	2	2	no	—
159	D₅₂	—	4	2	no	—
160	C₄×D₁₃	—	2	1	no	—
161	C₄×D₁₃	—	2	2	no	—
162	C₄×D₁₃	—	2	1	no	—
163	C₄×D₁₃	—	2	2	no	—
164	C₄×D₁₃	—	2	2	no	—
165	C₄×D₁₃	—	4	1	no	—
166	Dic₂₆	—	4	2	no	—
167	C₈×C₁₃	—	2	1	no	—
168	C₈×C₁₃	—	2	2	no	—
169	C₈×C₁₃	—	2	1	no	—
170	#1055	—	2	2	no	—
171	#1055	—	2	1	no	—
172	#1055	H(5,𝔽₂)	1	1	yes	characteristic
173	#1055	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×#131	—	13	2	no	—
175	#532	—	13	1	no	—
176	C₂×M₃₂	—	13	2	no	—
177	C₂×M₃₂	—	13	2	no	—
178	C₂²×C₁₆	—	13	2	no	—
179	C₂²×C₁₆	—	13	2	no	—
180	#436	—	13	2	no	—
181	#436	—	13	1	no	—
182	#436	#261	1	1	yes	characteristic
183	C₈×D₄	—	13	1	no	—
184	C₈×D₄	—	13	1	no	—
185	C₈×D₄	—	13	2	no	—
186	#423	—	13	1	no	—
187	#408	—	13	2	no	—
188	C₂×C₄×C₈	—	13	1	no	—
189	C₄⋊C₁₆	—	13	1	no	—
190	C₄⋊C₁₆	—	13	1	no	—
191	C₄⋊C₁₆	—	13	1	no	—
192	C₄⋊C₁₆	—	13	1	no	—
193	C₂²⋊C₁₆	—	13	1	no	—
194	C₂²⋊C₁₆	—	13	1	no	—
195	C₂²⋊C₁₆	—	13	2	no	—
196	C₂²⋊C₁₆	—	13	2	no	—
197	C₂²⋊C₁₆	—	13	2	no	—
198	C₂²⋊C₁₆	—	13	2	no	—
199	C₂²⋊C₁₆	—	13	2	no	—
200	#346	—	13	2	no	—
201	C₄×C₁₆	—	13	1	no	—
202	C₄×C₁₆	—	13	1	no	—
203	C₂²×C₁₃	—	2	2	no	—
204	C₂²×C₁₃	—	2	2	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
207	C₂×D₁₃	—	2	1	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₂×D₁₃	—	2	1	no	—
210	C₂×D₁₃	—	2	2	no	—
211	C₂×D₁₃	—	2	1	no	—
212	C₂×D₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	2	2	no	—
215	C₄×C₁₃	#504	1	2	yes	—
216	C₄×C₁₃	#518	1	1	yes	characteristic
217	C₄×C₁₃	#568	1	1	yes	characteristic
218	Dic₁₃	—	2	1	no	—
219	Dic₁₃	—	2	1	no	—
220	Dic₁₃	—	2	1	no	—
221	Dic₁₃	—	2	2	no	—
222	C₂×#41	—	13	2	no	—
223	#131	—	52	2	no	—
224	#131	—	52	2	no	—
225	C₂×M₁₆	—	13	2	no	—
226	C₂×M₁₆	—	13	2	no	—
227	C₂×M₁₆	C₂×#261	1	2	yes	—
228	C₂²×C₈	—	13	2	no	—
229	C₂²×C₈	—	13	1	no	—
230	C₂²×C₈	—	13	2	no	—
231	C₂²×C₈	—	13	1	no	—
232	C₄⋊Q₈	—	13	1	no	—
233	#127	—	13	1	no	—
234	#124	—	13	2	no	—
235	#121	—	13	1	no	—
236	#121	—	13	1	no	—
237	#121	—	13	2	no	—
238	C₄×D₄	—	13	2	no	—
239	C₄×D₄	—	13	1	no	—
240	C₄×D₄	C₂×#261	1	1	yes	characteristic
241	C₂×C₄²	—	13	1	no	—
242	M₃₂	—	52	2	no	—
243	M₃₂	—	52	2	no	—
244	C₂×C₁₆	—	13	1	no	—
245	C₂×C₁₆	—	13	1	no	—
246	C₂×C₁₆	—	13	1	no	—
247	C₂×C₁₆	—	13	1	no	—
248	C₂×C₁₆	—	13	2	no	—
249	C₂×C₁₆	—	13	2	no	—
250	C₂×C₁₆	—	52	2	no	—
251	C₂×C₁₆	—	52	2	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	13	1	no	—
254	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
255	C₂²⋊C₈	—	13	2	no	—
256	C₂²⋊C₈	—	13	2	no	—
257	C₂²⋊C₈	—	13	2	no	—
258	C₂²⋊C₈	C₂×#261	1	2	yes	—
259	C₄×C₈	—	13	1	no	—
260	C₄×C₈	—	13	1	no	—
261	C₄×C₈	—	13	1	no	—
262	C₄×C₈	C₂×#261	1	1	yes	characteristic
263	C₂×C₁₃	—	4	2	no	—
264	C₂×C₁₃	#2972	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
266	C₂×C₁₃	#3407	1	1	yes	characteristic
267	D₁₃	—	4	1	no	—
268	D₁₃	—	4	2	no	—
269	#41	—	52	2	no	—
270	#41	—	52	2	no	—
271	C₂×Q₈	—	13	2	no	—
272	C₂×D₄	—	13	2	no	—
273	C₂×D₄	—	13	2	no	—
274	C₂×D₄	—	13	2	no	—
275	C₂×D₄	—	13	1	no	—
276	C₂×D₄	—	13	2	no	—
277	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
278	C₂²×C₄	—	13	1	no	—
279	C₂²×C₄	—	13	2	no	—
280	C₂²×C₄	—	13	1	no	—
281	C₂²×C₄	—	13	1	no	—
282	C₂²×C₄	C₂²×#261	1	2	yes	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	52	2	no	—
285	M₁₆	—	52	2	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	13	1	no	—
288	C₂×C₈	—	13	2	no	—
289	C₂×C₈	—	26	2	no	—
290	C₂×C₈	—	26	2	no	—
291	C₂×C₈	—	26	1	no	—
292	C₂×C₈	—	26	1	no	—
293	C₂×C₈	—	26	2	no	—
294	C₂×C₈	—	26	2	no	—
295	C₂×C₈	—	52	1	no	—
296	C₂×C₈	C₂²×#261	1	1	yes	characteristic
297	C₂×C₈	C₂²×#261	1	1	yes	characteristic
298	C₂×C₈	C₂²×#261	1	2	yes	—
299	C₄⋊C₄	—	13	2	no	—
300	C₄⋊C₄	—	13	1	no	—
301	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
302	C₂²⋊C₄	—	13	2	no	—
303	C₂²⋊C₄	—	13	2	no	—
304	C₂²⋊C₄	—	13	2	no	—
305	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
306	C₄²	—	13	1	no	—
307	C₄²	—	13	1	no	—
308	C₄²	—	13	1	no	—
309	C₄²	C₂²×#261	1	1	yes	characteristic
310	C₁₆	—	26	1	no	—
311	C₁₆	—	26	1	no	—
312	C₁₆	—	26	2	no	—
313	C₁₆	—	26	2	no	—
314	C₁₆	—	26	1	no	—
315	C₁₆	—	26	1	no	—
316	C₁₃	#33495	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	13	2	no	—
318	C₂³	—	13	1	no	—
319	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
320	Q₈	—	52	2	no	—
321	D₄	—	2	2	no	—
322	D₄	—	26	2	no	—
323	D₄	—	52	2	no	—
324	D₄	—	52	2	no	—
325	D₄	—	52	2	no	—
326	D₄	—	52	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	13	2	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	1	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
340	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
341	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
342	C₂×C₄	C₂³×#261	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	2	no	—
345	C₈	—	2	1	no	—
346	C₈	—	13	1	no	—
347	C₈	—	13	1	no	—
348	C₈	—	26	2	no	—
349	C₈	—	26	1	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	2	no	—
355	C₂²	—	26	1	no	—
356	C₂²	—	26	2	no	—
357	C₂²	—	26	1	no	—
358	C₂²	—	26	2	no	—
359	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	26	1	no	—
363	C₄	—	26	1	no	—
364	C₄	—	26	1	no	—
365	C₄	—	26	2	no	—
366	C₄	#11719096	1	2	yes	—
367	C₄	#11719623	1	1	yes	characteristic
368	C₄	#11719629	1	1	yes	characteristic
369	C₂	—	4	2	no	—
370	C₂	—	52	1	no	—
371	C₂	—	52	2	no	—
372	C₂	#13131684	1	1	yes	characteristic
373	C₂	C₂×#11719096	1	1	yes	characteristic
374	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#436	C₄
prime radical	#11719253	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#436	#261
hypocenter	C₁₃	#33495
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. #13131222	C₂	semidirect
2. #13131220	C₂	semidirect
3. #13131218	C₂	semidirect
4. #13131217	C₂	semidirect
5. #13131215	C₂	semidirect
6. D₄×#6092	C₂	semidirect
7. #13131200	C₂	semidirect
8. #13131195	C₂	semidirect
9. #13131068	C₂	semidirect
10. #13131067	C₂	semidirect
316. C₁₃	#33495	semidirect

Maximal subgroups

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

subgroup	index
#13131222	2
#13131220	2
#13131218	2
#13131217	2
#13131215	2
D₄×#6092	2
#13131200	2
#13131195	2
#13131068	2
#13131067	2
#13123545	2
#33495	13