Subgroups — #16020971

Order 3328.

566 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127849	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117100	C₂	1	2	yes	maximal, complemented
3	#13117085	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116891	C₂	1	2	yes	maximal, complemented
5	#13116873	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116871	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116868	C₂	1	2	yes	maximal, complemented
8	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116710	C₂	1	2	yes	maximal, complemented
10	C₄×#88067	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3391	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#550	—	2	1	no	—
13	#11719290	—	2	1	no	—
14	C₁₃×#493	C₂²	1	1	yes	characteristic
15	C₁₃×#493	C₂²	1	2	yes	—
16	C₁₃×#459	—	2	2	no	—
17	C₁₃×#459	—	2	1	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×D₈	C₂²	1	2	yes	—
21	C₂×#88067	C₂²	1	1	yes	characteristic
22	C₂×#88067	C₂²	1	2	yes	—
23	C₂×#88066	C₂²	1	2	yes	—
24	C₂×#88066	C₂²	1	2	yes	—
25	C₂×#88066	C₂²	1	2	yes	—
26	#11718509	—	2	1	no	—
27	#11718505	—	2	2	no	—
28	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
29	#11718357	C₂²	1	2	yes	—
30	#11718356	C₂²	1	1	yes	characteristic
31	C₄×D₁₀₄	C₂²	1	2	yes	—
32	#11718154	C₂²	1	1	yes	characteristic
33	#11718154	C₂²	1	2	yes	—
34	#11718084	C₂²	1	1	yes	characteristic
35	#11718083	C₂²	1	2	yes	—
36	#11718082	C₂²	1	1	yes	characteristic
37	C₄⋊#6090	C₂²	1	2	yes	—
38	C₄⋊#6090	C₂²	1	1	yes	characteristic
39	#11718053	C₂²	1	2	yes	—
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×#41	—	4	1	no	—
42	C₂×C₁₃×D₈	—	2	2	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#127	—	2	2	no	—
48	C₁₃×#121	—	4	1	no	—
49	C₁₃×#121	—	4	2	no	—
50	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×D₄×C₁₃	—	2	1	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	#88201	—	4	1	no	—
54	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
55	C₂×D₁₀₄	—	2	1	no	—
56	C₂×D₁₀₄	C₂³	1	2	yes	—
57	C₂×#6095	—	2	2	no	—
58	#88148	—	4	2	no	—
59	#88147	—	4	1	no	—
60	#88129	—	2	2	no	—
61	#88128	—	2	1	no	—
62	C₄×D₅₂	—	2	2	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
66	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×#102	—	2	1	no	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88067	—	2	2	no	—
72	#88067	—	4	2	no	—
73	#88066	—	2	4	no	—
74	#88066	—	4	2	no	—
75	#88066	—	4	2	no	—
76	#88061	—	2	2	no	—
77	#88058	C₂³	1	1	yes	characteristic
78	#88056	—	2	1	no	—
79	C₂×#6090	C₂³	1	1	yes	characteristic
80	C₂×#6090	C₂³	1	2	yes	—
81	C₂×#6090	C₂³	1	1	yes	characteristic
82	#33966	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	8	4	no	—
84	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	1	no	—
87	C₂×D₄×C₁₃	—	4	2	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	—	4	1	no	—
93	C₂×D₅₂	—	2	2	no	—
94	C₂×D₅₂	—	4	1	no	—
95	C₂×D₅₂	—	4	1	no	—
96	C₂×C₄×D₁₃	—	4	2	no	—
97	C₂×Dic₂₆	—	2	1	no	—
98	C₁₃×D₈	—	2	2	no	—
99	C₁₃×D₈	—	2	1	no	—
100	C₁₃×D₈	—	2	1	no	—
101	C₁₃×D₈	—	2	2	no	—
102	C₁₃×D₈	—	8	2	no	—
103	C₁₃×D₈	—	8	1	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
110	C₁₃×C₂²⋊C₄	—	4	1	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	2	yes	—
114	#6103	—	4	2	no	—
115	#6103	—	4	2	no	—
116	#6103	—	4	2	no	—
117	C₄⋊Dic₁₃	—	2	1	no	—
118	#6101	—	4	1	no	—
119	C₄×Dic₁₃	—	4	1	no	—
120	Dic₅₂	—	2	1	no	—
121	Dic₅₂	—	2	1	no	—
122	D₁₀₄	—	2	2	no	—
123	D₁₀₄	—	2	2	no	—
124	D₁₀₄	—	8	1	no	—
125	#6095	—	8	2	no	—
126	#6090	—	2	1	no	—
127	#6090	—	2	2	no	—
128	#6090	#41	1	2	yes	complemented
129	#3396	—	13	1	no	—
130	#3391	D₁₃	1	1	yes	characteristic, complemented
131	#2250	—	13	2	no	—
132	#2246	—	13	1	no	—
133	#2210	—	13	2	no	—
134	#2207	—	13	1	no	—
135	#2206	—	13	2	no	—
136	#2206	—	13	1	no	—
137	#2183	—	13	1	no	—
138	#2177	—	13	2	no	—
139	C₄×SD₃₂	—	13	1	no	—
140	C₂³×C₁₃	—	4	2	no	—
141	C₂³×C₁₃	—	4	1	no	—
142	C₂²×D₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	2	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	1	no	—
146	D₄×C₁₃	—	4	1	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	8	4	no	—
149	D₄×C₁₃	—	8	2	no	—
150	D₄×C₁₃	—	16	2	no	—
151	D₄×C₁₃	—	16	1	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	8	4	no	—
154	C₂×C₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×Dic₁₃	—	4	1	no	—
161	C₂×Dic₁₃	—	4	1	no	—
162	D₅₂	—	4	2	no	—
163	D₅₂	—	4	2	no	—
164	D₅₂	—	8	2	no	—
165	D₅₂	—	16	1	no	—
166	C₄×D₁₃	—	16	2	no	—
167	Dic₂₆	—	4	1	no	—
168	Dic₂₆	—	4	1	no	—
169	C₈×C₁₃	—	2	2	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
172	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
173	#559	—	26	1	no	—
174	#550	—	2	1	no	—
175	C₂×SD₃₂	—	13	1	no	—
176	C₂×SD₃₂	—	13	2	no	—
177	C₂×D₁₆	—	13	2	no	—
178	C₂×D₁₆	—	13	2	no	—
179	C₂×D₁₆	—	13	2	no	—
180	#495	—	13	2	no	—
181	#493	—	13	1	no	—
182	#493	C₂×D₁₃	1	2	yes	—
183	#493	C₂×D₁₃	1	1	yes	characteristic
184	#463	—	26	2	no	—
185	#460	—	26	1	no	—
186	#459	—	2	1	no	—
187	#459	—	2	2	no	—
188	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
189	C₄×Q₁₆	—	13	1	no	—
190	C₄×D₈	—	13	2	no	—
191	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
192	C₄×D₈	C₂×D₁₃	1	2	yes	—
193	#367	—	13	1	no	—
194	C₄⋊C₁₆	—	13	1	no	—
195	C₄⋊C₁₆	—	13	2	no	—
196	#358	—	13	1	no	—
197	#357	—	13	1	no	—
198	#357	—	13	2	no	—
199	#357	—	13	2	no	—
200	#346	—	13	2	no	—
201	C₄×C₁₆	—	13	1	no	—
202	C₂²×C₁₃	—	8	2	no	—
203	C₂²×C₁₃	—	8	1	no	—
204	C₂²×C₁₃	—	8	2	no	—
205	C₂²×C₁₃	—	8	1	no	—
206	C₂²×C₁₃	—	8	1	no	—
207	C₂²×C₁₃	—	8	2	no	—
208	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
209	C₂×D₁₃	—	8	2	no	—
210	C₂×D₁₃	—	8	2	no	—
211	C₂×D₁₃	—	8	2	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	#509	1	2	yes	—
217	C₄×C₁₃	#550	1	1	yes	characteristic
218	C₄×C₁₃	#576	1	1	yes	characteristic
219	Dic₁₃	—	8	1	no	—
220	Dic₁₃	—	8	1	no	—
221	C₂×#41	—	4	1	no	—
222	C₂×Q₁₆	—	13	1	no	—
223	C₂×SD₁₆	—	26	2	no	—
224	C₂×D₈	—	2	2	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	—	13	2	no	—
227	C₂×D₈	—	26	1	no	—
228	C₂×D₈	C₂²×D₁₃	1	2	yes	—
229	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
230	#127	—	2	2	no	—
231	#127	—	2	1	no	—
232	#127	—	26	1	no	—
233	#124	—	26	2	no	—
234	#124	—	52	1	no	—
235	#123	—	52	2	no	—
236	#121	—	4	1	no	—
237	#121	—	4	2	no	—
238	C₄×Q₈	—	26	1	no	—
239	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
240	C₄×D₄	—	2	2	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	26	2	no	—
243	C₄×D₄	—	52	1	no	—
244	SD₃₂	—	26	2	no	—
245	SD₃₂	—	52	2	no	—
246	D₁₆	—	26	4	no	—
247	D₁₆	—	52	2	no	—
248	D₁₆	—	52	2	no	—
249	C₂×C₁₆	—	13	1	no	—
250	C₂×C₁₆	—	13	1	no	—
251	C₂×C₁₆	—	13	2	no	—
252	#107	—	13	1	no	—
253	#107	C₂²×D₁₃	1	1	yes	characteristic
254	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
255	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
256	#103	—	26	1	no	—
257	#102	—	2	2	no	—
258	#102	—	2	1	no	—
259	#102	—	26	2	no	—
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₂×C₁₃	—	16	2	no	—
263	C₂×C₁₃	—	16	1	no	—
264	C₂×C₁₃	#3391	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
266	C₂×C₁₃	#3415	1	1	yes	characteristic
267	D₁₃	—	16	2	no	—
268	#41	—	8	4	no	—
269	C₂×Q₈	—	26	1	no	—
270	C₂×Q₈	#6104	1	1	yes	characteristic
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	26	2	no	—
278	C₂×D₄	—	52	1	no	—
279	C₂×D₄	—	52	1	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	52	2	no	—
283	Q₁₆	—	26	1	no	—
284	Q₁₆	—	26	1	no	—
285	SD₁₆	—	104	2	no	—
286	D₈	—	2	2	no	—
287	D₈	—	2	1	no	—
288	D₈	—	2	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	8	1	no	—
291	D₈	—	8	2	no	—
292	D₈	—	26	2	no	—
293	D₈	—	26	2	no	—
294	D₈	—	104	1	no	—
295	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	26	1	no	—
300	C₄⋊C₄	—	52	1	no	—
301	C₄⋊C₄	#6104	1	1	yes	characteristic
302	C₄⋊C₄	#6104	1	2	yes	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	52	2	no	—
306	C₂²⋊C₄	—	52	2	no	—
307	C₂²⋊C₄	—	52	2	no	—
308	C₄²	—	52	1	no	—
309	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
310	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
311	C₁₆	—	13	2	no	—
312	C₁₆	—	26	1	no	—
313	C₁₆	—	26	2	no	—
314	C₁₃	#33966	1	1	yes	characteristic, complemented, Sylow
315	C₂³	—	4	2	no	—
316	C₂³	—	4	1	no	—
317	C₂³	—	52	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	52	1	no	—
320	Q₈	—	52	1	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	1	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	1	no	—
325	D₄	—	8	4	no	—
326	D₄	—	8	2	no	—
327	D₄	—	16	2	no	—
328	D₄	—	16	1	no	—
329	D₄	—	52	2	no	—
330	D₄	—	52	2	no	—
331	D₄	—	104	2	no	—
332	D₄	—	208	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	8	4	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	52	1	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	208	2	no	—
339	C₂×C₄	C₂×#6104	1	2	yes	—
340	C₂×C₄	C₂×#6104	1	1	yes	characteristic
341	C₂×C₄	C₂×#6104	1	2	yes	—
342	C₂×C₄	C₂×#6104	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	2	no	—
346	C₈	#41×D₁₃	1	1	yes	characteristic
347	C₈	#88256	1	1	yes	characteristic
348	C₂²	—	8	2	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	1	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	104	2	no	—
355	C₂²	—	104	2	no	—
356	C₂²	—	104	2	no	—
357	C₂²	C₂²×#6104	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	2	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	104	1	no	—
363	C₄	—	104	1	no	—
364	C₄	#11718808	1	2	yes	—
365	C₄	#11719539	1	1	yes	characteristic
366	C₄	#11719540	1	1	yes	characteristic
367	C₂	—	16	2	no	—
368	C₂	—	16	1	no	—
369	C₂	—	208	2	no	—
370	C₂	#13127808	1	1	yes	characteristic
371	C₂	C₂×#11718808	1	1	yes	characteristic
372	C₂	#13127975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3391	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3391	D₁₃
hypocenter	C₁₃	#33966
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

Complements

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

N	H ≅ G/N	type
1. #13127849	C₂	semidirect
2. #13117100	C₂	semidirect
3. #13117085	C₂	semidirect
4. #13116891	C₂	semidirect
5. #13116873	C₂	semidirect
6. #13116871	C₂	semidirect
7. #13116868	C₂	semidirect
8. #13116745	C₂	semidirect
9. #13116710	C₂	semidirect
10. C₄×#88067	C₂	semidirect
11. C₁₃×#3391	C₂	semidirect
128. #6090	#41	semidirect
130. #3391	D₁₃	semidirect
314. C₁₃	#33966	semidirect

Maximal subgroups

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

subgroup	index
#13127849	2
#13117100	2
#13117085	2
#13116891	2
#13116873	2
#13116871	2
#13116868	2
#13116745	2
#13116710	2
C₄×#88067	2
C₁₃×#3391	2
#33966	13