Subgroups — #15854497

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127826	C₂	1	1	yes	maximal, characteristic
2	#13127688	C₂	1	1	yes	maximal, characteristic
3	#13127118	C₂	1	2	yes	maximal, complemented
4	#13114380	C₂	1	2	yes	maximal, complemented
5	#13114227	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114203	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114103	C₂	1	2	yes	maximal, complemented
8	#13114096	C₂	1	2	yes	maximal, complemented
9	#13113722	C₂	1	1	yes	maximal, characteristic
10	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719437	C₂²	1	2	yes	—
13	#11719432	C₂²	1	2	yes	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#498	C₂²	1	2	yes	—
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₁₃×#487	C₂²	1	2	yes	—
18	#11718852	—	2	1	no	—
19	#11718846	—	2	2	no	—
20	Q₁₆×Dic₁₃	—	2	1	no	—
21	#11718755	C₂²	1	2	yes	—
22	#11718751	C₂²	1	2	yes	—
23	#11718748	C₂²	1	2	yes	—
24	#11718744	C₂²	1	2	yes	—
25	#11718551	—	2	2	no	—
26	#11718535	—	2	1	no	—
27	#11718533	—	2	1	no	—
28	#11718477	—	2	2	no	—
29	#11718475	—	2	2	no	—
30	#11718469	—	2	2	no	—
31	#11718463	—	2	2	no	—
32	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
33	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
34	#11718349	C₂²	1	2	yes	—
35	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
37	C₈²×C₁₃	C₂²	1	1	yes	characteristic
38	#11718139	C₂²	1	2	yes	—
39	#11718139	C₂²	1	2	yes	—
40	#11718051	C₂²	1	2	yes	—
41	#11718050	C₂²	1	1	yes	characteristic
42	#11718050	C₂²	1	1	yes	characteristic
43	#11718048	C₂²	1	1	yes	characteristic
44	#11718048	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₁₃	—	2	2	no	—
46	C₂×C₁₃×Q₁₆	—	2	1	no	—
47	C₂×C₁₃×Q₁₆	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
50	#88201	—	2	2	no	—
51	#88200	—	2	2	no	—
52	Q₈×Dic₁₃	—	2	2	no	—
53	C₂×#6107	—	2	2	no	—
54	C₂×#6106	—	2	2	no	—
55	C₂×Dic₅₂	D₄	1	1	yes	characteristic
56	C₂×D₁₀₄	D₄	1	1	yes	characteristic
57	C₂×#6094	—	2	2	no	—
58	#88152	—	4	2	no	—
59	#88151	—	2	2	no	—
60	#88150	—	4	2	no	—
61	#88147	—	2	2	no	—
62	#88146	—	2	2	no	—
63	#88142	—	2	2	no	—
64	C₄×D₅₂	C₂³	1	2	yes	—
65	C₄×Dic₂₆	C₂³	1	1	yes	characteristic
66	C₄×Dic₂₆	C₂³	1	1	yes	characteristic
67	C₁₃×#107	—	2	1	no	—
68	C₁₃×#107	—	2	1	no	—
69	C₁₃×#106	—	4	2	no	—
70	C₁₃×#103	—	2	2	no	—
71	C₁₃×#103	—	2	2	no	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#88075	—	2	2	no	—
76	#88075	—	2	2	no	—
77	#88061	D₄	1	2	yes	—
78	#88060	—	2	2	no	—
79	#88058	D₄	1	1	yes	characteristic
80	#88058	D₄	1	1	yes	characteristic
81	#88056	D₄	1	1	yes	characteristic
82	#88056	D₄	1	1	yes	characteristic
83	#88054	—	2	1	no	—
84	#88054	—	2	1	no	—
85	C₈×Dic₁₃	—	2	1	no	—
86	C₈×Dic₁₃	—	2	1	no	—
87	#88050	—	2	2	no	—
88	#88049	—	2	2	no	—
89	#88048	—	2	2	no	—
90	#88047	—	2	2	no	—
91	C₄⋊#1055	C₂³	1	1	yes	characteristic
92	C₄⋊#1055	C₂³	1	1	yes	characteristic
93	C₄⋊#1055	C₂³	1	2	yes	—
94	#19907	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	—	2	2	no	—
96	C₂×Q₈×C₁₃	—	2	2	no	—
97	Q₈×D₁₃	—	4	4	no	—
98	Q₈×D₁₃	—	8	2	no	—
99	C₂×D₅₂	C₂×D₄	1	2	yes	—
100	C₂×C₄×D₁₃	—	2	2	no	—
101	C₂×C₄×D₁₃	—	4	2	no	—
102	C₂×Dic₂₆	—	4	2	no	—
103	C₂×Dic₂₆	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₆	C₂×D₄	1	1	yes	characteristic
105	C₁₃×Q₁₆	—	4	2	no	—
106	C₁₃×Q₁₆	—	8	1	no	—
107	C₂×C₈×C₁₃	—	2	1	no	—
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
117	#6107	—	8	2	no	—
118	#6106	—	8	2	no	—
119	#6103	—	2	2	no	—
120	#6103	—	4	2	no	—
121	#6103	—	4	2	no	—
122	#6103	—	4	2	no	—
123	C₄⋊Dic₁₃	—	4	2	no	—
124	C₄⋊Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
126	C₄⋊Dic₁₃	C₂×D₄	1	1	yes	characteristic
127	#6101	—	2	1	no	—
128	#6101	—	2	1	no	—
129	#6101	—	4	2	no	—
130	C₄×Dic₁₃	—	2	1	no	—
131	C₄×Dic₁₃	—	2	1	no	—
132	C₄×Dic₁₃	—	4	2	no	—
133	C₂×#1055	—	2	1	no	—
134	C₂×#1055	—	2	2	no	—
135	C₂×#1055	—	2	1	no	—
136	Dic₅₂	—	2	2	no	—
137	D₁₀₄	—	2	2	no	—
138	#6094	—	8	2	no	—
139	#3393	—	13	1	no	—
140	Q₈×D₈	—	13	1	no	—
141	#3328	—	13	2	no	—
142	#1707	D₁₃	1	1	yes	characteristic, complemented
143	#1699	—	13	2	no	—
144	#1674	—	13	1	no	—
145	#1670	—	13	1	no	—
146	#1652	—	13	2	no	—
147	#1651	—	13	2	no	—
148	#1589	—	13	1	no	—
149	C₈×Q₁₆	—	13	1	no	—
150	C₂²×D₁₃	—	2	2	no	—
151	Q₈×C₁₃	—	4	2	no	—
152	Q₈×C₁₃	—	4	2	no	—
153	Q₈×C₁₃	—	4	2	no	—
154	Q₈×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×Dic₁₃	—	2	1	no	—
161	C₂×Dic₁₃	—	2	1	no	—
162	C₂×Dic₁₃	—	2	1	no	—
163	C₂×Dic₁₃	—	2	1	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	D₅₂	—	2	2	no	—
166	D₅₂	—	2	2	no	—
167	C₄×D₁₃	—	4	2	no	—
168	C₄×D₁₃	—	4	2	no	—
169	C₄×D₁₃	—	8	4	no	—
170	Dic₂₆	—	2	1	no	—
171	Dic₂₆	—	2	1	no	—
172	Dic₂₆	—	2	1	no	—
173	Dic₂₆	—	8	4	no	—
174	Dic₂₆	#135	1	2	yes	—
175	C₈×C₁₃	—	2	2	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	—	2	1	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	#137	1	2	yes	—
180	#1055	—	4	2	no	—
181	#1055	—	4	1	no	—
182	#1055	—	4	1	no	—
183	#556	—	13	2	no	—
184	D₄×Q₈	—	13	2	no	—
185	#500	—	26	1	no	—
186	#500	C₂×D₁₃	1	1	yes	characteristic
187	#498	C₂×D₁₃	1	2	yes	—
188	#495	—	26	1	no	—
189	#494	C₂×D₁₃	1	1	yes	characteristic
190	#487	C₂×D₁₃	1	2	yes	—
191	#485	—	26	2	no	—
192	#475	—	13	2	no	—
193	#474	—	13	2	no	—
194	#474	—	26	2	no	—
195	#470	—	26	2	no	—
196	#464	—	26	2	no	—
197	#462	—	13	2	no	—
198	#461	—	13	2	no	—
199	#451	—	26	2	no	—
200	C₈×Q₈	—	13	1	no	—
201	C₈×Q₈	—	13	1	no	—
202	C₄×Q₁₆	—	13	1	no	—
203	C₄×Q₁₆	—	26	1	no	—
204	C₄×D₈	—	13	1	no	—
205	C₄×D₈	—	26	1	no	—
206	#436	—	13	2	no	—
207	#419	—	26	2	no	—
208	#335	—	13	1	no	—
209	#335	—	13	1	no	—
210	#332	—	13	2	no	—
211	#332	—	13	2	no	—
212	Q₈⋊C₈	—	13	1	no	—
213	Q₈⋊C₈	—	13	1	no	—
214	D₄⋊C₈	—	13	2	no	—
215	C₈²	C₂×D₁₃	1	1	yes	characteristic
216	C₂²×C₁₃	D₄²	1	1	yes	characteristic
217	C₂×D₁₃	—	4	2	no	—
218	C₂×D₁₃	—	4	2	no	—
219	C₂×D₁₃	—	4	2	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	8	2	no	—
222	C₄×C₁₃	—	8	2	no	—
223	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
224	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
225	C₄×C₁₃	#576	1	1	yes	characteristic
226	C₄×C₁₃	#577	1	1	yes	characteristic
227	Dic₁₃	—	2	2	no	—
228	Dic₁₃	—	4	1	no	—
229	Dic₁₃	—	4	2	no	—
230	Dic₁₃	—	4	1	no	—
231	Dic₁₃	—	4	1	no	—
232	C₂²×Q₈	—	26	2	no	—
233	C₂×Q₁₆	—	2	1	no	—
234	C₂×Q₁₆	—	2	1	no	—
235	C₂×Q₁₆	—	13	1	no	—
236	C₂×Q₁₆	—	26	2	no	—
237	C₂×SD₁₆	—	26	2	no	—
238	C₂×D₈	—	13	1	no	—
239	C₂×M₁₆	—	26	2	no	—
240	C₄⋊Q₈	—	26	2	no	—
241	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
242	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
243	#126	—	52	2	no	—
244	#124	—	26	2	no	—
245	C₂²⋊Q₈	—	26	2	no	—
246	C₂²⋊Q₈	—	26	2	no	—
247	C₂²⋊Q₈	—	52	2	no	—
248	C₄×Q₈	—	13	1	no	—
249	C₄×Q₈	—	13	1	no	—
250	C₄×Q₈	—	26	2	no	—
251	C₄×D₄	—	13	2	no	—
252	C₄×D₄	—	26	2	no	—
253	#117	—	26	2	no	—
254	#107	—	2	1	no	—
255	#107	—	2	1	no	—
256	#107	—	13	1	no	—
257	#107	—	13	1	no	—
258	#107	—	26	2	no	—
259	#106	—	4	2	no	—
260	#106	—	26	2	no	—
261	C₄⋊C₈	—	13	2	no	—
262	C₄⋊C₈	—	13	1	no	—
263	C₄⋊C₈	—	13	1	no	—
264	C₄⋊C₈	—	26	1	no	—
265	C₄⋊C₈	—	26	1	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	13	1	no	—
269	#103	—	13	1	no	—
270	#103	—	26	2	no	—
271	#103	—	26	2	no	—
272	#103	—	26	2	no	—
273	#102	—	13	2	no	—
274	#102	—	26	2	no	—
275	C₂²⋊C₈	—	26	2	no	—
276	C₄×C₈	—	26	1	no	—
277	C₄×C₈	—	26	1	no	—
278	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₂×C₁₃	#3278	1	1	yes	characteristic
282	C₂×C₁₃	#3280	1	1	yes	characteristic
283	C₂×C₁₃	#3293	1	1	yes	characteristic
284	D₁₃	—	8	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	13	1	no	—
288	C₂×Q₈	—	13	1	no	—
289	C₂×Q₈	—	52	2	no	—
290	C₂×Q₈	—	52	4	no	—
291	C₂×Q₈	—	104	2	no	—
292	C₂×D₄	—	13	2	no	—
293	C₂²×C₄	—	26	2	no	—
294	C₂²×C₄	—	52	2	no	—
295	Q₁₆	—	4	2	no	—
296	Q₁₆	—	8	1	no	—
297	Q₁₆	—	26	2	no	—
298	Q₁₆	—	104	2	no	—
299	SD₁₆	—	104	2	no	—
300	D₈	—	26	2	no	—
301	M₁₆	—	104	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	26	1	no	—
304	C₂×C₈	—	26	1	no	—
305	C₂×C₈	—	26	2	no	—
306	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
307	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
309	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	13	2	no	—
315	C₄⋊C₄	—	13	1	no	—
316	C₄⋊C₄	—	13	1	no	—
317	C₄⋊C₄	—	26	1	no	—
318	C₄⋊C₄	—	26	1	no	—
319	C₄⋊C₄	—	52	2	no	—
320	C₄⋊C₄	—	52	2	no	—
321	C₂²⋊C₄	—	26	2	no	—
322	C₂²⋊C₄	—	52	2	no	—
323	C₂²⋊C₄	—	52	2	no	—
324	C₂²⋊C₄	—	52	2	no	—
325	C₄²	—	26	1	no	—
326	C₄²	—	26	1	no	—
327	C₄²	—	52	2	no	—
328	C₄²	C₂³×D₁₃	1	1	yes	characteristic
329	C₁₃	#19907	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	26	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	13	2	no	—
336	Q₈	—	26	1	no	—
337	Q₈	—	26	1	no	—
338	Q₈	—	26	1	no	—
339	Q₈	—	104	4	no	—
340	D₄	—	26	2	no	—
341	D₄	—	26	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	26	1	no	—
345	C₂×C₄	—	26	1	no	—
346	C₂×C₄	—	26	1	no	—
347	C₂×C₄	—	26	1	no	—
348	C₂×C₄	—	26	2	no	—
349	C₂×C₄	—	52	2	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	104	4	no	—
352	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
354	C₂×C₄	#88251	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	52	1	no	—
360	C₈	—	52	1	no	—
361	C₈	—	52	2	no	—
362	C₈	#88169	1	2	yes	—
363	C₂²	—	52	2	no	—
364	C₂²	—	52	2	no	—
365	C₂²	—	52	2	no	—
366	C₂²	#11719422	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	26	2	no	—
371	C₄	—	52	2	no	—
372	C₄	—	52	1	no	—
373	C₄	—	52	1	no	—
374	C₄	—	52	1	no	—
375	C₄	C₂×#88169	1	1	yes	characteristic
376	C₄	#11719467	1	1	yes	characteristic
377	C₄	D₁₃×#135	1	1	yes	characteristic
378	C₄	#11719474	1	1	yes	characteristic
379	C₂	—	104	2	no	—
380	C₂	#13126501	1	1	yes	characteristic
381	C₂	#13126520	1	1	yes	characteristic
382	C₂	#13126640	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11719422
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1707	C₂
socle	C₂²×C₁₃	D₄²
prime radical	C₂×D₁₀₄	D₄
hypercenter	#1707	D₁₃
hypocenter	C₁₃	#19907
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
3. #13127118	C₂	semidirect
4. #13114380	C₂	semidirect
5. #13114227	C₂	semidirect
6. #13114203	C₂	semidirect
7. #13114103	C₂	semidirect
8. #13114096	C₂	semidirect
10. C₈×Dic₅₂	C₂	semidirect
11. C₁₃×#1707	C₂	semidirect
142. #1707	D₁₃	semidirect
329. C₁₃	#19907	semidirect

Maximal subgroups

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

subgroup	index
#13127826	2
#13127688	2
#13127118	2
#13114380	2
#13114227	2
#13114203	2
#13114103	2
#13114096	2
#13113722	2
C₈×Dic₅₂	2
C₁₃×#1707	2
#19907	13