Subgroups — #16019862

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13126596	C₂	1	1	yes	maximal, characteristic
2	#13117061	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117027	C₂	1	2	yes	maximal, complemented
4	#13116975	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116934	C₂	1	2	yes	maximal, complemented
6	#13116876	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116774	C₂	1	2	yes	maximal, complemented
8	C₄×#88069	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116541	C₂	1	2	yes	maximal, complemented
11	C₁₃×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#556	—	2	1	no	—
13	C₂×#88158	C₂²	1	2	yes	—
14	#11719258	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#485	—	2	2	no	—
17	C₁₃×#475	—	2	1	no	—
18	C₁₃×#467	C₂²	1	2	yes	—
19	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#427	C₂²	1	2	yes	—
22	C₂×#88069	C₂²	1	1	yes	characteristic
23	C₂×#88068	C₂²	1	2	yes	—
24	#11718776	C₂²	1	2	yes	—
25	C₂²×#6090	C₂²	1	2	yes	—
26	#11718509	—	2	1	no	—
27	#11718395	—	2	2	no	—
28	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
29	#11718356	C₂²	1	1	yes	characteristic
30	#11718156	C₂²	1	1	yes	characteristic
31	C₂²⋊#6090	C₂²	1	2	yes	—
32	#11718084	C₂²	1	1	yes	characteristic
33	#11718084	C₂²	1	2	yes	—
34	#11718083	C₂²	1	2	yes	—
35	#11718083	C₂²	1	2	yes	—
36	#11718083	C₂²	1	2	yes	—
37	#11718082	C₂²	1	2	yes	—
38	#11718081	C₂²	1	1	yes	characteristic
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6127	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×#126	—	4	2	no	—
46	C₁₃×#124	—	4	1	no	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	1	no	—
49	C₄×Q₈×C₁₃	—	4	1	no	—
50	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×#117	—	2	2	no	—
52	#88194	—	4	1	no	—
53	#88184	—	2	2	no	—
54	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
55	#88158	—	2	4	no	—
56	#88158	—	4	2	no	—
57	C₂×D₁₀₄	—	2	1	no	—
58	C₂×D₁₀₄	C₂³	1	2	yes	—
59	C₂×#6095	—	2	2	no	—
60	#88147	—	4	1	no	—
61	#88138	—	4	2	no	—
62	#88128	—	2	1	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#107	C₂³	1	1	yes	characteristic
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#106	—	2	2	no	—
68	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
69	C₁₃×#103	—	2	1	no	—
70	C₁₃×#103	—	2	2	no	—
71	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88069	—	2	2	no	—
74	#88068	—	4	2	no	—
75	#88061	—	2	2	no	—
76	#88058	C₂³	1	1	yes	characteristic
77	#88056	—	2	1	no	—
78	C₂×#6090	—	4	2	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	#33871	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	4	1	no	—
85	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₂²⋊D₁₃	—	4	2	no	—
88	C₂×C₂²⋊D₁₃	—	4	2	no	—
89	#6127	—	4	4	no	—
90	#6127	—	8	2	no	—
91	C₂×D₅₂	—	2	2	no	—
92	C₂×D₅₂	—	4	1	no	—
93	C₂×D₅₂	—	4	1	no	—
94	C₂×C₄×D₁₃	—	4	2	no	—
95	C₂×Dic₂₆	—	2	1	no	—
96	C₁₃×Q₁₆	—	2	1	no	—
97	C₁₃×Q₁₆	—	2	1	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
112	C₄²×C₁₃	—	4	2	no	—
113	C₄²×C₁₃	—	4	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	#6103	—	4	2	no	—
116	C₄⋊Dic₁₃	—	2	1	no	—
117	#6101	—	4	1	no	—
118	C₄×Dic₁₃	—	4	1	no	—
119	Dic₅₂	—	2	1	no	—
120	Dic₅₂	#41	1	2	yes	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	—	2	2	no	—
123	D₁₀₄	—	8	1	no	—
124	#6095	—	4	4	no	—
125	#6090	—	2	1	no	—
126	#6090	—	2	2	no	—
127	#6090	—	2	1	no	—
128	#3342	D₁₃	1	1	yes	characteristic, complemented
129	#3287	—	13	1	no	—
130	#2240	—	13	1	no	—
131	#2234	—	13	2	no	—
132	#2225	—	13	1	no	—
133	#2217	—	13	2	no	—
134	#2207	—	13	1	no	—
135	#2189	—	13	2	no	—
136	C₄×Q₃₂	—	13	1	no	—
137	D₄×C₁₆	—	13	1	no	—
138	#2138	—	13	2	no	—
139	C₂³×C₁₃	C₂×D₈	1	2	yes	—
140	C₂²×D₁₃	—	4	2	no	—
141	Q₈×C₁₃	—	4	1	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	Q₈×C₁₃	—	8	2	no	—
144	D₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₁₃	—	8	4	no	—
155	C₂²⋊D₁₃	—	16	2	no	—
156	C₂×Dic₁₃	—	4	1	no	—
157	C₂×Dic₁₃	—	4	1	no	—
158	D₅₂	—	4	2	no	—
159	D₅₂	—	4	2	no	—
160	D₅₂	—	8	2	no	—
161	D₅₂	—	16	1	no	—
162	C₄×D₁₃	—	8	4	no	—
163	Dic₂₆	—	2	2	no	—
164	Dic₂₆	—	4	1	no	—
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×#135	—	13	2	no	—
170	#556	—	2	1	no	—
171	#550	—	26	1	no	—
172	C₂×Q₃₂	—	13	1	no	—
173	C₂×SD₃₂	—	13	2	no	—
174	C₂²×C₁₆	—	13	2	no	—
175	#500	C₂×D₁₃	1	1	yes	characteristic
176	#493	—	13	1	no	—
177	#485	—	2	2	no	—
178	#475	—	2	1	no	—
179	#467	C₂×D₁₃	1	2	yes	—
180	#466	—	13	2	no	—
181	#460	—	26	1	no	—
182	#449	—	26	2	no	—
183	C₄×Q₁₆	—	13	1	no	—
184	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
185	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
186	#427	C₂×D₁₃	1	2	yes	—
187	#367	—	13	2	no	—
188	#366	—	13	1	no	—
189	C₄⋊C₁₆	—	13	1	no	—
190	#358	—	13	1	no	—
191	#358	—	13	1	no	—
192	#358	—	13	2	no	—
193	#357	—	13	2	no	—
194	#357	—	13	2	no	—
195	#357	—	13	2	no	—
196	C₂²⋊C₁₆	—	13	2	no	—
197	C₄×C₁₆	—	13	1	no	—
198	C₂²×C₁₃	—	2	2	no	—
199	C₂²×C₁₃	—	2	2	no	—
200	C₂²×C₁₃	—	2	2	no	—
201	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
202	C₂×D₁₃	—	8	2	no	—
203	C₂×D₁₃	—	8	2	no	—
204	C₂×D₁₃	—	8	2	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	2	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	—	8	1	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	#508	1	2	yes	—
211	C₄×C₁₃	#546	1	1	yes	characteristic
212	C₄×C₁₃	#577	1	1	yes	characteristic
213	Dic₁₃	—	4	2	no	—
214	Dic₁₃	—	8	1	no	—
215	C₂×#41	—	26	2	no	—
216	#135	—	26	4	no	—
217	#135	—	52	2	no	—
218	C₂×Q₁₆	—	13	1	no	—
219	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×SD₁₆	—	26	2	no	—
221	C₂×D₈	—	13	2	no	—
222	C₂×D₈	—	26	1	no	—
223	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
224	C₄⋊Q₈	—	2	1	no	—
225	#127	—	26	1	no	—
226	#127	—	52	1	no	—
227	#126	—	4	2	no	—
228	#124	—	4	1	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	#121	—	26	2	no	—
231	#121	—	52	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	4	1	no	—
234	C₄×Q₈	—	26	1	no	—
235	C₄×D₄	—	52	1	no	—
236	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
237	#117	—	2	2	no	—
238	Q₃₂	—	26	2	no	—
239	SD₃₂	—	52	2	no	—
240	C₂×C₁₆	—	13	1	no	—
241	C₂×C₁₆	—	13	1	no	—
242	C₂×C₁₆	—	13	2	no	—
243	C₂×C₁₆	—	52	2	no	—
244	#107	—	2	1	no	—
245	#107	—	13	1	no	—
246	#107	C₂²×D₁₃	1	1	yes	characteristic
247	#107	C₂²×D₁₃	1	2	yes	—
248	#106	—	2	2	no	—
249	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
250	#103	—	2	1	no	—
251	#103	—	2	2	no	—
252	#103	—	26	1	no	—
253	#102	—	26	2	no	—
254	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
255	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
256	C₂×C₁₃	—	4	2	no	—
257	C₂×C₁₃	#3294	1	1	yes	characteristic
258	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
259	C₂×C₁₃	#3416	1	1	yes	characteristic
260	D₁₃	—	16	2	no	—
261	#41	—	52	4	no	—
262	#41	—	104	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	26	1	no	—
266	C₂×D₄	—	26	2	no	—
267	C₂×D₄	—	52	2	no	—
268	C₂×D₄	—	52	2	no	—
269	C₂×D₄	—	52	1	no	—
270	C₂×D₄	—	52	1	no	—
271	C₂×D₄	#6104	1	1	yes	characteristic
272	C₂²×C₄	—	52	2	no	—
273	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	13	2	no	—
277	Q₁₆	—	26	1	no	—
278	SD₁₆	—	52	4	no	—
279	D₈	—	26	2	no	—
280	D₈	—	26	2	no	—
281	D₈	—	104	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
285	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
286	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	26	1	no	—
293	C₄⋊C₄	—	52	1	no	—
294	C₄⋊C₄	#6104	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	52	2	no	—
298	C₂²⋊C₄	#6104	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	52	1	no	—
302	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
303	C₁₆	—	26	1	no	—
304	C₁₆	—	26	1	no	—
305	C₁₆	—	26	2	no	—
306	C₁₃	#33871	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	52	2	no	—
308	C₂³	C₂×#6104	1	2	yes	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	Q₈	—	26	2	no	—
313	Q₈	—	52	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	52	2	no	—
316	D₄	—	52	2	no	—
317	D₄	—	104	4	no	—
318	D₄	—	104	2	no	—
319	D₄	—	208	1	no	—
320	D₄	—	208	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	52	1	no	—
327	C₂×C₄	—	52	1	no	—
328	C₂×C₄	—	104	4	no	—
329	C₂×C₄	C₂×#6104	1	2	yes	—
330	C₂×C₄	C₂×#6104	1	1	yes	characteristic
331	C₂×C₄	C₂×#6104	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	#41×D₁₃	1	1	yes	characteristic
336	C₈	#88256	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	104	2	no	—
341	C₂²	—	104	2	no	—
342	C₂²	—	104	2	no	—
343	C₂²	C₂²×#6104	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	52	2	no	—
350	C₄	—	104	1	no	—
351	C₄	#11718855	1	2	yes	—
352	C₄	#11719534	1	1	yes	characteristic
353	C₄	#11719542	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	208	2	no	—
356	C₂	#13126643	1	1	yes	characteristic
357	C₂	C₂×#11718855	1	1	yes	characteristic
358	C₂	#13127980	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3342	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13126596	C₂
hypercenter	#3342	D₁₃
hypocenter	C₁₃	#33871
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #13117061	C₂	semidirect
3. #13117027	C₂	semidirect
4. #13116975	C₂	semidirect
5. #13116934	C₂	semidirect
6. #13116876	C₂	semidirect
7. #13116774	C₂	semidirect
8. C₄×#88069	C₂	semidirect
9. D₄×#6090	C₂	semidirect
10. #13116541	C₂	semidirect
11. C₁₃×#3342	C₂	semidirect
128. #3342	D₁₃	semidirect
306. C₁₃	#33871	semidirect

Maximal subgroups

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

subgroup	index
#13126596	2
#13117061	2
#13117027	2
#13116975	2
#13116934	2
#13116876	2
#13116774	2
C₄×#88069	2
D₄×#6090	2
#13116541	2
C₁₃×#3342	2
#33871	13