Subgroups — #16020632

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127704	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117077	C₂	1	2	yes	maximal, complemented
3	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116993	C₂	1	2	yes	maximal, complemented
5	#13116983	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116868	C₂	1	2	yes	maximal, complemented
7	#13116868	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116710	C₂	1	2	yes	maximal, complemented
10	C₄×#88066	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3391	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#550	—	2	1	no	—
13	#11719284	—	2	1	no	—
14	C₁₃×#493	C₂²	1	1	yes	characteristic
15	C₁₃×#493	C₂²	1	2	yes	—
16	C₁₃×#459	—	2	1	no	—
17	C₁₃×#459	—	2	2	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×#88066	C₂²	1	1	yes	characteristic
22	C₂×#88066	C₂²	1	2	yes	—
23	#11718508	—	2	2	no	—
24	#11718503	—	2	1	no	—
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	#11718344	C₂²	1	2	yes	—
28	#11718343	C₂²	1	1	yes	characteristic
29	#11718154	C₂²	1	1	yes	characteristic
30	#11718154	C₂²	1	2	yes	—
31	#11718154	C₂²	1	2	yes	—
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718083	C₂²	1	1	yes	characteristic
35	#11718082	C₂²	1	2	yes	—
36	#11718081	C₂²	1	1	yes	characteristic
37	C₄⋊#6090	C₂²	1	1	yes	characteristic
38	C₄⋊#6090	C₂²	1	2	yes	—
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	1	yes	characteristic
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
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	#88200	—	4	1	no	—
54	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
55	#88150	—	4	2	no	—
56	C₄⋊C₄×D₁₃	—	4	1	no	—
57	C₄×D₅₂	—	2	1	no	—
58	C₄×D₅₂	—	2	2	no	—
59	#88124	—	2	2	no	—
60	#88123	—	2	1	no	—
61	C₁₃×#107	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₁₃×#102	—	2	2	no	—
65	C₁₃×#102	—	2	1	no	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88066	—	2	2	no	—
69	#88066	—	2	4	no	—
70	#88061	—	2	2	no	—
71	#88061	—	2	1	no	—
72	#88058	—	2	1	no	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88058	C₂³	1	2	yes	—
75	#88057	—	2	2	no	—
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	2	yes	—
79	#33933	—	13	1	no	maximal, Sylow
80	C₁₃×#41	—	8	4	no	—
81	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂×D₅₂	—	2	1	no	—
91	C₂×C₄×D₁₃	—	4	1	no	—
92	C₂×C₄×D₁₃	—	4	2	no	—
93	C₂×Dic₂₆	—	4	1	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	—	8	2	no	—
99	C₁₃×D₈	—	8	1	no	—
100	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	1	no	—
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	2	yes	—
110	#6103	—	4	1	no	—
111	#6103	—	4	2	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	1	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	#6101	—	4	2	no	—
116	#6101	—	4	2	no	—
117	D₁₀₄	—	2	1	no	—
118	D₁₀₄	#41	1	2	yes	complemented
119	#6090	—	2	1	no	—
120	#6090	—	2	2	no	—
121	#6090	—	2	1	no	—
122	#3391	D₁₃	1	1	yes	characteristic, complemented
123	#3380	—	13	1	no	—
124	#2244	—	13	2	no	—
125	#2240	—	13	1	no	—
126	#2228	—	13	2	no	—
127	#2226	—	13	1	no	—
128	#2206	—	13	2	no	—
129	#2206	—	13	1	no	—
130	#2183	—	13	1	no	—
131	#2177	—	13	2	no	—
132	C₄×D₁₆	—	13	1	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂³×C₁₃	—	4	1	no	—
135	C₂²×D₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	2	2	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	1	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	8	4	no	—
142	D₄×C₁₃	—	8	2	no	—
143	D₄×C₁₃	—	16	1	no	—
144	D₄×C₁₃	—	16	2	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	8	4	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×Dic₁₃	—	4	1	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	D₅₂	—	2	2	no	—
156	D₅₂	—	4	1	no	—
157	C₄×D₁₃	—	8	2	no	—
158	C₄×D₁₃	—	16	2	no	—
159	Dic₂₆	—	16	1	no	—
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
164	#554	—	26	1	no	—
165	#550	—	2	1	no	—
166	C₂×D₁₆	—	13	1	no	—
167	C₂×D₁₆	—	13	2	no	—
168	#500	—	13	1	no	—
169	#499	—	13	2	no	—
170	#493	C₂×D₁₃	1	1	yes	characteristic
171	#493	C₂×D₁₃	1	2	yes	—
172	#478	—	26	2	no	—
173	#476	—	26	1	no	—
174	#459	—	2	1	no	—
175	#459	—	2	2	no	—
176	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
177	C₄×D₈	—	13	1	no	—
178	C₄×D₈	—	13	2	no	—
179	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
180	C₄×D₈	C₂×D₁₃	1	2	yes	—
181	#367	—	13	2	no	—
182	#366	—	13	1	no	—
183	C₄⋊C₁₆	—	13	1	no	—
184	C₄⋊C₁₆	—	13	2	no	—
185	#357	—	13	1	no	—
186	#357	—	13	2	no	—
187	#357	—	13	2	no	—
188	#357	—	13	2	no	—
189	#357	—	13	2	no	—
190	#357	—	13	1	no	—
191	#346	—	13	2	no	—
192	C₄×C₁₆	—	13	1	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	1	no	—
197	C₂²×C₁₃	—	8	2	no	—
198	C₂²×C₁₃	—	8	1	no	—
199	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
200	C₂×D₁₃	—	4	2	no	—
201	C₂×D₁₃	—	8	1	no	—
202	C₂×D₁₃	—	8	1	no	—
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	2	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	#509	1	2	yes	—
208	C₄×C₁₃	#548	1	1	yes	characteristic
209	C₄×C₁₃	#577	1	1	yes	characteristic
210	Dic₁₃	—	8	1	no	—
211	Dic₁₃	—	8	2	no	—
212	C₂×#41	—	4	1	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	13	1	no	—
216	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₂×D₈	C₂²×D₁₃	1	2	yes	—
218	C₄⋊Q₈	—	26	1	no	—
219	#127	—	2	2	no	—
220	#127	—	2	1	no	—
221	#125	—	26	2	no	—
222	C₂²⋊Q₈	—	52	1	no	—
223	C₂²⋊Q₈	—	52	2	no	—
224	#121	—	4	1	no	—
225	#121	—	4	2	no	—
226	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	26	1	no	—
230	C₄×D₄	—	26	2	no	—
231	C₂×C₄⋊C₄	—	52	1	no	—
232	D₁₆	—	26	2	no	—
233	D₁₆	—	26	4	no	—
234	C₂×C₁₆	—	13	1	no	—
235	C₂×C₁₆	—	13	1	no	—
236	C₂×C₁₆	—	13	2	no	—
237	#107	—	13	1	no	—
238	#107	—	13	2	no	—
239	#107	—	26	1	no	—
240	#107	C₂²×D₁₃	1	1	yes	characteristic
241	#106	—	26	2	no	—
242	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
244	#102	—	2	2	no	—
245	#102	—	2	1	no	—
246	#102	—	26	2	no	—
247	#102	—	26	1	no	—
248	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₁₃	1	2	yes	—
250	C₂×C₁₃	—	16	2	no	—
251	C₂×C₁₃	—	16	1	no	—
252	C₂×C₁₃	#3358	1	1	yes	characteristic
253	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
254	C₂×C₁₃	#3415	1	1	yes	characteristic
255	D₁₃	—	8	2	no	—
256	#41	—	8	4	no	—
257	C₂×Q₈	—	52	1	no	—
258	C₂×Q₈	#6104	1	1	yes	characteristic
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	26	1	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	52	1	no	—
269	C₂²×C₄	—	52	2	no	—
270	D₈	—	2	1	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	2	no	—
274	D₈	—	8	2	no	—
275	D₈	—	8	1	no	—
276	D₈	—	13	2	no	—
277	D₈	—	26	1	no	—
278	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
279	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	26	1	no	—
283	C₄⋊C₄	—	26	2	no	—
284	C₄⋊C₄	—	52	2	no	—
285	C₄⋊C₄	—	52	1	no	—
286	C₄⋊C₄	—	52	2	no	—
287	C₄⋊C₄	#6104	1	1	yes	characteristic
288	C₄⋊C₄	#6104	1	2	yes	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	52	1	no	—
292	C₂²⋊C₄	—	52	2	no	—
293	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
294	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
295	C₁₆	—	26	1	no	—
296	C₁₆	—	26	2	no	—
297	C₁₆	—	26	1	no	—
298	C₁₃	#33933	1	1	yes	characteristic, complemented, Sylow
299	C₂³	—	4	2	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	52	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	208	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	16	2	no	—
311	D₄	—	16	1	no	—
312	D₄	—	26	2	no	—
313	D₄	—	52	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	52	1	no	—
318	C₂×C₄	—	52	2	no	—
319	C₂×C₄	—	104	2	no	—
320	C₂×C₄	—	208	2	no	—
321	C₂×C₄	C₂×#6104	1	1	yes	characteristic
322	C₂×C₄	C₂×#6104	1	2	yes	—
323	C₂×C₄	C₂×#6104	1	2	yes	—
324	C₂×C₄	C₂×#6104	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	#41×D₁₃	1	1	yes	characteristic
329	C₈	#88257	1	1	yes	characteristic
330	C₂²	—	8	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	52	2	no	—
337	C₂²	—	104	1	no	—
338	C₂²	—	104	1	no	—
339	C₂²	C₂²×#6104	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	104	1	no	—
345	C₄	—	104	2	no	—
346	C₄	#11718808	1	2	yes	—
347	C₄	#11719536	1	1	yes	characteristic
348	C₄	#11719541	1	1	yes	characteristic
349	C₂	—	16	2	no	—
350	C₂	—	16	1	no	—
351	C₂	—	104	2	no	—
352	C₂	#13127467	1	1	yes	characteristic
353	C₂	C₂×#11718808	1	1	yes	characteristic
354	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	#33933
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. #13127704	C₂	semidirect
2. #13117077	C₂	semidirect
3. #13117059	C₂	semidirect
4. #13116993	C₂	semidirect
5. #13116983	C₂	semidirect
6. #13116868	C₂	semidirect
7. #13116868	C₂	semidirect
8. #13116745	C₂	semidirect
9. #13116710	C₂	semidirect
10. C₄×#88066	C₂	semidirect
11. C₁₃×#3391	C₂	semidirect
118. D₁₀₄	#41	semidirect
122. #3391	D₁₃	semidirect
298. C₁₃	#33933	semidirect

Maximal subgroups

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

subgroup	index
#13127704	2
#13117077	2
#13117059	2
#13116993	2
#13116983	2
#13116868	2
#13116745	2
#13116710	2
C₄×#88066	2
C₁₃×#3391	2
#33933	13