Subgroups — #16018812

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117085	C₂	1	2	yes	maximal, complemented
3	#13117061	C₂	1	2	yes	maximal, complemented
4	#13116871	C₂	1	4	yes	maximal, complemented
5	#13116747	C₂	1	4	yes	maximal, complemented
6	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#532	—	2	1	no	—
9	C₂²×D₁₀₄	—	2	1	no	—
10	C₂²×D₁₀₄	C₂²	1	2	yes	—
11	C₂×#88128	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
13	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#486	—	2	1	no	—
15	C₁₃×#466	—	2	1	no	—
16	C₁₃×#466	C₂²	1	2	yes	—
17	C₄×C₁₃×D₈	—	2	1	no	—
18	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88066	C₂²	1	8	yes	complemented
21	#11718356	—	2	1	no	—
22	#11718356	—	2	1	no	—
23	#11718356	C₂²	1	2	yes	—
24	#11718356	C₂²	1	2	yes	—
25	C₂²⋊#6090	C₂²	1	2	yes	—
26	#11718083	C₂²	1	8	yes	complemented
27	C₄⋊#6090	C₂²	1	2	yes	complemented
28	C₄×#6090	C₂²	1	2	yes	complemented
29	C₂×C₁₃×#41	—	4	1	no	—
30	C₂²×D₅₂	—	2	2	no	—
31	C₂²×D₅₂	—	4	1	no	—
32	C₂²×D₅₂	—	4	1	no	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂×C₁₃×D₈	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×#127	—	2	1	no	—
39	C₁₃×#124	—	4	1	no	—
40	C₁₃×#121	—	2	2	no	—
41	C₁₃×#121	—	4	1	no	—
42	C₄×D₄×C₁₃	—	2	2	no	—
43	C₄×D₄×C₁₃	—	4	1	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×D₁₀₄	—	2	2	no	—
46	C₂×D₁₀₄	—	2	2	no	—
47	C₂×D₁₀₄	—	2	2	no	—
48	C₂×D₁₀₄	—	4	4	no	—
49	C₂×D₁₀₄	—	4	4	no	—
50	C₂×D₁₀₄	C₂³	1	4	yes	—
51	#88128	—	2	2	no	—
52	#88128	—	2	2	no	—
53	#88128	—	4	1	no	—
54	#88128	—	4	1	no	—
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	C₂³	1	2	yes	—
57	C₁₃×#102	—	2	2	no	—
58	C₁₃×#102	—	2	2	no	—
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	#88066	—	4	8	no	—
64	C₂×#6090	C₂³	1	4	yes	—
65	#33771	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	—	4	2	no	—
67	C₁₃×#41	—	8	4	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×D₅₂	—	2	4	no	—
78	C₂×D₅₂	—	4	4	no	—
79	C₂×D₅₂	—	4	4	no	—
80	C₂×D₅₂	—	4	2	no	—
81	C₂×D₅₂	—	4	2	no	—
82	C₂×D₅₂	—	4	2	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	8	4	no	—
86	C₂×D₅₂	—	8	4	no	—
87	C₁₃×D₈	—	2	4	no	—
88	C₁₃×D₈	—	4	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₄²×C₁₃	D₈	1	1	yes	characteristic
100	C₄²×C₁₃	D₈	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	D₁₀₄	—	2	8	no	—
104	D₁₀₄	—	4	4	no	—
105	D₁₀₄	—	4	4	no	—
106	D₁₀₄	—	4	4	no	—
107	#6090	—	2	4	no	—
108	#3169	D₁₃	1	1	yes	characteristic, complemented
109	C₂×#493	—	13	1	no	—
110	#2246	—	13	2	no	—
111	#2240	—	13	2	no	—
112	#2206	—	13	4	no	—
113	#2184	—	13	4	no	—
114	#2162	—	13	1	no	—
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₁₃	—	4	4	no	—
118	C₂²×D₁₃	—	8	4	no	—
119	C₂²×D₁₃	—	8	4	no	—
120	C₂²×D₁₃	—	8	4	no	—
121	Q₈×C₁₃	—	8	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	8	4	no	—
125	D₄×C₁₃	—	8	2	no	—
126	D₄×C₁₃	—	8	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	8	4	no	—
132	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	D₅₂	—	4	4	no	—
140	D₅₂	—	4	4	no	—
141	D₅₂	—	8	4	no	—
142	D₅₂	—	8	4	no	—
143	D₅₂	—	8	4	no	—
144	D₅₂	—	8	4	no	—
145	D₅₂	—	8	4	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
150	C₂²×D₈	—	13	2	no	—
151	C₂²×D₈	—	26	1	no	—
152	#532	—	2	1	no	—
153	C₂×#127	—	26	1	no	—
154	C₂×D₁₆	—	13	8	no	—
155	#500	C₂×D₁₃	1	1	yes	characteristic, complemented
156	#493	—	13	2	no	—
157	#493	—	13	2	no	—
158	#493	—	26	1	no	—
159	#493	—	26	1	no	—
160	#493	C₂×D₁₃	1	1	yes	characteristic, complemented
161	#486	—	2	1	no	—
162	#466	—	2	1	no	—
163	#466	C₂×D₁₃	1	2	yes	—
164	C₄×D₈	—	2	1	no	—
165	C₄×D₈	C₂×D₁₃	1	2	yes	complemented
166	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
167	C₄⋊C₁₆	—	13	2	no	—
168	#357	—	13	8	no	—
169	C₂²⋊C₁₆	—	13	2	no	—
170	C₄×C₁₆	—	13	2	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	8	2	no	—
175	C₂²×C₁₃	—	8	2	no	—
176	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
177	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×D₁₃	—	8	4	no	—
179	C₂×D₁₃	—	8	4	no	—
180	C₂×D₁₃	—	8	4	no	—
181	C₂×D₁₃	—	16	8	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
186	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
187	C₄×C₁₃	#509	1	2	yes	—
188	C₄×C₁₃	#535	1	1	yes	characteristic
189	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×#41	—	4	1	no	—
191	C₂²×D₄	—	26	2	no	—
192	C₂²×D₄	—	52	1	no	—
193	C₂²×D₄	—	52	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	13	4	no	—
196	C₂×D₈	—	26	2	no	—
197	C₂×D₈	—	26	2	no	—
198	C₂×D₈	—	26	2	no	—
199	C₂×D₈	—	52	4	no	—
200	C₂×D₈	—	52	4	no	—
201	C₂×D₈	C₂²×D₁₃	1	2	yes	—
202	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	1	no	—
205	#127	—	2	1	no	—
206	#127	—	26	2	no	—
207	#127	—	26	2	no	—
208	#127	—	52	1	no	—
209	#127	—	52	1	no	—
210	#124	—	4	1	no	—
211	#121	—	2	2	no	—
212	#121	—	4	1	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	4	1	no	—
215	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
216	D₁₆	—	52	8	no	—
217	C₂×C₁₆	—	13	4	no	—
218	#107	—	2	1	no	—
219	#107	C₂²×D₁₃	1	2	yes	—
220	#102	—	2	2	no	—
221	#102	—	2	2	no	—
222	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	—	16	2	no	—
228	C₂×C₁₃	#3205	1	1	yes	characteristic
229	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
231	D₁₃	—	16	4	no	—
232	C₂⁴	—	52	2	no	—
233	#41	—	8	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	26	4	no	—
240	C₂×D₄	—	52	4	no	—
241	C₂×D₄	—	52	4	no	—
242	C₂×D₄	—	52	2	no	—
243	C₂×D₄	—	52	2	no	—
244	C₂×D₄	—	52	2	no	—
245	C₂×D₄	—	52	2	no	—
246	C₂×D₄	—	52	2	no	—
247	C₂×D₄	—	104	4	no	—
248	C₂×D₄	—	104	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	#6104	1	1	yes	characteristic
251	C₂²×C₄	#6104	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	D₈	—	2	4	no	—
254	D₈	—	4	2	no	—
255	D₈	—	26	8	no	—
256	D₈	—	52	4	no	—
257	D₈	—	52	4	no	—
258	D₈	—	52	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	#6104	1	1	yes	characteristic
270	C₄²	#6104	1	1	yes	characteristic
271	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₁₆	—	26	4	no	—
274	C₁₃	#33771	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	4	2	no	—
276	C₂³	—	52	4	no	—
277	C₂³	—	104	4	no	—
278	C₂³	—	104	4	no	—
279	C₂³	—	104	4	no	—
280	C₂³	C₂×#6104	1	1	yes	characteristic
281	Q₈	—	8	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	4	no	—
285	D₄	—	8	2	no	—
286	D₄	—	8	2	no	—
287	D₄	—	52	4	no	—
288	D₄	—	52	4	no	—
289	D₄	—	104	4	no	—
290	D₄	—	104	4	no	—
291	D₄	—	104	4	no	—
292	D₄	—	104	4	no	—
293	D₄	—	104	4	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	#88066	1	4	yes	complemented
300	C₂×C₄	C₂×#6104	1	1	yes	characteristic
301	C₂×C₄	C₂×#6104	1	1	yes	characteristic
302	C₂×C₄	C₂×#6104	1	1	yes	characteristic
303	C₂×C₄	C₂×#6104	1	1	yes	characteristic
304	C₂×C₄	C₂×#6104	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	#88203	1	2	yes	—
309	C₈	#88256	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	104	4	no	—
316	C₂²	—	104	4	no	—
317	C₂²	—	104	4	no	—
318	C₂²	—	208	8	no	—
319	C₂²	C₂×#88066	1	2	yes	—
320	C₂²	C₂²×#6104	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×#88066	1	2	yes	—
325	C₄	C₂×#88066	1	2	yes	—
326	C₄	#11718854	1	2	yes	—
327	C₄	C₂×#88203	1	1	yes	characteristic
328	C₄	#11719535	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	16	2	no	—
331	C₂	—	208	4	no	—
332	C₂	#13125751	1	1	yes	characteristic
333	C₂	C₂²×#88066	1	1	yes	characteristic
334	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#33771
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. C₂×#11718356	C₂	semidirect
2. #13117085	C₂	semidirect
3. #13117061	C₂	semidirect
4. #13116871	C₂	semidirect
5. #13116747	C₂	semidirect
6. #13116635	C₂	semidirect
7. C₁₃×#3169	C₂	semidirect
12. C₁₃×#500	C₂²	semidirect
13. C₁₃×#493	C₂²	semidirect
18. C₄×C₁₃×D₈	C₂²	semidirect
20. C₂×#88066	C₂²	semidirect
26. #11718083	C₂²	semidirect
27. C₄⋊#6090	C₂²	semidirect
28. C₄×#6090	C₂²	semidirect
108. #3169	D₁₃	semidirect
132. C₂×C₄×C₁₃	D₁₆	semidirect
155. #500	C₂×D₁₃	semidirect
160. #493	C₂×D₁₃	semidirect
165. C₄×D₈	C₂×D₁₃	semidirect
274. C₁₃	#33771	semidirect
299. C₂×C₄	#88066	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718356	2
#13117085	2
#13117061	2
#13116871	2
#13116747	2
#13116635	2
C₁₃×#3169	2
#33771	13