Subgroups — #16017207

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718808	C₂	1	2	yes	maximal, complemented
2	C₄×#88158	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116873	C₂	1	4	yes	maximal, complemented
4	#13116868	C₂	1	4	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116552	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#127	—	2	1	no	—
11	C₂×#88158	C₂²	1	1	yes	characteristic
12	C₄×#6127	—	2	1	no	—
13	C₁₃×#493	—	2	2	no	—
14	C₁₃×#493	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88067	C₂²	1	4	yes	complemented
17	#11718808	—	2	8	no	—
18	C₂×#88066	C₂²	1	4	yes	complemented
19	#11718774	—	2	1	no	—
20	#11718767	C₂²	1	1	yes	characteristic
21	C₂×#88052	C₂²	1	2	yes	—
22	C₄×Dic₅₂	C₂²	1	2	yes	complemented
23	C₄×D₁₀₄	C₂²	1	2	yes	complemented
24	C₄×#6095	—	2	2	no	—
25	#11718154	C₂²	1	8	yes	complemented
26	C₄⋊#6090	C₂²	1	4	yes	complemented
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	4	2	no	—
29	C₂×#6127	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	4	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	—	2	4	no	—
33	C₂×C₁₃×D₈	—	2	4	no	—
34	C₂×C₁₃×D₈	—	4	4	no	—
35	C₂×C₁₃×D₈	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#127	—	2	4	no	—
39	C₁₃×#127	—	4	1	no	—
40	C₁₃×#127	—	4	1	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₁₃	—	4	2	no	—
43	#88180	—	2	1	no	—
44	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
45	#88158	—	2	2	no	—
46	#88158	D₄	1	4	yes	complemented
47	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
48	C₂×#6095	—	2	1	no	—
49	C₄×D₅₂	—	2	2	no	—
50	#88126	—	4	1	no	—
51	C₄²×D₁₃	—	4	1	no	—
52	C₄×Dic₂₆	—	2	2	no	—
53	C₄×C₈×C₁₃	D₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	#88067	—	2	8	no	—
56	#88066	—	2	8	no	—
57	#88061	—	2	2	no	—
58	#88058	C₂³	1	2	yes	—
59	#88057	—	2	1	no	—
60	#88056	—	2	2	no	—
61	#88052	—	2	4	no	—
62	C₂×#6090	C₂³	1	4	yes	—
63	#33582	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	—	4	2	no	—
65	C₂×D₄×C₁₃	—	2	4	no	—
66	C₂×D₄×C₁₃	—	4	4	no	—
67	C₂×D₄×C₁₃	—	4	4	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	8	8	no	—
72	C₂²×C₄×C₁₃	D₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₃	—	4	1	no	—
75	#6127	—	2	4	no	—
76	#6127	—	4	2	no	—
77	C₂×D₅₂	—	2	1	no	—
78	C₂×C₄×D₁₃	—	4	1	no	—
79	C₂×C₄×D₁₃	—	4	2	no	—
80	C₂×Dic₂₆	—	2	1	no	—
81	C₁₃×D₈	—	2	4	no	—
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×D₈	—	4	8	no	—
84	C₁₃×D₈	—	4	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₄²×C₁₃	D₈	1	1	yes	characteristic
92	C₄²×C₁₃	D₈	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	2	yes	—
94	C₂²⋊Dic₁₃	—	4	1	no	—
95	#6103	—	4	2	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	#6101	—	4	2	no	—
98	C₄×Dic₁₃	—	4	2	no	—
99	C₄×Dic₁₃	—	4	1	no	—
100	Dic₅₂	—	2	1	no	—
101	Dic₅₂	C₂×D₄	1	2	yes	complemented
102	D₁₀₄	—	2	1	no	—
103	D₁₀₄	C₂×D₄	1	2	yes	complemented
104	#6095	—	2	4	no	—
105	#6090	—	2	4	no	—
106	C₂×#509	—	13	2	no	—
107	C₂×#493	D₁₃	1	1	yes	characteristic, complemented
108	C₄×#135	—	13	1	no	—
109	#2207	—	13	4	no	—
110	#2206	—	13	4	no	—
111	#2150	—	13	1	no	—
112	#2140	—	13	2	no	—
113	C₂³×C₁₃	—	4	4	no	—
114	C₂³×C₁₃	—	8	8	no	—
115	C₂³×C₁₃	—	8	4	no	—
116	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₁₃	—	4	1	no	—
118	D₄×C₁₃	—	4	4	no	—
119	D₄×C₁₃	—	4	4	no	—
120	D₄×C₁₃	—	8	8	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	8	8	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	#121	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	—	4	2	no	—
133	D₅₂	—	2	2	no	—
134	D₅₂	—	4	1	no	—
135	C₄×D₁₃	—	4	4	no	—
136	C₄×D₁₃	—	8	4	no	—
137	Dic₂₆	—	2	2	no	—
138	Dic₂₆	—	4	1	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	#121	1	2	yes	—
141	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×#135	—	13	1	no	—
144	C₂²×D₈	—	2	1	no	—
145	C₂²×D₈	C₂×D₁₃	1	2	yes	—
146	C₂×#127	—	2	1	no	—
147	C₄×#41	—	26	1	no	—
148	#509	—	26	8	no	—
149	C₂×SD₃₂	—	13	4	no	—
150	C₂×D₁₆	—	13	4	no	—
151	C₂×M₃₂	—	13	2	no	—
152	#493	—	2	2	no	—
153	#493	C₂×D₁₃	1	4	yes	—
154	C₄×Q₁₆	—	13	2	no	—
155	C₄×SD₁₆	—	26	2	no	—
156	C₄×D₈	—	13	2	no	—
157	#427	—	13	1	no	—
158	#416	—	26	1	no	—
159	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
160	C₄⋊C₁₆	—	13	4	no	—
161	#357	—	13	8	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	8	8	no	—
164	C₂²×C₁₃	—	8	4	no	—
165	C₂²×C₁₃	—	16	8	no	—
166	C₂²×C₁₃	#459	1	2	yes	—
167	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₁₃	—	4	2	no	—
169	C₂×D₁₃	—	8	2	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	#459	1	2	yes	—
172	C₄×C₁₃	#509	1	4	yes	complemented
173	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
174	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	Dic₁₃	—	4	2	no	—
176	Dic₁₃	—	8	2	no	—
177	C₂×#41	—	26	1	no	—
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	—	4	2	no	—
180	#135	—	13	4	no	—
181	#135	—	26	2	no	—
182	C₂×Q₁₆	—	13	1	no	—
183	C₂×SD₁₆	—	26	1	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	2	4	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	4	4	no	—
189	C₂×D₈	—	13	1	no	—
190	C₂×D₈	C₂²×D₁₃	1	4	yes	—
191	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	#127	—	2	4	no	—
194	#127	—	4	1	no	—
195	#127	—	4	1	no	—
196	C₄×Q₈	—	26	2	no	—
197	C₄×D₄	—	26	2	no	—
198	C₄×D₄	—	52	2	no	—
199	#117	—	26	1	no	—
200	#117	—	52	1	no	—
201	C₂×C₄²	—	52	1	no	—
202	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
203	SD₃₂	—	26	8	no	—
204	D₁₆	—	26	8	no	—
205	M₃₂	—	26	4	no	—
206	C₂×C₁₆	—	13	4	no	—
207	#107	—	13	2	no	—
208	#106	—	26	1	no	—
209	#103	—	26	2	no	—
210	#102	—	26	2	no	—
211	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
212	C₄×C₈	C₂²×D₁₃	1	2	yes	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	—	16	4	no	—
215	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#509	1	2	yes	—
217	D₁₃	—	8	2	no	—
218	C₂⁴	—	4	2	no	—
219	#41	—	26	4	no	—
220	#41	—	52	2	no	—
221	C₂×Q₈	—	26	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	8	8	no	—
229	C₂×D₄	—	26	1	no	—
230	C₂×D₄	—	52	1	no	—
231	C₂²×C₄	—	52	1	no	—
232	C₂²×C₄	—	52	2	no	—
233	C₂²×C₄	#6104	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
235	Q₁₆	—	13	2	no	—
236	Q₁₆	—	26	1	no	—
237	SD₁₆	—	26	4	no	—
238	D₈	—	2	4	no	—
239	D₈	—	2	4	no	—
240	D₈	—	4	8	no	—
241	D₈	—	4	4	no	—
242	D₈	—	13	2	no	—
243	D₈	—	26	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	D₄×D₁₃	1	2	yes	—
246	C₂×C₈	#6129	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
250	C₄⋊C₄	—	26	2	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₂²⋊C₄	—	52	1	no	—
253	C₂²⋊C₄	—	52	2	no	—
254	C₄²	—	52	1	no	—
255	C₄²	—	52	2	no	—
256	C₄²	#6104	1	1	yes	characteristic
257	C₄²	#6104	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
259	C₁₆	—	26	4	no	—
260	C₁₃	#33582	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	4	4	no	—
262	C₂³	—	8	8	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	52	1	no	—
265	C₂³	C₂×#6104	1	1	yes	characteristic
266	Q₈	—	26	2	no	—
267	Q₈	—	52	1	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	8	no	—
273	D₄	—	26	2	no	—
274	D₄	—	52	4	no	—
275	D₄	—	52	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	52	1	no	—
278	C₂×C₄	—	52	4	no	—
279	C₂×C₄	—	52	2	no	—
280	C₂×C₄	—	104	4	no	—
281	C₂×C₄	C₂×#6104	1	2	yes	—
282	C₂×C₄	C₂×#6104	1	1	yes	characteristic
283	C₂×C₄	C₂×#6104	1	2	yes	—
284	C₂×C₄	#88186	1	2	yes	—
285	C₂×C₄	#88192	1	2	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	#88192	1	2	yes	—
289	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
290	C₈	C₂×#6129	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	8	8	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	16	8	no	—
295	C₂²	—	52	2	no	—
296	C₂²	—	104	2	no	—
297	C₂²	#11718733	1	2	yes	—
298	C₂²	C₂²×#6104	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	52	2	no	—
301	C₄	—	104	2	no	—
302	C₄	#11718733	1	2	yes	—
303	C₄	#11718808	1	4	yes	complemented
304	C₄	C₂×#88186	1	1	yes	characteristic
305	C₄	C₂×#88192	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	—	104	2	no	—
309	C₂	C₂×#11718733	1	1	yes	characteristic
310	C₂	C₂×#11718808	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#493	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	#33582
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#11718808	C₂	semidirect
2. C₄×#88158	C₂	semidirect
3. #13116873	C₂	semidirect
4. #13116868	C₂	semidirect
5. #13116592	C₂	semidirect
6. #13116552	C₂	semidirect
7. C₂×C₁₃×#493	C₂	semidirect
16. C₂×#88067	C₂²	semidirect
18. C₂×#88066	C₂²	semidirect
22. C₄×Dic₅₂	C₂²	semidirect
23. C₄×D₁₀₄	C₂²	semidirect
25. #11718154	C₂²	semidirect
26. C₄⋊#6090	C₂²	semidirect
46. #88158	D₄	semidirect
101. Dic₅₂	C₂×D₄	semidirect
103. D₁₀₄	C₂×D₄	semidirect
107. C₂×#493	D₁₃	semidirect
172. C₄×C₁₃	#509	semidirect
260. C₁₃	#33582	semidirect
303. C₄	#11718808	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718808	2
C₄×#88158	2
#13116873	2
#13116868	2
#13116592	2
#13116552	2
C₂×C₁₃×#493	2
#33582	13