Subgroups — #16017358

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718565	C₂	1	1	yes	maximal, characteristic, complemented
2	#13124417	C₂	1	2	yes	maximal, complemented
3	#13116932	C₂	1	4	yes	maximal, complemented
4	#13116927	C₂	1	2	yes	maximal, complemented
5	#13116922	C₂	1	2	yes	maximal, complemented
6	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116544	C₂	1	2	yes	maximal, complemented
8	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11719496	—	2	2	no	—
11	C₂×#88158	C₂²	1	2	yes	—
12	C₂²×C₁₃×C₁₆	—	2	4	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	#11718777	C₂²	1	4	yes	—
18	#11718776	C₂²	1	4	yes	—
19	#11718775	—	2	4	no	—
20	#11718774	—	2	2	no	—
21	#11718767	C₂²	1	2	yes	—
22	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
23	#11718565	—	2	4	no	—
24	C₂×#88040	C₂²	1	2	yes	complemented
25	C₂×D₂₀₈	C₂²	1	1	yes	characteristic, complemented
26	#11718101	C₂²	1	4	yes	complemented
27	#11718097	C₂²	1	2	yes	complemented
28	#11718096	C₂²	1	2	yes	complemented
29	#11718095	C₂²	1	4	yes	complemented
30	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂×#6127	—	2	2	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	#88207	—	4	2	no	—
39	#88184	—	2	4	no	—
40	#88183	—	4	2	no	—
41	C₄×C₂²⋊D₁₃	—	4	4	no	—
42	#88180	—	2	2	no	—
43	#88178	—	2	4	no	—
44	C₂×Dic₅₂	C₂³	1	2	yes	—
45	#88158	—	2	4	no	—
46	#88158	—	2	4	no	—
47	C₂×D₁₀₄	C₂³	1	2	yes	—
48	C₂×#6095	—	2	2	no	—
49	C₂×C₁₃×C₁₆	—	2	2	no	—
50	C₂×C₁₃×C₁₆	—	2	4	no	—
51	C₂×C₁₃×C₁₆	—	2	4	no	—
52	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
53	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
56	#88061	—	2	4	no	—
57	#88058	C₂³	1	4	yes	—
58	#88057	—	2	2	no	—
59	#88056	—	2	4	no	—
60	Dic₁₀₄	—	2	2	no	—
61	#88040	—	2	4	no	—
62	D₂₀₈	—	2	2	no	—
63	#33600	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	D₈	1	2	yes	—
68	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₁₃	—	4	4	no	—
72	C₂×C₂²⋊D₁₃	—	4	2	no	—
73	#6127	—	4	4	no	—
74	#6127	—	4	4	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×C₄×D₁₃	—	4	2	no	—
77	C₂×Dic₂₆	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊Dic₁₃	—	4	4	no	—
88	C₂²⋊Dic₁₃	—	4	2	no	—
89	#6103	—	4	4	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	#6101	—	4	4	no	—
92	C₄×Dic₁₃	—	4	2	no	—
93	Dic₅₂	—	2	2	no	—
94	Dic₅₂	—	2	2	no	—
95	D₁₀₄	—	2	2	no	—
96	D₁₀₄	—	2	2	no	—
97	#6095	—	4	4	no	—
98	C₁₃×C₁₆	—	2	2	no	—
99	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₂×#508	—	13	1	no	—
102	C₂³×C₁₆	D₁₃	1	1	yes	characteristic, complemented
103	#3050	—	13	2	no	—
104	#2217	—	13	4	no	—
105	#2216	—	13	2	no	—
106	#2215	—	13	2	no	—
107	#2158	—	13	1	no	—
108	#2138	—	13	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	—	2	4	no	—
111	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂³×C₁₃	C₂×D₈	1	2	yes	—
113	C₂²×D₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	#121	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#135	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂²⋊D₁₃	—	8	4	no	—
124	C₂²⋊D₁₃	—	8	8	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	D₅₂	—	4	2	no	—
128	D₅₂	—	4	2	no	—
129	C₄×D₁₃	—	8	4	no	—
130	Dic₂₆	—	4	2	no	—
131	Dic₂₆	—	4	2	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	#121	1	2	yes	—
134	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂×#135	—	13	2	no	—
137	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
138	#525	—	26	2	no	—
139	#508	—	26	4	no	—
140	C₂×Q₃₂	—	13	1	no	—
141	C₂×SD₃₂	—	13	2	no	—
142	C₂×D₁₆	—	13	1	no	—
143	C₂²×C₁₆	—	2	4	no	—
144	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
145	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
146	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
147	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
148	#467	—	13	4	no	—
149	#466	—	13	4	no	—
150	#465	—	26	4	no	—
151	#427	—	13	2	no	—
152	#416	—	26	2	no	—
153	#367	—	13	2	no	—
154	#366	—	13	2	no	—
155	#358	—	13	4	no	—
156	#357	—	13	4	no	—
157	C₂²×C₁₃	—	2	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	#466	1	2	yes	—
162	C₂²×C₁₃	#508	1	4	yes	complemented
163	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₁₃	—	8	4	no	—
165	C₂×D₁₃	—	8	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	#466	1	2	yes	—
168	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
169	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
170	Dic₁₃	—	8	4	no	—
171	Dic₁₃	—	8	2	no	—
172	C₂×#41	—	26	2	no	—
173	C₂³×C₄	D₅₂	1	1	yes	characteristic
174	#135	—	26	4	no	—
175	#135	—	26	4	no	—
176	C₂×Q₁₆	—	13	2	no	—
177	C₂×SD₁₆	—	26	2	no	—
178	C₂×D₈	—	13	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	D₅₂	1	1	yes	characteristic
182	C₂²×C₈	D₅₂	1	2	yes	—
183	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
184	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	#123	—	52	2	no	—
186	C₂²⋊Q₈	—	26	4	no	—
187	#121	—	26	4	no	—
188	#120	—	52	2	no	—
189	C₄×D₄	—	52	4	no	—
190	#117	—	26	2	no	—
191	Q₃₂	—	26	2	no	—
192	SD₃₂	—	26	4	no	—
193	D₁₆	—	26	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	4	no	—
196	C₂×C₁₆	—	2	4	no	—
197	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	complemented
198	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
199	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
201	#107	—	13	4	no	—
202	#106	—	26	2	no	—
203	#103	—	26	4	no	—
204	#102	—	26	4	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	—	2	4	no	—
207	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#508	1	2	yes	—
209	D₁₃	—	16	2	no	—
210	C₂⁴	D₁₀₄	1	1	yes	characteristic
211	#41	—	52	4	no	—
212	#41	—	52	4	no	—
213	C₂×Q₈	—	26	2	no	—
214	C₂×D₄	—	26	2	no	—
215	C₂×D₄	—	52	4	no	—
216	C₂×D₄	—	52	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	52	2	no	—
220	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
221	C₂²×C₄	D₁₀₄	1	2	yes	—
222	C₂²×C₄	C₂×D₅₂	1	2	yes	—
223	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
224	Q₁₆	—	26	2	no	—
225	Q₁₆	—	26	2	no	—
226	SD₁₆	—	52	4	no	—
227	D₈	—	26	2	no	—
228	D₈	—	26	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×D₅₂	1	2	yes	—
234	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
235	C₂×C₈	#6127	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
237	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	26	4	no	—
239	C₄⋊C₄	—	52	4	no	—
240	C₂²⋊C₄	—	52	2	no	—
241	C₂²⋊C₄	—	52	4	no	—
242	C₂²⋊C₄	—	52	4	no	—
243	C₄²	—	52	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
246	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
247	C₁₃	#33600	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	2	2	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	52	2	no	—
251	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
252	C₂³	C₂×D₁₀₄	1	2	yes	—
253	Q₈	—	52	2	no	—
254	Q₈	—	52	2	no	—
255	D₄	—	52	2	no	—
256	D₄	—	52	2	no	—
257	D₄	—	104	4	no	—
258	D₄	—	104	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	52	4	no	—
264	C₂×C₄	—	52	2	no	—
265	C₂×C₄	—	104	4	no	—
266	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
267	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
268	C₂×C₄	#88158	1	2	yes	—
269	C₂×C₄	#88184	1	2	yes	—
270	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	#88184	1	2	yes	—
273	C₈	C₂×#6127	1	1	yes	characteristic
274	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	104	4	no	—
280	C₂²	—	104	2	no	—
281	C₂²	#11718565	1	4	yes	complemented
282	C₂²	#11718776	1	2	yes	—
283	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	104	4	no	—
286	C₄	—	104	2	no	—
287	C₄	#11718776	1	2	yes	—
288	C₄	C₂×#88158	1	1	yes	characteristic
289	C₄	C₂×#88184	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	4	no	—
292	C₂	—	208	2	no	—
293	C₂	C₂×#11718776	1	1	yes	characteristic
294	C₂	C₂×#11718565	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂³×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂³×C₁₆	D₁₃
hypocenter	C₁₃	#33600
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11718565	C₂	semidirect
2. #13124417	C₂	semidirect
3. #13116932	C₂	semidirect
4. #13116927	C₂	semidirect
5. #13116922	C₂	semidirect
6. #13116620	C₂	semidirect
7. #13116544	C₂	semidirect
8. C₂³×C₁₃×C₁₆	C₂	semidirect
22. C₂×Dic₁₀₄	C₂²	semidirect
24. C₂×#88040	C₂²	semidirect
25. C₂×D₂₀₈	C₂²	semidirect
26. #11718101	C₂²	semidirect
27. #11718097	C₂²	semidirect
28. #11718096	C₂²	semidirect
29. #11718095	C₂²	semidirect
52. C₂×C₁₃×C₁₆	D₄	semidirect
102. C₂³×C₁₆	D₁₃	semidirect
162. C₂²×C₁₃	#508	semidirect
197. C₂×C₁₆	C₂²⋊D₁₃	semidirect
247. C₁₃	#33600	semidirect
281. C₂²	#11718565	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718565	2
#13124417	2
#13116932	2
#13116927	2
#13116922	2
#13116620	2
#13116544	2
C₂³×C₁₃×C₁₆	2
#33600	13