Subgroups — #16016231

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 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
2	C₄×#88158	C₂	1	2	yes	maximal, complemented
3	#13116713	C₂	1	2	yes	maximal, complemented
4	#13116708	C₂	1	2	yes	maximal, complemented
5	#13116702	C₂	1	4	yes	maximal, complemented
6	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116544	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₂²	1	2	yes	—
10	C₄×#6127	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718774	—	2	2	no	—
15	#11718767	C₂²	1	2	yes	—
16	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
17	#11718565	C₄	1	8	yes	complemented
18	C₂×#88040	C₂²	1	2	yes	—
19	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₅₂	C₂²	1	4	yes	—
21	C₄×D₁₀₄	C₂²	1	4	yes	—
22	C₄×#6095	—	2	4	no	—
23	C₁₃×#346	C₂²	1	2	yes	—
24	C₁₃×#346	C₂²	1	2	yes	—
25	#11718101	C₂²	1	4	yes	—
26	#11718097	C₂²	1	2	yes	—
27	#11718096	C₂²	1	2	yes	—
28	#11718095	C₂²	1	4	yes	—
29	C₂×#6127	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃	—	4	4	no	—
34	#88180	—	2	2	no	—
35	C₂×Dic₅₂	C₂³	1	2	yes	—
36	#88158	—	2	4	no	—
37	#88158	C₂×C₄	1	8	yes	complemented
38	C₂×D₁₀₄	C₂³	1	2	yes	—
39	C₂×#6095	—	2	2	no	—
40	C₄×D₅₂	—	2	4	no	—
41	#88126	—	4	2	no	—
42	C₄²×D₁₃	—	4	2	no	—
43	C₄×Dic₂₆	—	2	4	no	—
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	#88061	—	2	4	no	—
52	#88058	C₂³	1	4	yes	—
53	#88057	—	2	2	no	—
54	#88056	—	2	4	no	—
55	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
56	#88040	C₂×C₄	1	8	yes	complemented
57	D₂₀₈	C₂×C₄	1	4	yes	complemented
58	#33451	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	D₈	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	4	2	no	—
62	#6127	—	2	8	no	—
63	#6127	—	4	4	no	—
64	C₂×D₅₂	—	2	2	no	—
65	C₂×C₄×D₁₃	—	4	2	no	—
66	C₂×C₄×D₁₃	—	4	4	no	—
67	C₂×Dic₂₆	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₄²×C₁₃	D₈	1	2	yes	—
75	C₄²×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²⋊Dic₁₃	—	4	2	no	—
77	#6103	—	4	4	no	—
78	C₄⋊Dic₁₃	—	2	4	no	—
79	#6101	—	4	4	no	—
80	C₄×Dic₁₃	—	4	4	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	Dic₅₂	—	2	2	no	—
83	Dic₅₂	C₂²×C₄	1	4	yes	—
84	D₁₀₄	—	2	2	no	—
85	D₁₀₄	C₂²×C₄	1	4	yes	—
86	#6095	—	2	8	no	—
87	C₁₃×C₁₆	—	2	2	no	—
88	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	C₂×#508	—	13	1	no	—
91	C₄×#135	—	13	2	no	—
92	#2177	—	13	2	no	—
93	#2176	—	13	2	no	—
94	#2175	—	13	4	no	—
95	#2158	—	13	1	no	—
96	#2138	—	13	2	no	—
97	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
98	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂²×D₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#135	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²⋊D₁₃	—	4	8	no	—
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	—	4	4	no	—
109	D₅₂	—	2	4	no	—
110	D₅₂	—	4	2	no	—
111	C₄×D₁₃	—	4	8	no	—
112	C₄×D₁₃	—	8	8	no	—
113	Dic₂₆	—	2	4	no	—
114	Dic₂₆	—	4	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	C₄×D₄	1	2	yes	—
117	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂×#135	—	13	2	no	—
120	C₄×#41	—	26	2	no	—
121	#508	—	13	8	no	—
122	C₂×Q₃₂	—	13	1	no	—
123	C₂×SD₃₂	—	13	2	no	—
124	C₂×D₁₆	—	13	1	no	—
125	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
126	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
127	C₄×Q₁₆	—	13	4	no	—
128	C₄×SD₁₆	—	26	4	no	—
129	C₄×D₈	—	13	4	no	—
130	#427	—	13	2	no	—
131	#416	—	26	2	no	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	#367	—	13	2	no	—
134	#366	—	13	2	no	—
135	#358	—	13	4	no	—
136	#357	—	13	4	no	—
137	#346	C₂×D₁₃	1	2	yes	—
138	#346	C₂×D₁₃	1	2	yes	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	C₄×D₈	1	2	yes	—
141	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₁₃	—	4	4	no	—
143	C₂×D₁₃	—	8	4	no	—
144	C₄×C₁₃	—	2	4	no	—
145	C₄×C₁₃	C₄×D₈	1	2	yes	—
146	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
148	Dic₁₃	—	4	4	no	—
149	Dic₁₃	—	8	4	no	—
150	C₂×#41	—	26	2	no	—
151	#135	—	13	8	no	—
152	#135	—	26	4	no	—
153	C₂×Q₁₆	—	13	2	no	—
154	C₂×SD₁₆	—	26	2	no	—
155	C₂×D₈	—	13	2	no	—
156	C₂²×C₈	D₅₂	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
158	C₄×Q₈	—	26	4	no	—
159	C₄×D₄	—	26	4	no	—
160	C₄×D₄	—	52	4	no	—
161	#117	—	26	2	no	—
162	#117	—	52	2	no	—
163	C₂×C₄²	—	52	2	no	—
164	C₂×C₄²	D₅₂	1	1	yes	characteristic
165	Q₃₂	—	13	4	no	—
166	SD₃₂	—	13	8	no	—
167	D₁₆	—	13	4	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
170	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
171	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
173	#107	—	13	4	no	—
174	#106	—	26	2	no	—
175	#103	—	26	4	no	—
176	#102	—	26	4	no	—
177	C₄×C₈	D₅₂	1	2	yes	—
178	C₄×C₈	C₂²×D₁₃	1	2	yes	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
181	C₂×C₁₃	#3414	1	2	yes	—
182	D₁₃	—	8	4	no	—
183	#41	—	26	8	no	—
184	#41	—	52	4	no	—
185	C₂×Q₈	—	26	2	no	—
186	C₂×D₄	—	26	2	no	—
187	C₂×D₄	—	52	2	no	—
188	C₂²×C₄	—	52	2	no	—
189	C₂²×C₄	—	52	4	no	—
190	C₂²×C₄	D₁₀₄	1	2	yes	—
191	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
192	Q₁₆	—	13	4	no	—
193	Q₁₆	—	26	2	no	—
194	SD₁₆	—	26	8	no	—
195	D₈	—	13	4	no	—
196	D₈	—	26	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
199	C₂×C₈	C₂×D₅₂	1	2	yes	—
200	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
201	C₂×C₈	#6127	1	2	yes	—
202	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	26	4	no	—
204	C₄⋊C₄	—	52	4	no	—
205	C₂²⋊C₄	—	52	2	no	—
206	C₂²⋊C₄	—	52	4	no	—
207	C₄²	—	52	4	no	—
208	C₄²	—	52	2	no	—
209	C₄²	D₁₀₄	1	2	yes	—
210	C₄²	C₂×D₅₂	1	2	yes	—
211	C₁₆	—	2	2	no	—
212	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
213	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
214	C₁₃	#33451	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	52	2	no	—
216	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
217	Q₈	—	26	4	no	—
218	Q₈	—	52	2	no	—
219	D₄	—	26	4	no	—
220	D₄	—	52	8	no	—
221	D₄	—	52	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	52	2	no	—
224	C₂×C₄	—	52	8	no	—
225	C₂×C₄	—	52	4	no	—
226	C₂×C₄	—	104	8	no	—
227	C₂×C₄	C₄×D₅₂	1	2	yes	—
228	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
229	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
230	C₂×C₄	#88158	1	2	yes	—
231	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₅₂	1	2	yes	—
234	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
235	C₈	C₂×#6127	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	52	4	no	—
238	C₂²	—	104	4	no	—
239	C₂²	C₄×D₁₀₄	1	2	yes	—
240	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	52	4	no	—
243	C₄	—	104	4	no	—
244	C₄	C₄×D₁₀₄	1	2	yes	—
245	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
246	C₄	C₂×#88158	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	104	4	no	—
249	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
250	C₂	#13127972	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718565	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#33451
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₄×#88158	C₂	semidirect
3. #13116713	C₂	semidirect
4. #13116708	C₂	semidirect
5. #13116702	C₂	semidirect
6. #13116620	C₂	semidirect
7. #13116544	C₂	semidirect
8. C₂×C₁₃×#346	C₂	semidirect
17. #11718565	C₄	semidirect
37. #88158	C₂×C₄	semidirect
45. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
55. Dic₁₀₄	C₂×C₄	semidirect
56. #88040	C₂×C₄	semidirect
57. D₂₀₈	C₂×C₄	semidirect
97. C₂×#346	D₁₃	semidirect
169. C₂×C₁₆	C₄×D₁₃	semidirect
214. C₁₃	#33451	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
C₄×#88158	2
#13116713	2
#13116708	2
#13116702	2
#13116620	2
#13116544	2
C₂×C₁₃×#346	2
#33451	13