Subgroups — #16732471

Order 3328.

3114 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719458	C₂	1	4	yes	maximal, complemented
2	#13128621	C₂	1	2	yes	maximal, complemented
3	#13125330	C₂	1	12	yes	maximal, complemented
4	#13125295	C₂	1	6	yes	maximal, complemented
5	#13125264	C₂	1	3	yes	maximal, complemented
6	C₂×#11718360	C₂	1	3	yes	maximal, complemented
7	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88257	C₂²	1	4	yes	—
9	C₂×C₁₃×#131	C₂²	1	2	yes	—
10	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
11	#11719458	—	2	32	no	—
12	C₂×#88163	C₂²	1	24	yes	—
13	C₂²×Dic₅₂	C₂²	1	12	yes	—
14	C₂×#88158	C₂²	1	12	yes	—
15	Q₈×D₅₂	—	2	4	no	—
16	#11719260	—	2	12	no	—
17	#11719231	C₂²	1	12	yes	—
18	#11719182	C₂²	1	6	yes	—
19	C₂×#88123	C₂²	1	6	yes	—
20	C₁₃×#405	C₂²	1	6	yes	—
21	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
23	#11718376	C₂²	1	24	yes	—
24	#11718360	C₂²	1	3	yes	—
25	#11718360	C₂²	1	6	yes	—
26	#11718360	C₂²	1	12	yes	—
27	#11718357	C₂²	1	12	yes	—
28	#11718356	C₂²	1	1	yes	characteristic
29	#11718354	C₂²	1	6	yes	—
30	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
31	#88257	—	2	32	no	—
32	C₂×Q₈×D₁₃	—	2	4	no	—
33	C₂×#6129	—	2	12	no	—
34	C₂×#6127	C₂³	1	12	yes	—
35	C₂²×Dic₂₆	C₂³	1	12	yes	—
36	C₁₃×#131	D₄	1	16	yes	complemented
37	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
39	C₄×Q₈×C₁₃	D₄	1	2	yes	—
40	C₄×D₄×C₁₃	D₄	1	6	yes	—
41	C₁₃×#117	C₂³	1	3	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
43	#88184	—	2	12	no	—
44	C₂×C₄⋊Dic₁₃	—	2	12	no	—
45	#88163	—	2	96	no	—
46	C₂×Dic₅₂	—	2	48	no	—
47	C₂×Dic₅₂	C₂³	1	12	yes	—
48	C₂×Dic₅₂	C₂³	1	24	yes	—
49	#88158	—	2	48	no	—
50	C₂×D₁₀₄	C₂³	1	4	yes	—
51	C₂×#6095	C₂³	1	24	yes	—
52	#88151	—	2	24	no	—
53	#88140	—	2	24	no	—
54	#88129	C₂³	1	12	yes	—
55	#88128	C₂³	1	2	yes	—
56	C₄×D₅₂	—	2	12	no	—
57	#88123	—	2	12	no	—
58	#88123	C₂³	1	6	yes	—
59	#88123	C₂³	1	12	yes	—
60	C₁₃×#97	C₂³	1	6	yes	—
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	3	yes	—
63	C₄×C₈×C₁₃	C₂³	1	6	yes	—
64	#60648	—	13	1	no	maximal, Sylow
65	C₁₃×#41	C₂×D₄	1	8	yes	—
66	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₁₃	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
70	C₂×C₂²⋊D₁₃	—	2	12	no	—
71	C₂²×Dic₁₃	—	2	12	no	—
72	Q₈×D₁₃	—	4	16	no	—
73	#6129	—	4	48	no	—
74	#6127	—	2	48	no	—
75	C₂×D₅₂	—	2	4	no	—
76	C₂×D₅₂	C₂⁴	1	4	yes	—
77	C₂×C₄×D₁₃	—	2	12	no	—
78	C₂×Dic₂₆	—	2	48	no	—
79	C₂×Dic₂₆	—	2	12	no	—
80	C₂×Dic₂₆	C₂⁴	1	12	yes	—
81	C₁₃×M₁₆	C₂×D₄	1	24	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
84	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
85	C₁₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
86	C₁₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
87	C₄²×C₁₃	C₂×D₄	1	6	yes	—
88	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₄²×C₁₃	C₂⁴	1	3	yes	—
90	#6103	—	2	24	no	—
91	C₄⋊Dic₁₃	—	2	12	no	—
92	C₄⋊Dic₁₃	—	2	24	no	—
93	Dic₅₂	—	2	24	no	—
94	Dic₅₂	—	2	48	no	—
95	D₁₀₄	—	2	8	no	—
96	#6095	—	2	48	no	—
97	C₂×#578	—	13	4	no	—
98	#3499	—	13	2	no	—
99	#3156	—	13	12	no	—
100	#3152	—	13	6	no	—
101	#3147	—	13	3	no	—
102	C₂×#494	—	13	3	no	—
103	C₄×#131	D₁₃	1	1	yes	characteristic, complemented
104	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
105	C₂²×D₁₃	—	2	4	no	—
106	Q₈×C₁₃	#127	1	4	yes	—
107	D₄×C₁₃	#127	1	12	yes	—
108	C₂×C₄×C₁₃	—	2	12	no	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
110	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
112	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
114	C₂²⋊D₁₃	—	4	24	no	—
115	C₂×Dic₁₃	—	2	12	no	—
116	C₂×Dic₁₃	—	4	24	no	—
117	D₅₂	—	2	8	no	—
118	D₅₂	—	4	8	no	—
119	C₄×D₁₃	—	4	24	no	—
120	Dic₂₆	—	2	24	no	—
121	Dic₂₆	—	4	24	no	—
122	C₈×C₁₃	C₂²×D₄	1	12	yes	—
123	C₈×C₁₃	C₂²×D₄	1	4	yes	—
124	C₂×2₋¹⁺⁴	—	13	4	no	—
125	#578	—	26	32	no	—
126	C₂×#137	—	13	24	no	—
127	C₂×#135	—	13	12	no	—
128	C₂²×Q₁₆	—	13	12	no	—
129	C₂×#131	C₂×D₁₃	1	2	yes	—
130	D₄×Q₈	—	26	4	no	—
131	#547	—	26	12	no	—
132	#536	—	13	12	no	—
133	#532	—	13	6	no	—
134	C₂×C₄⋊Q₈	—	13	6	no	—
135	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
136	#497	—	13	24	no	—
137	#495	—	13	12	no	—
138	#494	—	13	3	no	—
139	#494	—	13	6	no	—
140	#494	—	13	12	no	—
141	#493	—	13	1	no	—
142	#492	—	13	6	no	—
143	#405	C₂×D₁₃	1	6	yes	—
144	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
145	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
146	C₂²×C₁₃	—	2	6	no	—
147	C₂²×C₁₃	C₂×#127	1	6	yes	—
148	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
149	C₂×D₁₃	—	4	4	no	—
150	C₂×D₁₃	—	4	8	no	—
151	C₄×C₁₃	—	2	6	no	—
152	C₄×C₁₃	C₂×#127	1	6	yes	—
153	C₄×C₁₃	#578	1	4	yes	complemented
154	C₄×C₁₃	C₂³×D₄	1	2	yes	—
155	Dic₁₃	—	4	12	no	—
156	2₋¹⁺⁴	—	26	32	no	—
157	C₂×#41	—	13	12	no	—
158	C₂×#41	—	26	12	no	—
159	C₂×#41	C₂²×D₁₃	1	1	yes	characteristic
160	C₂²×Q₈	—	13	12	no	—
161	C₂²×Q₈	—	26	4	no	—
162	#137	—	26	96	no	—
163	#135	—	26	48	no	—
164	C₂×Q₁₆	—	13	12	no	—
165	C₂×Q₁₆	—	13	24	no	—
166	C₂×Q₁₆	—	26	48	no	—
167	C₂×SD₁₆	—	13	24	no	—
168	C₂×D₈	—	13	4	no	—
169	#131	D₅₂	1	16	yes	complemented
170	C₂×M₁₆	C₂²×D₁₃	1	6	yes	—
171	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
172	C₄⋊Q₈	—	13	6	no	—
173	C₄⋊Q₈	—	13	12	no	—
174	C₄⋊Q₈	—	26	12	no	—
175	#127	—	13	2	no	—
176	#124	—	13	12	no	—
177	#123	—	26	24	no	—
178	C₂²⋊Q₈	—	26	24	no	—
179	#121	—	26	12	no	—
180	C₄×Q₈	D₅₂	1	2	yes	—
181	C₄×D₄	—	26	12	no	—
182	C₄×D₄	D₅₂	1	6	yes	—
183	#117	C₂²×D₁₃	1	3	yes	—
184	C₂×C₄⋊C₄	—	26	12	no	—
185	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
186	#97	C₂²×D₁₃	1	6	yes	—
187	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	3	yes	—
189	C₄×C₈	C₂²×D₁₃	1	6	yes	—
190	C₂×C₁₃	—	2	6	no	—
191	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#578	1	2	yes	—
193	D₁₃	—	8	4	no	—
194	#41	—	26	48	no	—
195	#41	—	52	48	no	—
196	#41	C₂×D₅₂	1	8	yes	—
197	C₂×Q₈	—	13	12	no	—
198	C₂×Q₈	—	26	12	no	—
199	C₂×Q₈	—	26	48	no	—
200	C₂×Q₈	—	52	16	no	—
201	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
202	C₂×D₄	—	13	4	no	—
203	C₂×D₄	—	26	12	no	—
204	C₂×D₄	—	26	4	no	—
205	C₂×D₄	C₂×D₅₂	1	3	yes	—
206	C₂²×C₄	—	26	12	no	—
207	C₂²×C₄	—	26	12	no	—
208	C₂²×C₄	C₂×D₅₂	1	6	yes	—
209	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
210	Q₁₆	—	26	24	no	—
211	Q₁₆	—	26	48	no	—
212	SD₁₆	—	26	48	no	—
213	D₈	—	26	8	no	—
214	M₁₆	C₂×D₅₂	1	24	yes	—
215	C₂×C₈	C₂×D₅₂	1	24	yes	—
216	C₂×C₈	C₂³×D₁₃	1	2	yes	—
217	C₂×C₈	C₂³×D₁₃	1	6	yes	—
218	C₄⋊C₄	—	26	12	no	—
219	C₄⋊C₄	—	26	24	no	—
220	C₄⋊C₄	C₂×D₅₂	1	6	yes	—
221	C₂²⋊C₄	—	26	24	no	—
222	C₂²⋊C₄	C₂×D₅₂	1	6	yes	—
223	C₄²	C₂×D₅₂	1	6	yes	—
224	C₄²	C₂³×D₁₃	1	1	yes	characteristic
225	C₄²	C₂³×D₁₃	1	3	yes	—
226	C₁₃	#60648	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	26	4	no	—
228	C₂³	C₂²×D₅₂	1	3	yes	—
229	Q₈	—	26	24	no	—
230	Q₈	—	52	24	no	—
231	Q₈	#88128	1	4	yes	—
232	D₄	—	26	8	no	—
233	D₄	—	52	24	no	—
234	D₄	—	52	8	no	—
235	D₄	#88128	1	12	yes	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	26	12	no	—
238	C₂×C₄	—	52	24	no	—
239	C₂×C₄	—	52	24	no	—
240	C₂×C₄	C₂²×D₅₂	1	6	yes	—
241	C₂×C₄	C₂²×D₅₂	1	3	yes	—
242	C₂×C₄	C₂²×D₅₂	1	12	yes	—
243	C₂×C₄	C₂²×D₅₂	1	2	yes	—
244	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
245	C₈	C₂²×D₅₂	1	4	yes	—
246	C₈	C₂²×D₅₂	1	12	yes	—
247	C₂²	—	2	6	no	—
248	C₂²	—	52	8	no	—
249	C₂²	—	52	4	no	—
250	C₂²	C₂×#88128	1	6	yes	—
251	C₂²	C₂³×D₅₂	1	1	yes	characteristic
252	C₄	—	2	6	no	—
253	C₄	—	52	12	no	—
254	C₄	C₂×#88128	1	6	yes	—
255	C₄	#11719458	1	4	yes	complemented
256	C₄	C₂³×D₅₂	1	2	yes	—
257	C₂	—	2	6	no	—
258	C₂	—	104	4	no	—
259	C₂	C₂²×#88128	1	1	yes	characteristic
260	C₂	C₂×#11719458	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂³×D₄
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#60648
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#11719458	C₂	semidirect
2. #13128621	C₂	semidirect
3. #13125330	C₂	semidirect
4. #13125295	C₂	semidirect
5. #13125264	C₂	semidirect
6. C₂×#11718360	C₂	semidirect
7. C₄×C₁₃×#131	C₂	semidirect
36. C₁₃×#131	D₄	semidirect
103. C₄×#131	D₁₃	semidirect
153. C₄×C₁₃	#578	semidirect
169. #131	D₅₂	semidirect
226. C₁₃	#60648	semidirect
255. C₄	#11719458	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719458	2
#13128621	2
#13125330	2
#13125295	2
#13125264	2
C₂×#11718360	2
C₄×C₁₃×#131	2
#60648	13