Subgroups — #16732464

Order 3328.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719457	C₂	1	4	yes	maximal, complemented
2	#13128614	C₂	1	2	yes	maximal, complemented
3	#13125322	C₂	1	12	yes	maximal, complemented
4	#13125288	C₂	1	6	yes	maximal, complemented
5	#13125264	C₂	1	3	yes	maximal, complemented
6	C₂×#11718356	C₂	1	3	yes	maximal, complemented
7	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88256	C₂²	1	4	yes	—
9	C₂×C₁₃×#131	C₂²	1	2	yes	—
10	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
11	#11719457	—	2	32	no	—
12	C₂×#88162	C₂²	1	24	yes	complemented
13	C₂×#88158	C₂²	1	12	yes	complemented
14	C₂²×D₁₀₄	C₂²	1	12	yes	complemented
15	#11719282	—	2	4	no	—
16	D₄×D₅₂	—	2	12	no	—
17	#11719230	C₂²	1	12	yes	complemented
18	#11719182	C₂²	1	6	yes	complemented
19	C₂×#88128	C₂²	1	6	yes	complemented
20	C₁₃×#405	C₂²	1	6	yes	complemented
21	C₄×C₁₃×M₁₆	C₂²	1	3	yes	complemented
22	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	complemented
23	#11718373	C₂²	1	24	yes	complemented
24	#11718360	C₂²	1	1	yes	characteristic, complemented
25	#11718357	C₂²	1	12	yes	complemented
26	#11718356	C₂²	1	3	yes	complemented
27	#11718356	C₂²	1	6	yes	complemented
28	#11718356	C₂²	1	12	yes	complemented
29	#11718354	C₂²	1	6	yes	complemented
30	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
31	#88256	—	2	32	no	—
32	C₂×#6131	—	2	4	no	—
33	C₂×D₄×D₁₃	—	2	12	no	—
34	C₂×#6127	C₂³	1	12	yes	—
35	C₂²×D₅₂	—	2	12	no	—
36	C₂²×D₅₂	C₂³	1	12	yes	—
37	C₁₃×#131	D₄	1	16	yes	complemented
38	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
40	C₄×Q₈×C₁₃	D₄	1	2	yes	—
41	C₄×D₄×C₁₃	D₄	1	6	yes	—
42	C₁₃×#117	C₂³	1	3	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
44	#88184	—	2	12	no	—
45	#88162	—	2	96	no	—
46	C₂×Dic₅₂	C₂³	1	4	yes	—
47	#88158	—	2	48	no	—
48	C₂×D₁₀₄	—	2	48	no	—
49	C₂×D₁₀₄	C₂³	1	12	yes	—
50	C₂×D₁₀₄	C₂³	1	24	yes	—
51	C₂×#6095	C₂³	1	24	yes	—
52	#88149	—	2	24	no	—
53	#88136	—	2	24	no	—
54	#88129	C₂³	1	12	yes	—
55	#88128	—	2	12	no	—
56	#88128	C₂³	1	6	yes	—
57	#88128	C₂³	1	12	yes	—
58	C₄×D₅₂	—	2	12	no	—
59	#88123	C₂³	1	2	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	#60647	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	—	2	12	no	—
66	C₁₃×#41	C₂×D₄	1	8	yes	—
67	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×D₄×C₁₃	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
71	C₂×C₂²⋊D₁₃	—	2	12	no	—
72	#6131	—	4	16	no	—
73	D₄×D₁₃	—	4	48	no	—
74	#6127	—	2	48	no	—
75	C₂×D₅₂	—	2	48	no	—
76	C₂×D₅₂	—	2	12	no	—
77	C₂×D₅₂	—	2	12	no	—
78	C₂×D₅₂	—	2	24	no	—
79	C₂×D₅₂	—	8	24	no	—
80	C₂×D₅₂	C₂⁴	1	12	yes	—
81	C₂×C₄×D₁₃	—	2	12	no	—
82	C₂×Dic₂₆	C₂⁴	1	4	yes	—
83	C₁₃×M₁₆	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	complemented
85	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
87	C₁₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
88	C₁₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
89	C₄²×C₁₃	C₂×D₄	1	6	yes	—
90	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₄²×C₁₃	C₂⁴	1	3	yes	—
92	#6103	—	2	24	no	—
93	C₄⋊Dic₁₃	—	2	4	no	—
94	Dic₅₂	—	2	8	no	—
95	D₁₀₄	—	2	24	no	—
96	D₁₀₄	—	2	48	no	—
97	#6095	—	2	48	no	—
98	C₂×#576	—	13	4	no	—
99	#3498	—	13	2	no	—
100	#3155	—	13	12	no	—
101	#3151	—	13	6	no	—
102	#3147	—	13	3	no	—
103	C₂×#493	—	13	3	no	—
104	C₄×#131	D₁₃	1	1	yes	characteristic, complemented
105	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
106	C₂²×D₁₃	—	2	12	no	—
107	C₂²×D₁₃	—	4	24	no	—
108	C₂²×D₁₃	—	8	24	no	—
109	Q₈×C₁₃	#127	1	4	yes	—
110	D₄×C₁₃	#127	1	12	yes	—
111	C₂×C₄×C₁₃	—	2	12	no	—
112	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
113	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
117	C₂²⋊D₁₃	—	4	24	no	—
118	C₂×Dic₁₃	—	2	4	no	—
119	D₅₂	—	2	24	no	—
120	D₅₂	—	4	24	no	—
121	D₅₂	—	4	24	no	—
122	D₅₂	—	8	24	no	—
123	C₄×D₁₃	—	4	24	no	—
124	Dic₂₆	—	2	8	no	—
125	C₈×C₁₃	C₂²×D₄	1	12	yes	—
126	C₈×C₁₃	C₂²×D₄	1	4	yes	—
127	C₂×H(5,𝔽₂)	—	13	4	no	—
128	#576	—	26	32	no	—
129	C₂×AGL(1,ℤ/8)	—	13	24	no	—
130	C₂×#135	—	13	12	no	—
131	C₂²×D₈	—	13	12	no	—
132	C₂×#131	C₂×D₁₃	1	2	yes	—
133	#550	—	26	4	no	—
134	D₄²	—	26	12	no	—
135	#535	—	13	12	no	—
136	#532	—	13	6	no	—
137	C₂×#127	—	13	6	no	—
138	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
139	#496	—	13	24	no	—
140	#495	—	13	12	no	—
141	#494	—	13	1	no	—
142	#493	—	13	3	no	—
143	#493	—	13	6	no	—
144	#493	—	13	12	no	—
145	#492	—	13	6	no	—
146	#405	C₂×D₁₃	1	6	yes	complemented
147	C₄×M₁₆	C₂×D₁₃	1	3	yes	complemented
148	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	complemented
149	C₂²×C₁₃	—	2	6	no	—
150	C₂²×C₁₃	C₂×#127	1	6	yes	—
151	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
152	C₂×D₁₃	—	4	12	no	—
153	C₂×D₁₃	—	4	24	no	—
154	C₂×D₁₃	—	16	24	no	—
155	C₄×C₁₃	—	2	6	no	—
156	C₄×C₁₃	C₂×#127	1	6	yes	—
157	C₄×C₁₃	#576	1	4	yes	complemented
158	C₄×C₁₃	C₂³×D₄	1	2	yes	—
159	Dic₁₃	—	4	4	no	—
160	H(5,𝔽₂)	—	26	32	no	—
161	C₂×#41	—	13	12	no	—
162	C₂×#41	—	26	4	no	—
163	C₂×#41	C₂²×D₁₃	1	1	yes	characteristic
164	C₂²×D₄	—	13	12	no	—
165	C₂²×D₄	—	26	12	no	—
166	C₂²×D₄	—	26	12	no	—
167	AGL(1,ℤ/8)	—	26	96	no	—
168	#135	—	26	48	no	—
169	C₂×Q₁₆	—	13	4	no	—
170	C₂×SD₁₆	—	13	24	no	—
171	C₂×D₈	—	13	12	no	—
172	C₂×D₈	—	13	24	no	—
173	C₂×D₈	—	26	48	no	—
174	#131	D₅₂	1	16	yes	complemented
175	C₂×M₁₆	C₂²×D₁₃	1	6	yes	—
176	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
177	C₄⋊Q₈	—	13	2	no	—
178	#127	—	13	6	no	—
179	#127	—	13	12	no	—
180	#127	—	26	12	no	—
181	#124	—	13	12	no	—
182	#121	—	26	12	no	—
183	#121	—	26	24	no	—
184	#120	—	26	24	no	—
185	C₄×Q₈	D₅₂	1	2	yes	—
186	C₄×D₄	—	26	12	no	—
187	C₄×D₄	D₅₂	1	6	yes	—
188	#117	C₂²×D₁₃	1	3	yes	—
189	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
190	#97	C₂²×D₁₃	1	6	yes	—
191	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₁₃	1	3	yes	—
193	C₄×C₈	C₂²×D₁₃	1	6	yes	—
194	C₂×C₁₃	—	2	6	no	—
195	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#576	1	2	yes	—
197	D₁₃	—	8	12	no	—
198	C₂⁴	—	26	12	no	—
199	#41	—	26	48	no	—
200	#41	—	52	16	no	—
201	#41	C₂×D₅₂	1	8	yes	—
202	C₂×Q₈	—	13	4	no	—
203	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
204	C₂×D₄	—	13	12	no	—
205	C₂×D₄	—	26	12	no	—
206	C₂×D₄	—	26	48	no	—
207	C₂×D₄	—	26	12	no	—
208	C₂×D₄	—	26	12	no	—
209	C₂×D₄	—	26	24	no	—
210	C₂×D₄	—	52	48	no	—
211	C₂×D₄	—	104	24	no	—
212	C₂×D₄	C₂×D₅₂	1	3	yes	—
213	C₂²×C₄	—	26	12	no	—
214	C₂²×C₄	C₂×D₅₂	1	6	yes	—
215	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
216	Q₁₆	—	26	8	no	—
217	SD₁₆	—	26	48	no	—
218	D₈	—	26	24	no	—
219	D₈	—	26	48	no	—
220	M₁₆	C₂×D₅₂	1	24	yes	complemented
221	C₂×C₈	C₂×D₅₂	1	24	yes	complemented
222	C₂×C₈	C₂³×D₁₃	1	2	yes	—
223	C₂×C₈	C₂³×D₁₃	1	6	yes	—
224	C₄⋊C₄	—	26	4	no	—
225	C₄⋊C₄	C₂×D₅₂	1	6	yes	—
226	C₂²⋊C₄	—	26	24	no	—
227	C₂²⋊C₄	C₂×D₅₂	1	6	yes	—
228	C₄²	C₂×D₅₂	1	6	yes	—
229	C₄²	C₂³×D₁₃	1	1	yes	characteristic
230	C₄²	C₂³×D₁₃	1	3	yes	—
231	C₁₃	#60647	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	26	12	no	—
233	C₂³	—	52	24	no	—
234	C₂³	—	104	24	no	—
235	C₂³	C₂²×D₅₂	1	3	yes	—
236	Q₈	—	26	8	no	—
237	Q₈	#88128	1	4	yes	—
238	D₄	—	26	24	no	—
239	D₄	—	52	24	no	—
240	D₄	—	52	24	no	—
241	D₄	—	52	24	no	—
242	D₄	—	104	24	no	—
243	D₄	#88128	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	26	4	no	—
246	C₂×C₄	—	52	24	no	—
247	C₂×C₄	C₂²×D₅₂	1	6	yes	—
248	C₂×C₄	C₂²×D₅₂	1	3	yes	—
249	C₂×C₄	C₂²×D₅₂	1	12	yes	—
250	C₂×C₄	C₂²×D₅₂	1	2	yes	—
251	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
252	C₈	C₂²×D₅₂	1	4	yes	—
253	C₈	C₂²×D₅₂	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	52	24	no	—
256	C₂²	—	52	12	no	—
257	C₂²	—	208	24	no	—
258	C₂²	C₂×#88128	1	6	yes	—
259	C₂²	C₂³×D₅₂	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	52	4	no	—
262	C₄	C₂×#88128	1	6	yes	—
263	C₄	#11719457	1	4	yes	complemented
264	C₄	C₂³×D₅₂	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	104	12	no	—
267	C₂	C₂²×#88128	1	1	yes	characteristic
268	C₂	C₂×#11719457	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	#60647
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×#11719457	C₂	semidirect
2. #13128614	C₂	semidirect
3. #13125322	C₂	semidirect
4. #13125288	C₂	semidirect
5. #13125264	C₂	semidirect
6. C₂×#11718356	C₂	semidirect
7. C₄×C₁₃×#131	C₂	semidirect
12. C₂×#88162	C₂²	semidirect
13. C₂×#88158	C₂²	semidirect
14. C₂²×D₁₀₄	C₂²	semidirect
17. #11719230	C₂²	semidirect
18. #11719182	C₂²	semidirect
19. C₂×#88128	C₂²	semidirect
20. C₁₃×#405	C₂²	semidirect
21. C₄×C₁₃×M₁₆	C₂²	semidirect
22. C₂×C₄×C₈×C₁₃	C₂²	semidirect
23. #11718373	C₂²	semidirect
24. #11718360	C₂²	semidirect
25. #11718357	C₂²	semidirect
26. #11718356	C₂²	semidirect
27. #11718356	C₂²	semidirect
28. #11718356	C₂²	semidirect
29. #11718354	C₂²	semidirect
37. C₁₃×#131	D₄	semidirect
83. C₁₃×M₁₆	C₂×D₄	semidirect
84. C₂×C₈×C₁₃	C₂×D₄	semidirect
104. C₄×#131	D₁₃	semidirect
146. #405	C₂×D₁₃	semidirect
147. C₄×M₁₆	C₂×D₁₃	semidirect
148. C₂×C₄×C₈	C₂×D₁₃	semidirect
157. C₄×C₁₃	#576	semidirect
174. #131	D₅₂	semidirect
220. M₁₆	C₂×D₅₂	semidirect
221. C₂×C₈	C₂×D₅₂	semidirect
231. C₁₃	#60647	semidirect
263. C₄	#11719457	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719457	2
#13128614	2
#13125322	2
#13125288	2
#13125264	2
C₂×#11718356	2
C₄×C₁₃×#131	2
#60647	13