Subgroups — C₂×#14521815 #16860420

Order 3968.

1460 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521815	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11758484	C₂	1	4	yes	maximal, complemented
4	C₂×#11758478	C₂	1	4	yes	maximal, complemented
5	C₂²×#92474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×D₃₁	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₂²⋊D₃₁	C₄	1	2	yes	—
10	C₂×#92495	C₄	1	2	yes	—
11	C₂×#92495	C₄	1	2	yes	—
12	C₂×#92495	C₄	1	8	yes	—
13	C₂×#92495	C₂²	1	2	yes	—
14	C₂×#92495	C₂²	1	2	yes	—
15	C₂×#92495	C₂²	1	8	yes	—
16	C₂²×D₁₂₄	C₄	1	1	yes	characteristic
17	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
18	C₂²×Dic₆₂	C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
21	C₂×M₁₆×C₃₁	C₂²	1	8	yes	complemented
22	C₂²×C₈×C₃₁	C₂²	1	2	yes	complemented
23	C₂×#92474	C₂²	1	2	yes	—
24	C₂×#92474	C₂²	1	2	yes	complemented
25	C₂×#92474	C₂²	1	8	yes	complemented
26	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	complemented
27	#11758484	C₂²	1	32	yes	complemented
28	M₁₆×D₃₁	C₂²	1	32	yes	complemented
29	#11758478	C₂²	1	32	yes	complemented
30	C₂×#92469	C₂²	1	8	yes	complemented
31	C₂×C₈×D₃₁	C₂²	1	8	yes	complemented
32	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×D₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊D₃₁	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊D₃₁	C₂×C₄	1	16	yes	—
45	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
46	#92495	C₂×C₄	1	8	yes	—
47	#92495	C₂×C₄	1	8	yes	—
48	#92495	C₂×C₄	1	16	yes	—
49	#92495	C₂×C₄	1	16	yes	—
50	#92495	C₂³	1	8	yes	—
51	#92495	C₂³	1	8	yes	—
52	C₂×D₁₂₄	C₂×C₄	1	2	yes	—
53	C₂×D₁₂₄	C₂×C₄	1	2	yes	—
54	C₂×D₁₂₄	C₂×C₄	1	8	yes	—
55	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
56	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	—
57	C₂×C₄×D₃₁	C₂³	1	4	yes	—
58	C₂×C₄×D₃₁	C₂³	1	8	yes	—
59	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
60	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
61	C₂×Dic₆₂	C₂×C₄	1	8	yes	—
62	M₁₆×C₃₁	C₂³	1	8	yes	—
63	M₁₆×C₃₁	C₂³	1	8	yes	complemented
64	C₂×C₈×C₃₁	C₂³	1	4	yes	—
65	C₂×C₈×C₃₁	C₂³	1	8	yes	complemented
66	#92474	C₂³	1	8	yes	—
67	#92474	C₂³	1	8	yes	complemented
68	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
69	C₂×C₃₁⋊C₈	C₂³	1	8	yes	complemented
70	#92469	C₂³	1	32	yes	complemented
71	C₈×D₃₁	C₂³	1	32	yes	complemented
72	C₂³×C₃₁	—	4	2	no	—
73	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
77	C₂²×D₃₁	—	4	4	no	—
78	C₂²×D₃₁	C₂²×C₄	1	4	yes	—
79	C₂²×D₃₁	C₂²×C₄	1	8	yes	—
80	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
88	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
90	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
91	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
92	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₃₁	C₂²×C₄	1	16	yes	—
94	C₂²⋊D₃₁	C₂²×C₄	1	16	yes	—
95	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
96	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
97	D₁₂₄	C₂²×C₄	1	8	yes	—
98	D₁₂₄	C₂²×C₄	1	8	yes	—
99	C₄×D₃₁	C₂²×C₄	1	8	yes	—
100	C₄×D₃₁	C₂²×C₄	1	16	yes	—
101	C₄×D₃₁	C₂⁴	1	8	yes	—
102	Dic₆₂	C₂²×C₄	1	8	yes	—
103	Dic₆₂	C₂²×C₄	1	8	yes	—
104	C₈×C₃₁	C₂⁴	1	8	yes	—
105	C₃₁⋊C₈	C₂⁴	1	8	yes	—
106	C₂×#568	—	31	1	no	maximal, Sylow
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	4	4	no	—
111	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
112	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
113	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
114	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₃₁	—	2	8	no	—
116	C₂×D₃₁	—	4	8	no	—
117	C₂×D₃₁	C₂³×C₄	1	8	yes	—
118	C₄×C₃₁	C₂³×C₄	1	2	yes	—
119	C₄×C₃₁	C₂³×C₄	1	2	yes	—
120	C₄×C₃₁	C₂³×C₄	1	2	yes	—
121	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
123	Dic₃₁	C₂³×C₄	1	8	yes	—
124	C₂²×#41	—	31	1	no	—
125	#568	—	31	16	no	—
126	C₂×#131	—	31	4	no	—
127	C₂×#131	—	31	4	no	—
128	C₂²×M₁₆	—	31	1	no	—
129	C₂²×M₁₆	—	31	4	no	—
130	C₂²×M₁₆	D₃₁	1	1	yes	characteristic, complemented
131	C₂×C₃₁	—	2	2	no	—
132	C₂×C₃₁	—	2	2	no	—
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	#568	1	2	yes	complemented
135	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
136	D₃₁	—	2	8	no	—
137	C₂×#41	—	31	2	no	—
138	C₂×#41	—	31	2	no	—
139	C₂×#41	—	31	8	no	—
140	C₂×#41	—	31	2	no	—
141	C₂×#41	—	31	2	no	—
142	C₂×#41	—	31	8	no	—
143	C₂²×Q₈	—	31	1	no	—
144	C₂²×D₄	—	31	2	no	—
145	C₂²×D₄	—	31	1	no	—
146	C₂³×C₄	—	31	2	no	—
147	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
148	#131	—	31	32	no	—
149	#131	—	31	32	no	—
150	C₂×M₁₆	—	31	2	no	—
151	C₂×M₁₆	—	31	2	no	—
152	C₂×M₁₆	—	31	8	no	—
153	C₂×M₁₆	—	31	32	no	—
154	C₂×M₁₆	—	31	8	no	—
155	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
156	C₂×M₁₆	C₂×D₃₁	1	2	yes	complemented
157	C₂×M₁₆	C₂×D₃₁	1	8	yes	complemented
158	C₂²×C₈	—	31	2	no	—
159	C₂²×C₈	—	31	8	no	—
160	C₂²×C₈	C₂×D₃₁	1	2	yes	complemented
161	C₃₁	C₂×#568	1	1	yes	characteristic, complemented, Sylow
162	C₂⁴	—	31	2	no	—
163	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
164	#41	—	31	8	no	—
165	#41	—	31	8	no	—
166	#41	—	31	16	no	—
167	#41	—	31	16	no	—
168	#41	—	31	8	no	—
169	#41	—	31	8	no	—
170	C₂×Q₈	—	31	2	no	—
171	C₂×Q₈	—	31	2	no	—
172	C₂×Q₈	—	31	8	no	—
173	C₂×D₄	—	31	4	no	—
174	C₂×D₄	—	31	4	no	—
175	C₂×D₄	—	31	16	no	—
176	C₂×D₄	—	31	2	no	—
177	C₂×D₄	—	31	2	no	—
178	C₂×D₄	—	31	8	no	—
179	C₂²×C₄	—	31	2	no	—
180	C₂²×C₄	—	31	4	no	—
181	C₂²×C₄	—	31	8	no	—
182	C₂²×C₄	—	31	8	no	—
183	C₂²×C₄	—	31	4	no	—
184	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₃₁	1	4	yes	—
188	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
189	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
192	M₁₆	—	31	8	no	—
193	M₁₆	—	31	8	no	—
194	M₁₆	—	31	32	no	—
195	M₁₆	C₂²×D₃₁	1	8	yes	—
196	M₁₆	C₂²×D₃₁	1	8	yes	complemented
197	C₂×C₈	—	31	8	no	—
198	C₂×C₈	—	31	4	no	—
199	C₂×C₈	—	31	32	no	—
200	C₂×C₈	C₂²×D₃₁	1	4	yes	—
201	C₂×C₈	C₂²×D₃₁	1	8	yes	complemented
202	C₂³	—	4	2	no	—
203	C₂³	—	31	4	no	—
204	C₂³	—	31	8	no	—
205	C₂³	—	124	4	no	—
206	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
207	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
208	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
209	C₂³	C₂×C₄×D₃₁	1	4	yes	—
210	Q₈	—	31	8	no	—
211	Q₈	—	31	8	no	—
212	D₄	—	31	16	no	—
213	D₄	—	31	16	no	—
214	D₄	—	31	8	no	—
215	D₄	—	31	8	no	—
216	C₂×C₄	—	31	8	no	—
217	C₂×C₄	—	31	4	no	—
218	C₂×C₄	—	31	16	no	—
219	C₂×C₄	—	31	8	no	—
220	C₂×C₄	—	31	8	no	—
221	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
223	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
224	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
227	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
228	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
230	C₂×C₄	C₂³×D₃₁	1	2	yes	—
231	C₂×C₄	C₂³×D₃₁	1	2	yes	—
232	C₂×C₄	C₂³×D₃₁	1	2	yes	—
233	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
234	C₈	—	31	8	no	—
235	C₈	C₂³×D₃₁	1	8	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	31	8	no	—
241	C₂²	—	62	8	no	—
242	C₂²	—	124	8	no	—
243	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
244	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
245	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
246	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
247	C₄	—	31	8	no	—
248	C₄	C₂²×C₄×D₃₁	1	2	yes	—
249	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
250	C₄	C₂²×C₄×D₃₁	1	2	yes	—
251	C₄	C₂²×C₄×D₃₁	1	2	yes	—
252	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	—	62	8	no	—
257	C₂	#14521815	1	2	yes	direct product factor
258	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂²×#92495	C₂
cosocle	C₄×C₃₁	C₂⁵
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	C₂×#568
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #14521815	C₂	direct (N × H)
3. C₂×#11758484	C₂	semidirect
4. C₂×#11758478	C₂	semidirect
5. C₂²×#92474	C₂	semidirect
6. C₂×M₁₆×D₃₁	C₂	semidirect
7. C₂²×M₁₆×C₃₁	C₂	semidirect
20. C₂×M₁₆×C₃₁	C₂²	semidirect
21. C₂×M₁₆×C₃₁	C₂²	semidirect
22. C₂²×C₈×C₃₁	C₂²	semidirect
24. C₂×#92474	C₂²	semidirect
25. C₂×#92474	C₂²	semidirect
26. C₂²×C₃₁⋊C₈	C₂²	semidirect
27. #11758484	C₂²	semidirect
28. M₁₆×D₃₁	C₂²	semidirect
29. #11758478	C₂²	semidirect
30. C₂×#92469	C₂²	semidirect
31. C₂×C₈×D₃₁	C₂²	semidirect
63. M₁₆×C₃₁	C₂³	semidirect
65. C₂×C₈×C₃₁	C₂³	semidirect
67. #92474	C₂³	semidirect
69. C₂×C₃₁⋊C₈	C₂³	semidirect
70. #92469	C₂³	semidirect
71. C₈×D₃₁	C₂³	semidirect
130. C₂²×M₁₆	D₃₁	semidirect
134. C₂×C₃₁	#568	semidirect
156. C₂×M₁₆	C₂×D₃₁	semidirect
157. C₂×M₁₆	C₂×D₃₁	semidirect
160. C₂²×C₈	C₂×D₃₁	semidirect
161. C₃₁	C₂×#568	semidirect
196. M₁₆	C₂²×D₃₁	semidirect
201. C₂×C₈	C₂²×D₃₁	semidirect
257. C₂	#14521815	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#92495	2
#14521815	2
C₂×#11758484	2
C₂×#11758478	2
C₂²×#92474	2
C₂×M₁₆×D₃₁	2
C₂²×M₁₆×C₃₁	2
C₂×#568	31

Subgroups — C2×#14521815 #16860420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521815 #16860420