Subgroups — C₂×#14521852 #16860462

Order 3968.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758566	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521852	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11758509	C₂	1	3	yes	maximal, complemented
4	C₂×#11758526	C₂	1	3	yes	maximal, complemented
5	C₂×#11758527	C₂	1	3	yes	maximal, complemented
6	C₂²×#92479	C₂	1	3	yes	maximal, complemented
7	C₂×#11758525	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×H(5,𝔽₂)	—	2	4	no	—
10	C₃₁×H(5,𝔽₂)	C₂²	1	8	yes	complemented
11	C₂×#41×C₃₁	—	2	1	no	—
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#41×C₃₁	C₂²	1	3	yes	complemented
14	C₂²×D₄×C₃₁	—	2	3	no	—
15	C₂²×D₄×C₃₁	C₂²	1	3	yes	complemented
16	#11758566	—	2	4	no	—
17	#11758566	C₂²	1	8	yes	complemented
18	C₂×#92499	—	2	1	no	—
19	C₂×D₄×D₃₁	—	2	3	no	—
20	C₂×#92495	C₂²	1	3	yes	complemented
21	C₂²×D₁₂₄	C₂²	1	3	yes	complemented
22	#11758527	C₂²	1	24	yes	complemented
23	#11758526	C₂²	1	24	yes	complemented
24	#11758525	C₂²	1	8	yes	complemented
25	C₂×#92482	C₂²	1	1	yes	characteristic, complemented
26	C₂×#92481	C₂²	1	3	yes	complemented
27	C₂×#92480	C₂²	1	3	yes	complemented
28	#11758509	C₂²	1	24	yes	complemented
29	C₂×#92479	C₂²	1	3	yes	complemented
30	C₂×#92479	C₂²	1	6	yes	complemented
31	C₂×#92479	C₂²	1	24	yes	complemented
32	C₂×#92474	C₂²	1	3	yes	complemented
33	C₂²×C₃₁⋊C₈	C₂²	1	3	yes	complemented
34	C₂⁴×C₃₁	—	2	3	no	—
35	C₂³×D₃₁	—	2	3	no	—
36	#41×C₃₁	—	2	8	no	—
37	#41×C₃₁	—	2	6	no	—
38	#41×C₃₁	D₄	1	4	yes	—
39	#41×C₃₁	C₂³	1	4	yes	—
40	#41×C₃₁	C₂³	1	12	yes	complemented
41	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×D₄×C₃₁	—	2	3	no	—
44	C₂×D₄×C₃₁	—	2	24	no	—
45	C₂×D₄×C₃₁	—	2	3	no	—
46	C₂×D₄×C₃₁	—	2	6	no	—
47	C₂×D₄×C₃₁	D₄	1	3	yes	—
48	C₂×D₄×C₃₁	C₂³	1	3	yes	—
49	C₂×D₄×C₃₁	C₂³	1	12	yes	complemented
50	C₂²×C₄×C₃₁	—	2	3	no	—
51	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₃₁	—	2	3	no	—
53	#92499	—	2	8	no	—
54	D₄×D₃₁	—	2	24	no	—
55	#92495	—	2	6	no	—
56	#92495	C₂³	1	12	yes	complemented
57	C₂×D₁₂₄	—	2	3	no	—
58	C₂×D₁₂₄	—	2	6	no	—
59	C₂×D₁₂₄	C₂³	1	3	yes	—
60	C₂×D₁₂₄	C₂³	1	12	yes	complemented
61	C₂×C₄×D₃₁	—	2	3	no	—
62	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
63	#92482	C₂³	1	4	yes	complemented
64	#92481	C₂³	1	12	yes	complemented
65	#92480	C₂³	1	12	yes	complemented
66	#92479	C₂³	1	12	yes	complemented
67	#92479	C₂³	1	24	yes	complemented
68	#92474	C₂³	1	12	yes	complemented
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃₁⋊C₈	C₂³	1	3	yes	—
71	C₂×C₃₁⋊C₈	C₂³	1	12	yes	complemented
72	C₂³×C₃₁	—	2	3	no	—
73	C₂³×C₃₁	—	2	12	no	—
74	C₂³×C₃₁	—	8	6	no	—
75	C₂³×C₃₁	C₂×D₄	1	3	yes	—
76	C₂²×D₃₁	—	2	3	no	—
77	C₂²×D₃₁	—	2	12	no	—
78	C₂²×D₃₁	—	8	6	no	—
79	Q₈×C₃₁	—	2	1	no	—
80	Q₈×C₃₁	C₂×D₄	1	4	yes	—
81	Q₈×C₃₁	C₂⁴	1	2	yes	—
82	D₄×C₃₁	—	2	12	no	—
83	D₄×C₃₁	—	2	3	no	—
84	D₄×C₃₁	—	2	12	no	—
85	D₄×C₃₁	C₂×D₄	1	12	yes	—
86	D₄×C₃₁	C₂⁴	1	6	yes	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	12	no	—
89	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
90	C₂×C₄×C₃₁	C₂×D₄	1	6	yes	—
91	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
93	C₂²⋊D₃₁	—	2	12	no	—
94	C₂×Dic₃₁	—	2	1	no	—
95	D₁₂₄	—	2	3	no	—
96	D₁₂₄	—	2	12	no	—
97	D₁₂₄	C₂⁴	1	6	yes	—
98	C₄×D₃₁	—	2	12	no	—
99	Dic₆₂	—	2	1	no	—
100	Dic₆₂	C₂⁴	1	2	yes	—
101	C₃₁⋊C₈	C₂⁴	1	2	yes	—
102	C₃₁⋊C₈	C₂⁴	1	6	yes	—
103	C₂×#576	—	31	1	no	maximal, Sylow
104	C₂²×C₃₁	—	2	6	no	—
105	C₂²×C₃₁	—	2	6	no	—
106	C₂²×C₃₁	—	4	6	no	—
107	C₂²×C₃₁	—	8	12	no	—
108	C₂²×C₃₁	C₂²×D₄	1	6	yes	—
109	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×D₃₁	—	2	6	no	—
111	C₂×D₃₁	—	4	6	no	—
112	C₂×D₃₁	—	8	12	no	—
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₃₁	C₂²×D₄	1	6	yes	—
116	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
117	Dic₃₁	—	2	2	no	—
118	C₂×H(5,𝔽₂)	—	31	1	no	—
119	C₂×H(5,𝔽₂)	D₃₁	1	1	yes	characteristic, complemented
120	#576	—	31	16	no	—
121	C₂×AGL(1,ℤ/8)	—	31	3	no	—
122	C₂×AGL(1,ℤ/8)	—	31	3	no	—
123	C₂×#135	—	31	3	no	—
124	C₂²×D₈	—	31	3	no	—
125	C₂×#131	—	31	1	no	—
126	C₂×C₃₁	—	2	6	no	—
127	C₂×C₃₁	—	4	6	no	—
128	C₂×C₃₁	#576	1	2	yes	complemented
129	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
130	D₃₁	—	4	6	no	—
131	H(5,𝔽₂)	—	2	4	no	—
132	H(5,𝔽₂)	—	31	8	no	—
133	H(5,𝔽₂)	—	62	4	no	—
134	H(5,𝔽₂)	C₂×D₃₁	1	8	yes	complemented
135	C₂×#41	—	2	1	no	—
136	C₂×#41	—	31	3	no	—
137	C₂×#41	—	62	1	no	—
138	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
139	C₂×#41	C₂×D₃₁	1	3	yes	complemented
140	C₂²×D₄	—	2	3	no	—
141	C₂²×D₄	—	31	3	no	—
142	C₂²×D₄	—	62	3	no	—
143	C₂²×D₄	C₂×D₃₁	1	3	yes	complemented
144	AGL(1,ℤ/8)	—	31	24	no	—
145	AGL(1,ℤ/8)	—	31	24	no	—
146	#135	—	31	24	no	—
147	C₂×Q₁₆	—	31	1	no	—
148	C₂×SD₁₆	—	31	3	no	—
149	C₂×SD₁₆	—	31	3	no	—
150	C₂×D₈	—	31	3	no	—
151	C₂×D₈	—	31	6	no	—
152	C₂×D₈	—	31	24	no	—
153	#131	—	31	8	no	—
154	C₂×M₁₆	—	31	3	no	—
155	C₂²×C₈	—	31	3	no	—
156	C₃₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	62	3	no	—
159	#41	—	2	8	no	—
160	#41	—	2	6	no	—
161	#41	—	31	12	no	—
162	#41	—	62	8	no	—
163	#41	—	62	6	no	—
164	#41	C₂²⋊D₃₁	1	4	yes	—
165	#41	C₂²×D₃₁	1	4	yes	—
166	#41	C₂²×D₃₁	1	12	yes	complemented
167	C₂×Q₈	—	31	1	no	—
168	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
170	C₂×D₄	—	2	3	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	31	3	no	—
175	C₂×D₄	—	31	12	no	—
176	C₂×D₄	—	62	3	no	—
177	C₂×D₄	—	62	24	no	—
178	C₂×D₄	—	62	3	no	—
179	C₂×D₄	—	62	6	no	—
180	C₂×D₄	C₂²⋊D₃₁	1	3	yes	—
181	C₂×D₄	C₂²×D₃₁	1	3	yes	—
182	C₂×D₄	C₂²×D₃₁	1	12	yes	complemented
183	C₂²×C₄	—	2	3	no	—
184	C₂²×C₄	—	62	3	no	—
185	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
186	Q₁₆	—	31	4	no	—
187	SD₁₆	—	31	12	no	—
188	SD₁₆	—	31	12	no	—
189	D₈	—	31	12	no	—
190	D₈	—	31	24	no	—
191	M₁₆	—	31	12	no	—
192	C₂×C₈	—	31	1	no	—
193	C₂×C₈	—	31	3	no	—
194	C₂×C₈	—	31	12	no	—
195	C₂³	—	2	3	no	—
196	C₂³	—	2	12	no	—
197	C₂³	—	8	6	no	—
198	C₂³	—	62	3	no	—
199	C₂³	—	62	12	no	—
200	C₂³	—	248	6	no	—
201	C₂³	C₂×C₂²⋊D₃₁	1	3	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	31	2	no	—
204	Q₈	—	62	1	no	—
205	Q₈	C₂×C₂²⋊D₃₁	1	4	yes	—
206	Q₈	C₂³×D₃₁	1	2	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	2	3	no	—
209	D₄	—	2	12	no	—
210	D₄	—	31	6	no	—
211	D₄	—	62	12	no	—
212	D₄	—	62	12	no	—
213	D₄	—	62	3	no	—
214	D₄	C₂×C₂²⋊D₃₁	1	12	yes	—
215	D₄	C₂³×D₃₁	1	6	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	62	1	no	—
219	C₂×C₄	—	62	12	no	—
220	C₂×C₄	C₂×C₂²⋊D₃₁	1	6	yes	—
221	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
222	C₂×C₄	C₂³×D₃₁	1	6	yes	—
223	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
224	C₈	—	31	2	no	—
225	C₈	—	31	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	2	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	62	6	no	—
231	C₂²	—	124	6	no	—
232	C₂²	—	248	12	no	—
233	C₂²	C₂²×C₂²⋊D₃₁	1	6	yes	—
234	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	62	2	no	—
237	C₄	C₂²×C₂²⋊D₃₁	1	6	yes	—
238	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
239	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	124	6	no	—
243	C₂	#14521852	1	2	yes	direct product factor
244	C₂	C₂³×C₂²⋊D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂×C₃₁×H(5,𝔽₂)	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₃₁
hypocenter	C₃₁	C₂×#576
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#11758566	C₂	semidirect
2. #14521852	C₂	direct (N × H)
3. C₂×#11758509	C₂	semidirect
4. C₂×#11758526	C₂	semidirect
5. C₂×#11758527	C₂	semidirect
6. C₂²×#92479	C₂	semidirect
7. C₂×#11758525	C₂	semidirect
8. C₂×C₃₁×H(5,𝔽₂)	C₂	semidirect
10. C₃₁×H(5,𝔽₂)	C₂²	semidirect
13. C₂×#41×C₃₁	C₂²	semidirect
15. C₂²×D₄×C₃₁	C₂²	semidirect
17. #11758566	C₂²	semidirect
20. C₂×#92495	C₂²	semidirect
21. C₂²×D₁₂₄	C₂²	semidirect
22. #11758527	C₂²	semidirect
23. #11758526	C₂²	semidirect
24. #11758525	C₂²	semidirect
25. C₂×#92482	C₂²	semidirect
26. C₂×#92481	C₂²	semidirect
27. C₂×#92480	C₂²	semidirect
28. #11758509	C₂²	semidirect
29. C₂×#92479	C₂²	semidirect
30. C₂×#92479	C₂²	semidirect
31. C₂×#92479	C₂²	semidirect
32. C₂×#92474	C₂²	semidirect
33. C₂²×C₃₁⋊C₈	C₂²	semidirect
40. #41×C₃₁	C₂³	semidirect
49. C₂×D₄×C₃₁	C₂³	semidirect
56. #92495	C₂³	semidirect
60. C₂×D₁₂₄	C₂³	semidirect
63. #92482	C₂³	semidirect
64. #92481	C₂³	semidirect
65. #92480	C₂³	semidirect
66. #92479	C₂³	semidirect
67. #92479	C₂³	semidirect
68. #92474	C₂³	semidirect
71. C₂×C₃₁⋊C₈	C₂³	semidirect
119. C₂×H(5,𝔽₂)	D₃₁	semidirect
128. C₂×C₃₁	#576	semidirect
134. H(5,𝔽₂)	C₂×D₃₁	semidirect
139. C₂×#41	C₂×D₃₁	semidirect
143. C₂²×D₄	C₂×D₃₁	semidirect
156. C₃₁	C₂×#576	semidirect
166. #41	C₂²×D₃₁	semidirect
182. C₂×D₄	C₂²×D₃₁	semidirect
243. C₂	#14521852	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11758566	2
#14521852	2
C₂×#11758509	2
C₂×#11758526	2
C₂×#11758527	2
C₂²×#92479	2
C₂×#11758525	2
C₂×C₃₁×H(5,𝔽₂)	2
C₂×#576	31

Subgroups — C2×#14521852 #16860462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521852 #16860462