Subgroups — C₂×#14521860 #16860471

Order 3968.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758567	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521860	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11758528	C₂	1	3	yes	maximal, complemented
4	C₂×#11758509	C₂	1	3	yes	maximal, complemented
5	C₂×#11758527	C₂	1	3	yes	maximal, complemented
6	C₂²×#92480	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	3	yes	—
13	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₃₁	—	2	3	no	—
15	C₂²×D₄×C₃₁	C₂²	1	3	yes	—
16	#11758567	—	2	4	no	—
17	#11758567	C₂²	1	8	yes	complemented
18	C₂×Q₈×D₃₁	—	2	1	no	—
19	C₂×#92497	—	2	3	no	—
20	C₂×#92495	C₂²	1	3	yes	complemented
21	C₂²×Dic₆₂	C₂²	1	3	yes	complemented
22	#11758528	C₂²	1	24	yes	complemented
23	#11758527	C₂²	1	24	yes	complemented
24	#11758525	C₂²	1	8	yes	complemented
25	C₂×#92482	C₂²	1	3	yes	complemented
26	C₂×#92481	C₂²	1	1	yes	characteristic, complemented
27	C₂×#92480	C₂²	1	3	yes	complemented
28	C₂×#92480	C₂²	1	6	yes	complemented
29	C₂×#92480	C₂²	1	24	yes	complemented
30	#11758509	C₂²	1	24	yes	complemented
31	C₂×#92479	C₂²	1	3	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	#41×C₃₁	—	2	8	no	—
36	#41×C₃₁	—	2	6	no	—
37	#41×C₃₁	D₄	1	4	yes	—
38	#41×C₃₁	C₂³	1	12	yes	—
39	#41×C₃₁	C₂³	1	4	yes	—
40	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₃₁	—	2	3	no	—
43	C₂×D₄×C₃₁	—	2	24	no	—
44	C₂×D₄×C₃₁	—	2	3	no	—
45	C₂×D₄×C₃₁	—	2	6	no	—
46	C₂×D₄×C₃₁	D₄	1	3	yes	—
47	C₂×D₄×C₃₁	C₂³	1	3	yes	—
48	C₂×D₄×C₃₁	C₂³	1	12	yes	—
49	C₂²×C₄×C₃₁	—	2	3	no	—
50	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
51	C₂×C₂²⋊D₃₁	—	2	3	no	—
52	C₂²×Dic₃₁	—	2	3	no	—
53	Q₈×D₃₁	—	2	8	no	—
54	#92497	—	2	24	no	—
55	#92495	—	2	6	no	—
56	#92495	C₂³	1	12	yes	complemented
57	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₃₁	—	2	3	no	—
59	C₂×Dic₆₂	—	2	3	no	—
60	C₂×Dic₆₂	—	2	6	no	—
61	C₂×Dic₆₂	C₂³	1	3	yes	—
62	C₂×Dic₆₂	C₂³	1	12	yes	complemented
63	#92482	C₂³	1	12	yes	complemented
64	#92481	C₂³	1	4	yes	complemented
65	#92480	C₂³	1	24	yes	complemented
66	#92480	C₂³	1	12	yes	complemented
67	#92479	C₂³	1	12	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	1	no	—
77	Q₈×C₃₁	—	2	1	no	—
78	Q₈×C₃₁	C₂×D₄	1	4	yes	—
79	Q₈×C₃₁	C₂⁴	1	2	yes	—
80	D₄×C₃₁	—	2	12	no	—
81	D₄×C₃₁	—	2	3	no	—
82	D₄×C₃₁	—	2	12	no	—
83	D₄×C₃₁	C₂×D₄	1	12	yes	—
84	D₄×C₃₁	C₂⁴	1	6	yes	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	12	no	—
87	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
88	C₂×C₄×C₃₁	C₂×D₄	1	6	yes	—
89	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
91	C₂²⋊D₃₁	—	2	12	no	—
92	C₂×Dic₃₁	—	2	3	no	—
93	C₂×Dic₃₁	—	2	12	no	—
94	D₁₂₄	—	2	1	no	—
95	D₁₂₄	C₂⁴	1	2	yes	—
96	C₄×D₃₁	—	2	12	no	—
97	Dic₆₂	—	2	3	no	—
98	Dic₆₂	—	2	12	no	—
99	Dic₆₂	C₂⁴	1	6	yes	—
100	C₃₁⋊C₈	C₂⁴	1	2	yes	—
101	C₃₁⋊C₈	C₂⁴	1	6	yes	—
102	C₂×#577	—	31	1	no	maximal, Sylow
103	C₂²×C₃₁	—	2	6	no	—
104	C₂²×C₃₁	—	2	6	no	—
105	C₂²×C₃₁	—	4	6	no	—
106	C₂²×C₃₁	—	8	12	no	—
107	C₂²×C₃₁	C₂²×D₄	1	6	yes	—
108	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₃₁	—	2	2	no	—
110	C₂×D₃₁	—	4	2	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₃₁	C₂²×D₄	1	6	yes	—
114	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
115	Dic₃₁	—	2	6	no	—
116	C₂×2₋¹⁺⁴	—	31	1	no	—
117	C₂×H(5,𝔽₂)	D₃₁	1	1	yes	characteristic, complemented
118	#577	—	31	16	no	—
119	C₂×#137	—	31	3	no	—
120	C₂×AGL(1,ℤ/8)	—	31	3	no	—
121	C₂×#135	—	31	3	no	—
122	C₂²×SD₁₆	—	31	3	no	—
123	C₂×#131	—	31	1	no	—
124	C₂×C₃₁	—	2	6	no	—
125	C₂×C₃₁	—	4	6	no	—
126	C₂×C₃₁	#577	1	2	yes	complemented
127	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
128	D₃₁	—	4	2	no	—
129	2₋¹⁺⁴	—	31	8	no	—
130	2₋¹⁺⁴	—	62	4	no	—
131	H(5,𝔽₂)	—	2	4	no	—
132	H(5,𝔽₂)	C₂×D₃₁	1	8	yes	complemented
133	C₂×#41	—	2	1	no	—
134	C₂×#41	—	31	3	no	—
135	C₂×#41	—	62	3	no	—
136	C₂×#41	C₂×D₃₁	1	3	yes	—
137	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
138	C₂²×Q₈	—	31	3	no	—
139	C₂²×Q₈	—	62	1	no	—
140	C₂²×D₄	—	2	3	no	—
141	C₂²×D₄	C₂×D₃₁	1	3	yes	—
142	#137	—	31	24	no	—
143	AGL(1,ℤ/8)	—	31	24	no	—
144	#135	—	31	24	no	—
145	C₂×Q₁₆	—	31	3	no	—
146	C₂×SD₁₆	—	31	1	no	—
147	C₂×SD₁₆	—	31	3	no	—
148	C₂×SD₁₆	—	31	6	no	—
149	C₂×SD₁₆	—	31	24	no	—
150	C₂×D₈	—	31	3	no	—
151	#131	—	31	8	no	—
152	C₂×M₁₆	—	31	3	no	—
153	C₂²×C₈	—	31	3	no	—
154	C₃₁	C₂×#577	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	—	2	3	no	—
156	#41	—	2	8	no	—
157	#41	—	2	6	no	—
158	#41	—	31	12	no	—
159	#41	—	62	24	no	—
160	#41	—	62	6	no	—
161	#41	C₂²⋊D₃₁	1	4	yes	—
162	#41	C₂²×D₃₁	1	12	yes	—
163	#41	C₂²×D₃₁	1	4	yes	—
164	C₂×Q₈	—	31	3	no	—
165	C₂×Q₈	—	31	12	no	—
166	C₂×Q₈	—	62	8	no	—
167	C₂×Q₈	—	62	3	no	—
168	C₂×Q₈	—	62	6	no	—
169	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	3	no	—
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	31	1	no	—
176	C₂×D₄	—	62	3	no	—
177	C₂×D₄	C₂²⋊D₃₁	1	3	yes	—
178	C₂×D₄	C₂²×D₃₁	1	3	yes	—
179	C₂×D₄	C₂²×D₃₁	1	12	yes	—
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	62	3	no	—
182	C₂²×C₄	—	62	3	no	—
183	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
184	Q₁₆	—	31	12	no	—
185	SD₁₆	—	31	4	no	—
186	SD₁₆	—	31	12	no	—
187	SD₁₆	—	31	24	no	—
188	D₈	—	31	12	no	—
189	M₁₆	—	31	12	no	—
190	C₂×C₈	—	31	1	no	—
191	C₂×C₈	—	31	3	no	—
192	C₂×C₈	—	31	12	no	—
193	C₂³	—	2	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	8	6	no	—
196	C₂³	—	62	1	no	—
197	C₂³	C₂×C₂²⋊D₃₁	1	3	yes	—
198	Q₈	—	2	1	no	—
199	Q₈	—	31	6	no	—
200	Q₈	—	62	12	no	—
201	Q₈	—	62	3	no	—
202	Q₈	C₂×C₂²⋊D₃₁	1	4	yes	—
203	Q₈	C₂³×D₃₁	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	3	no	—
207	D₄	—	31	2	no	—
208	D₄	—	62	12	no	—
209	D₄	—	62	1	no	—
210	D₄	C₂×C₂²⋊D₃₁	1	12	yes	—
211	D₄	C₂³×D₃₁	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	62	12	no	—
215	C₂×C₄	—	62	3	no	—
216	C₂×C₄	—	62	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₃₁	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
219	C₂×C₄	C₂³×D₃₁	1	6	yes	—
220	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
221	C₈	—	31	2	no	—
222	C₈	—	31	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	8	12	no	—
227	C₂²	—	62	2	no	—
228	C₂²	—	124	2	no	—
229	C₂²	C₂²×C₂²⋊D₃₁	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	62	6	no	—
233	C₄	C₂²×C₂²⋊D₃₁	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
235	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	124	2	no	—
239	C₂	#14521860	1	2	yes	direct product factor
240	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×#577
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×#11758567	C₂	semidirect
2. #14521860	C₂	direct (N × H)
3. C₂×#11758528	C₂	semidirect
4. C₂×#11758509	C₂	semidirect
5. C₂×#11758527	C₂	semidirect
6. C₂²×#92480	C₂	semidirect
7. C₂×#11758525	C₂	semidirect
8. C₂×C₃₁×H(5,𝔽₂)	C₂	semidirect
10. C₃₁×H(5,𝔽₂)	C₂²	semidirect
17. #11758567	C₂²	semidirect
20. C₂×#92495	C₂²	semidirect
21. C₂²×Dic₆₂	C₂²	semidirect
22. #11758528	C₂²	semidirect
23. #11758527	C₂²	semidirect
24. #11758525	C₂²	semidirect
25. C₂×#92482	C₂²	semidirect
26. C₂×#92481	C₂²	semidirect
27. C₂×#92480	C₂²	semidirect
28. C₂×#92480	C₂²	semidirect
29. C₂×#92480	C₂²	semidirect
30. #11758509	C₂²	semidirect
31. C₂×#92479	C₂²	semidirect
32. C₂×#92474	C₂²	semidirect
33. C₂²×C₃₁⋊C₈	C₂²	semidirect
56. #92495	C₂³	semidirect
62. C₂×Dic₆₂	C₂³	semidirect
63. #92482	C₂³	semidirect
64. #92481	C₂³	semidirect
65. #92480	C₂³	semidirect
66. #92480	C₂³	semidirect
67. #92479	C₂³	semidirect
68. #92474	C₂³	semidirect
71. C₂×C₃₁⋊C₈	C₂³	semidirect
117. C₂×H(5,𝔽₂)	D₃₁	semidirect
126. C₂×C₃₁	#577	semidirect
132. H(5,𝔽₂)	C₂×D₃₁	semidirect
154. C₃₁	C₂×#577	semidirect
239. C₂	#14521860	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×#11758567	2
#14521860	2
C₂×#11758528	2
C₂×#11758509	2
C₂×#11758527	2
C₂²×#92480	2
C₂×#11758525	2
C₂×C₃₁×H(5,𝔽₂)	2
C₂×#577	31

Subgroups — C2×#14521860 #16860471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521860 #16860471