Subgroups — C₂×#14520811 #16845424

Order 3968.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	—	2	1	no	—
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×Dic₃₁	—	2	1	no	—
8	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
9	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₃₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃₁	—	2	1	no	—
13	C₂×C₄⋊Dic₃₁	—	2	2	no	—
14	C₂×C₄⋊Dic₃₁	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
16	C₂²×C₃₁⋊C₈	—	2	2	no	—
17	C₂²×C₃₁⋊C₈	—	2	2	no	—
18	C₂²×C₃₁⋊C₈	C₄	1	4	yes	complemented
19	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
20	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	—	2	4	no	—
22	C₂²×C₄×C₃₁	—	2	2	no	—
23	C₂²×C₄×C₃₁	—	4	2	no	—
24	C₂²×C₄×C₃₁	—	4	2	no	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	D₄	1	2	yes	—
30	C₂²×C₄×C₃₁	D₄	1	2	yes	—
31	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃₁	—	2	4	no	—
34	C₂²×Dic₃₁	—	2	2	no	—
35	C₂²×Dic₃₁	—	4	2	no	—
36	C₂²×Dic₃₁	—	4	2	no	—
37	C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	complemented
40	C₄⋊Dic₃₁	—	2	4	no	—
41	C₄⋊Dic₃₁	—	2	2	no	—
42	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	complemented
43	C₂×C₃₁⋊C₈	—	2	8	no	—
44	C₂×C₃₁⋊C₈	—	2	2	no	—
45	C₂×C₃₁⋊C₈	—	2	2	no	—
46	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	complemented
47	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
50	C₂³×C₃₁	D₈	1	2	yes	—
51	C₂³×C₃₁	SD₁₆	1	2	yes	—
52	C₂³×C₃₁	SD₁₆	1	2	yes	—
53	C₂³×C₃₁	Q₁₆	1	2	yes	—
54	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	—	4	8	no	—
57	C₂×C₄×C₃₁	—	4	2	no	—
58	C₂×C₄×C₃₁	—	4	2	no	—
59	C₂×C₄×C₃₁	C₄²	1	4	yes	—
60	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
62	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
66	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
72	C₂×Dic₃₁	—	2	4	no	—
73	C₂×Dic₃₁	—	4	8	no	—
74	C₂×Dic₃₁	—	4	2	no	—
75	C₂×Dic₃₁	—	4	2	no	—
76	C₃₁⋊C₈	—	2	4	no	—
77	C₂×#340	—	31	1	no	maximal, Sylow
78	C₂²×C₃₁	#95	1	4	yes	—
79	C₂²×C₃₁	#102	1	4	yes	—
80	C₂²×C₃₁	#102	1	4	yes	—
81	C₂²×C₃₁	#103	1	4	yes	—
82	C₂²×C₃₁	#103	1	4	yes	—
83	C₂²×C₃₁	#106	1	4	yes	—
84	C₂²×C₃₁	#107	1	4	yes	—
85	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
92	C₄×C₃₁	—	4	4	no	—
93	C₄×C₃₁	#95	1	4	yes	—
94	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
95	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	Dic₃₁	—	4	4	no	—
99	C₂³×C₈	—	31	1	no	—
100	C₂²×C₄⋊C₄	—	31	1	no	—
101	C₂²×C₄⋊C₄	D₃₁	1	1	yes	characteristic
102	#340	—	31	4	no	—
103	C₂×C₃₁	#340	1	8	yes	complemented
104	C₂×C₃₁	C₂×#95	1	1	yes	characteristic
105	C₂×C₃₁	C₂×#102	1	1	yes	characteristic
106	C₂×C₃₁	C₂×#102	1	1	yes	characteristic
107	C₂×C₃₁	C₂×#103	1	1	yes	characteristic
108	C₂×C₃₁	C₂×#103	1	1	yes	characteristic
109	C₂×C₃₁	C₂×#106	1	1	yes	characteristic
110	C₂×C₃₁	C₂×#107	1	1	yes	characteristic
111	C₂³×C₄	—	2	1	no	—
112	C₂³×C₄	—	62	1	no	—
113	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
114	C₂²×C₈	—	31	4	no	—
115	C₂²×C₈	—	31	2	no	—
116	C₂²×C₈	—	62	2	no	—
117	C₂²×C₈	—	62	2	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	31	4	no	—
121	C₂×C₄⋊C₄	—	31	2	no	—
122	C₂×C₄⋊C₄	—	62	2	no	—
123	C₂×C₄⋊C₄	—	62	1	no	—
124	C₂×C₄⋊C₄	Dic₃₁	1	4	yes	complemented
125	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
126	C₃₁	C₂×#340	1	1	yes	characteristic, complemented, Sylow
127	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	62	4	no	—
133	C₂²×C₄	—	62	2	no	—
134	C₂²×C₄	—	124	2	no	—
135	C₂²×C₄	—	124	2	no	—
136	C₂²×C₄	Dic₆₂	1	2	yes	—
137	C₂²×C₄	C₄×D₃₁	1	2	yes	—
138	C₂²×C₄	C₄×D₃₁	1	2	yes	—
139	C₂²×C₄	D₁₂₄	1	2	yes	—
140	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
141	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
143	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
144	C₂×C₈	—	31	4	no	—
145	C₂×C₈	—	62	8	no	—
146	C₂×C₈	—	62	2	no	—
147	C₂×C₈	—	62	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	31	4	no	—
151	C₄⋊C₄	—	62	4	no	—
152	C₄⋊C₄	—	62	2	no	—
153	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	complemented
154	C₂³	Dic₃₁⋊C₄	1	2	yes	—
155	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
156	C₂³	#92479	1	2	yes	—
157	C₂³	#92480	1	2	yes	—
158	C₂³	#92481	1	2	yes	—
159	C₂³	#92482	1	2	yes	—
160	C₂³	C₂²⋊Dic₃₁	1	2	yes	—
161	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	62	4	no	—
167	C₂×C₄	—	124	8	no	—
168	C₂×C₄	—	124	2	no	—
169	C₂×C₄	—	124	2	no	—
170	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
171	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
172	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
173	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
174	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
175	C₂×C₄	(C₂×D₃₁)⋊C₄	1	4	yes	—
176	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
177	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
180	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
183	C₈	—	62	4	no	—
184	C₂²	#11758395	1	4	yes	—
185	C₂²	#11758396	1	4	yes	—
186	C₂²	#11758397	1	4	yes	—
187	C₂²	#11758398	1	4	yes	—
188	C₂²	C₃₁⋊#95	1	4	yes	—
189	C₂²	#11758420	1	4	yes	—
190	C₂²	#11758423	1	4	yes	—
191	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
192	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
193	C₂²	C₂×#92479	1	1	yes	characteristic
194	C₂²	C₂×#92480	1	1	yes	characteristic
195	C₂²	C₂×#92481	1	1	yes	characteristic
196	C₂²	C₂×#92482	1	1	yes	characteristic
197	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
198	C₄	—	4	4	no	—
199	C₄	—	124	4	no	—
200	C₄	C₃₁⋊#95	1	4	yes	—
201	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
202	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
203	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
204	C₄	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
205	C₂	#14520811	1	8	yes	direct product factor
206	C₂	C₂×C₃₁⋊#95	1	1	yes	characteristic
207	C₂	C₂×#11758397	1	1	yes	characteristic
208	C₂	C₂×#11758420	1	1	yes	characteristic
209	C₂	C₂×#11758398	1	1	yes	characteristic
210	C₂	C₂×#11758423	1	1	yes	characteristic
211	C₂	C₂×#11758396	1	1	yes	characteristic
212	C₂	C₂×#11758395	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×C₄⋊C₄×C₃₁	C₂
socle, prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₃₁
hypocenter	C₃₁	C₂×#340
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #14520811	C₂	direct (N × H)
10. C₂×C₄⋊C₄×C₃₁	C₄	semidirect
14. C₂×C₄⋊Dic₃₁	C₄	semidirect
18. C₂²×C₃₁⋊C₈	C₄	semidirect
39. C₄⋊C₄×C₃₁	C₂×C₄	semidirect
42. C₄⋊Dic₃₁	C₂×C₄	semidirect
46. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
103. C₂×C₃₁	#340	semidirect
124. C₂×C₄⋊C₄	Dic₃₁	semidirect
126. C₃₁	C₂×#340	semidirect
153. C₄⋊C₄	C₂×Dic₃₁	semidirect
205. C₂	#14520811	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃₁⋊C₈	2
C₂²×C₄⋊Dic₃₁	2
#14520811	2
C₂²×C₄⋊C₄×C₃₁	2
C₂×#340	31

Subgroups — C2×#14520811 #16845424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14520811 #16845424