Subgroups — #16854276

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758478	C₂	1	2	yes	maximal
2	C₄×#92495	C₂	1	1	yes	maximal, characteristic
3	#14521075	C₂	1	2	yes	maximal, complemented
4	#14521074	C₂	1	4	yes	maximal, complemented
5	#14521073	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521071	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₈×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#92495	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₃₁	C₄	1	2	yes	—
13	C₄×C₂²⋊D₃₁	C₄	1	2	yes	—
14	#11758503	C₂²	1	1	yes	characteristic
15	C₂×#92474	C₂²	1	2	yes	—
16	#11758478	—	2	8	no	—
17	C₂×#92469	C₂²	1	4	yes	—
18	C₂×C₈×D₃₁	C₂²	1	4	yes	—
19	C₄×D₁₂₄	C₄	1	1	yes	characteristic
20	C₄×D₁₂₄	C₄	1	1	yes	characteristic
21	#11758449	C₂²	1	2	yes	—
22	C₄²×D₃₁	C₂²	1	2	yes	—
23	C₄×Dic₆₂	C₄	1	1	yes	characteristic
24	C₄×Dic₆₂	C₄	1	1	yes	characteristic
25	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
28	Dic₃₁⋊C₈	C₂²	1	4	yes	—
29	Dic₃₁⋊C₈	C₂²	1	4	yes	—
30	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
31	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
36	#92495	D₄	1	4	yes	—
37	#92495	Q₈	1	4	yes	—
38	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₃₁	C₂³	1	2	yes	—
42	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	4	no	—
44	C₂×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂×C₈×C₃₁	C₂³	1	2	yes	—
46	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
51	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
52	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
55	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
58	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
59	C₄×Dic₃₁	C₂³	1	2	yes	—
60	#92474	—	2	4	no	—
61	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
62	#92469	—	2	8	no	—
63	C₈×D₃₁	—	2	8	no	—
64	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	4	1	no	—
68	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₃₁	C₄⋊C₄	1	8	yes	—
77	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
80	D₁₂₄	C₄⋊C₄	1	4	yes	—
81	C₄×D₃₁	—	2	4	no	—
82	C₄×D₃₁	—	4	2	no	—
83	C₄×D₃₁	C₂×D₄	1	4	yes	—
84	C₄×D₃₁	C₂×Q₈	1	4	yes	—
85	Dic₆₂	C₄⋊C₄	1	4	yes	—
86	C₈×C₃₁	—	2	2	no	—
87	C₈×C₃₁	—	2	2	no	—
88	C₃₁⋊C₈	—	2	4	no	—
89	#2906	—	31	1	no	maximal, Sylow
90	C₂²×C₃₁	—	2	1	no	—
91	C₂²×C₃₁	—	2	1	no	—
92	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₃₁	—	2	2	no	—
95	C₂×D₃₁	—	2	2	no	—
96	C₂×D₃₁	C₂×C₄⋊C₄	1	4	yes	—
97	C₄×C₃₁	—	2	1	no	—
98	C₄×C₃₁	—	2	1	no	—
99	C₄×C₃₁	—	2	1	no	—
100	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₄×C₃₁	#131	1	2	yes	complemented
102	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
104	Dic₃₁	—	2	2	no	—
105	Dic₃₁	—	2	2	no	—
106	Dic₃₁	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×#131	—	31	2	no	—
108	C₄×#41	—	31	1	no	—
109	#424	—	31	2	no	—
110	#424	—	31	4	no	—
111	#423	—	31	1	no	—
112	#423	—	31	2	no	—
113	C₂×C₄⋊C₈	—	31	2	no	—
114	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic, complemented
115	C₂×C₃₁	—	2	2	no	—
116	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
118	C₂×C₃₁	#568	1	1	yes	characteristic
119	D₃₁	—	2	4	no	—
120	C₂×#41	—	31	1	no	—
121	#131	—	62	8	no	—
122	C₂×M₁₆	—	31	2	no	—
123	C₂×M₁₆	—	31	4	no	—
124	C₂²×C₈	—	31	4	no	—
125	C₂²×C₈	C₂×D₃₁	1	2	yes	—
126	C₄×Q₈	—	31	1	no	—
127	C₄×Q₈	—	31	1	no	—
128	C₄×D₄	—	31	2	no	—
129	C₄×D₄	—	31	2	no	—
130	C₄×D₄	—	31	1	no	—
131	C₄×D₄	—	31	1	no	—
132	#117	—	31	1	no	—
133	#117	—	31	2	no	—
134	C₂×C₄²	—	31	2	no	—
135	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
136	C₄⋊C₈	—	31	4	no	—
137	C₄⋊C₈	—	31	4	no	—
138	C₄⋊C₈	—	31	2	no	—
139	C₄⋊C₈	—	31	2	no	—
140	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
141	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
142	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
143	C₃₁	#2906	1	1	yes	characteristic, complemented, Sylow
144	#41	—	31	4	no	—
145	#41	—	31	4	no	—
146	C₂×Q₈	—	31	1	no	—
147	C₂×D₄	—	31	2	no	—
148	C₂×D₄	—	31	1	no	—
149	C₂²×C₄	—	31	2	no	—
150	C₂²×C₄	—	31	2	no	—
151	C₂²×C₄	—	31	2	no	—
152	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
155	M₁₆	—	62	8	no	—
156	M₁₆	—	62	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	31	4	no	—
159	C₂×C₈	—	62	8	no	—
160	C₂×C₈	C₂²×D₃₁	1	2	yes	—
161	C₂×C₈	C₂²×D₃₁	1	2	yes	—
162	C₄⋊C₄	—	31	2	no	—
163	C₄⋊C₄	—	31	2	no	—
164	C₄⋊C₄	—	31	1	no	—
165	C₄⋊C₄	—	31	1	no	—
166	C₂²⋊C₄	—	31	2	no	—
167	C₂²⋊C₄	—	31	2	no	—
168	C₂²⋊C₄	—	31	2	no	—
169	C₄²	—	31	2	no	—
170	C₄²	—	31	2	no	—
171	C₄²	—	31	2	no	—
172	C₄²	C₄×D₃₁	1	1	yes	characteristic
173	C₄²	C₄×D₃₁	1	1	yes	characteristic
174	C₄²	C₂²×D₃₁	1	1	yes	characteristic
175	C₄²	C₂²×D₃₁	1	1	yes	characteristic
176	C₂³	—	31	2	no	—
177	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
178	Q₈	—	31	4	no	—
179	D₄	—	31	8	no	—
180	D₄	—	31	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	31	2	no	—
184	C₂×C₄	—	31	4	no	—
185	C₂×C₄	—	31	4	no	—
186	C₂×C₄	—	31	2	no	—
187	C₂×C₄	—	31	2	no	—
188	C₂×C₄	—	62	4	no	—
189	C₂×C₄	—	124	2	no	—
190	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
195	C₂×C₄	D₄×D₃₁	1	2	yes	—
196	C₂×C₄	Q₈×D₃₁	1	2	yes	—
197	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	62	4	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	31	4	no	—
204	C₂²	—	62	2	no	—
205	C₂²	—	62	2	no	—
206	C₂²	C₄⋊C₄×D₃₁	1	2	yes	—
207	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	31	4	no	—
212	C₄	—	62	2	no	—
213	C₄	—	62	2	no	—
214	C₄	C₄⋊C₄×D₃₁	1	2	yes	—
215	C₄	#11758478	1	2	yes	complemented
216	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	62	4	no	—
220	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic
221	C₂	C₂×#11758478	1	1	yes	characteristic
222	C₂	#14521815	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	#2906
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #14521075	C₂	semidirect
4. #14521074	C₂	semidirect
5. #14521073	C₂	semidirect
6. #14521071	C₂	semidirect
7. C₄⋊C₈×D₃₁	C₂	semidirect
8. C₂×C₃₁×C₄⋊C₈	C₂	semidirect
101. C₄×C₃₁	#131	semidirect
114. C₂×C₄⋊C₈	D₃₁	semidirect
143. C₃₁	#2906	semidirect
215. C₄	#11758478	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758478	2
C₄×#92495	2
#14521075	2
#14521074	2
#14521073	2
#14521071	2
C₄⋊C₈×D₃₁	2
C₂×C₃₁×C₄⋊C₈	2
#2906	31