Subgroups — #16848621

Order 3968.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₃₁	C₂	1	2	yes	maximal
2	#14520948	C₂	1	2	yes	maximal, complemented
3	#14520924	C₂	1	4	yes	maximal, complemented
4	#14520915	C₂	1	1	yes	maximal, characteristic, complemented
5	#14520909	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁⋊#95	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
11	C₂³×Dic₃₁	C₂²	1	4	yes	—
12	C₄×D₄×C₃₁	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
17	D₄×Dic₃₁	—	2	8	no	—
18	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
21	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
22	C₃₁⋊#95	C₂²	1	1	yes	characteristic
23	C₃₁⋊#95	C₂²	1	2	yes	—
24	C₃₁⋊#95	C₂²	1	4	yes	—
25	C₃₁⋊#95	C₂²	1	1	yes	characteristic
26	C₃₁⋊#95	C₂²	1	2	yes	—
27	C₂⁴×C₃₁	C₂³	1	2	yes	—
28	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₃₁	D₄	1	4	yes	—
30	C₂×D₄×C₃₁	Q₈	1	4	yes	—
31	C₂²×C₄×C₃₁	—	4	4	no	—
32	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₃₁	—	2	8	no	—
38	C₂²×Dic₃₁	—	2	8	no	—
39	C₂²×Dic₃₁	—	4	4	no	—
40	C₂²×Dic₃₁	C₂³	1	4	yes	—
41	C₂²×Dic₃₁	C₂³	1	2	yes	—
42	C₂²×Dic₃₁	C₂³	1	2	yes	—
43	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	complemented
44	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
45	C₄²×C₃₁	C₂×C₄	1	4	yes	complemented
46	C₂²⋊Dic₃₁	—	2	8	no	—
47	C₄⋊Dic₃₁	—	2	4	no	—
48	C₄⋊Dic₃₁	—	2	4	no	—
49	C₄⋊Dic₃₁	—	4	4	no	—
50	C₄×Dic₃₁	—	2	4	no	—
51	C₂³×C₃₁	—	2	4	no	—
52	C₂³×C₃₁	—	2	4	no	—
53	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
54	C₂³×C₃₁	C₂×D₄	1	4	yes	—
55	C₂³×C₃₁	C₂×Q₈	1	4	yes	—
56	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
57	D₄×C₃₁	C₄⋊C₄	1	16	yes	complemented
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	4	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	4	8	no	—
62	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
68	C₂×Dic₃₁	—	2	2	no	—
69	C₂×Dic₃₁	—	2	2	no	—
70	C₂×Dic₃₁	—	2	2	no	—
71	C₂×Dic₃₁	—	2	4	no	—
72	C₂×Dic₃₁	—	2	2	no	—
73	C₂×Dic₃₁	—	2	4	no	—
74	C₂×Dic₃₁	—	2	2	no	—
75	C₂×Dic₃₁	—	2	4	no	—
76	C₂×Dic₃₁	—	4	16	no	—
77	C₂×Dic₃₁	#41	1	4	yes	—
78	#2349	—	31	1	no	maximal, Sylow
79	C₂²×C₃₁	—	2	4	no	—
80	C₂²×C₃₁	—	2	4	no	—
81	C₂²×C₃₁	—	2	8	no	—
82	C₂²×C₃₁	—	2	8	no	—
83	C₂²×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
84	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
87	C₂²×C₃₁	C₂×#41	1	2	yes	—
88	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
89	C₄×C₃₁	—	2	2	no	—
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	2	4	no	—
92	C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
93	Dic₃₁	—	2	4	no	—
94	Dic₃₁	—	4	2	no	—
95	Dic₃₁	—	4	4	no	—
96	C₂×C₄×D₄	—	31	2	no	—
97	C₂×C₄×D₄	D₃₁	1	1	yes	characteristic
98	#389	—	31	2	no	—
99	#385	—	31	4	no	—
100	#382	—	31	1	no	—
101	#380	—	31	2	no	—
102	C₄×C₄⋊C₄	—	31	1	no	—
103	C₂×#95	—	31	2	no	—
104	C₂×C₃₁	—	2	8	no	—
105	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
107	C₂×C₃₁	#518	1	1	yes	characteristic
108	C₂×C₃₁	#546	1	2	yes	—
109	C₂×C₃₁	#554	1	2	yes	—
110	C₂²×D₄	C₂×D₃₁	1	1	yes	characteristic
111	C₂³×C₄	—	31	4	no	—
112	C₂³×C₄	C₂×D₃₁	1	2	yes	—
113	C₄×D₄	—	62	8	no	—
114	C₄×D₄	Dic₃₁	1	8	yes	complemented
115	C₂×C₄⋊C₄	—	31	2	no	—
116	C₂×C₄⋊C₄	—	31	2	no	—
117	C₂×C₄⋊C₄	—	31	4	no	—
118	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	—	31	4	no	—
120	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
121	C₂×C₄²	—	31	2	no	—
122	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
123	#95	—	31	1	no	—
124	#95	—	31	2	no	—
125	#95	—	31	4	no	—
126	#95	—	31	1	no	—
127	#95	—	31	2	no	—
128	C₃₁	#2349	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	C₂²×D₃₁	1	2	yes	—
130	C₂×D₄	Dic₆₂	1	4	yes	—
131	C₂×D₄	D₁₂₄	1	4	yes	—
132	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
133	C₂²×C₄	—	4	4	no	—
134	C₂²×C₄	—	31	2	no	—
135	C₂²×C₄	—	31	4	no	—
136	C₂²×C₄	—	31	2	no	—
137	C₂²×C₄	—	62	8	no	—
138	C₂²×C₄	—	62	8	no	—
139	C₂²×C₄	—	124	4	no	—
140	C₂²×C₄	C₂×Dic₃₁	1	8	yes	complemented
141	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
144	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
145	C₄⋊C₄	—	62	4	no	—
146	C₄⋊C₄	—	62	4	no	—
147	C₄⋊C₄	—	124	4	no	—
148	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	complemented
149	C₂²⋊C₄	—	62	8	no	—
150	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	complemented
151	C₄²	—	62	4	no	—
152	C₄²	C₂×Dic₃₁	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×Dic₆₂	1	4	yes	—
156	C₂³	C₂×D₁₂₄	1	4	yes	—
157	C₂³	C₂²×Dic₃₁	1	4	yes	—
158	C₂³	C₂³×D₃₁	1	1	yes	characteristic
159	D₄	C₄⋊Dic₃₁	1	16	yes	complemented
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	31	4	no	—
165	C₂×C₄	—	62	2	no	—
166	C₂×C₄	—	62	2	no	—
167	C₂×C₄	—	62	2	no	—
168	C₂×C₄	—	62	4	no	—
169	C₂×C₄	—	62	2	no	—
170	C₂×C₄	—	62	4	no	—
171	C₂×C₄	—	62	2	no	—
172	C₂×C₄	—	62	4	no	—
173	C₂×C₄	—	124	16	no	—
174	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
175	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
176	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
177	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
178	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
179	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂×C₄⋊Dic₃₁	1	8	yes	—
185	C₂²	C₂²×Dic₆₂	1	1	yes	characteristic
186	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
187	C₂²	#11758561	1	2	yes	—
188	C₂²	#41×D₃₁	1	2	yes	—
189	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	62	4	no	—
194	C₄	—	124	4	no	—
195	C₄	—	124	2	no	—
196	C₄	C₂×C₄⋊Dic₃₁	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊Dic₃₁	1	1	yes	characteristic
199	C₂	#41×Dic₃₁	1	1	yes	characteristic
200	C₂	#14521524	1	1	yes	characteristic
201	C₂	#14521694	1	2	yes	—
202	C₂	#14521758	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
prime radical	C₂²×D₄×C₃₁	C₂²
hypercenter	C₂×C₄×D₄	D₃₁
hypocenter	C₃₁	#2349
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #14520948	C₂	semidirect
3. #14520924	C₂	semidirect
4. #14520915	C₂	semidirect
5. #14520909	C₂	semidirect
6. C₄×C₄⋊Dic₃₁	C₂	semidirect
7. C₂×C₃₁⋊#95	C₂	semidirect
12. C₄×D₄×C₃₁	C₄	semidirect
32. C₂²×C₄×C₃₁	C₂×C₄	semidirect
43. C₄⋊C₄×C₃₁	C₂×C₄	semidirect
44. C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
45. C₄²×C₃₁	C₂×C₄	semidirect
57. D₄×C₃₁	C₄⋊C₄	semidirect
114. C₄×D₄	Dic₃₁	semidirect
128. C₃₁	#2349	semidirect
140. C₂²×C₄	C₂×Dic₃₁	semidirect
148. C₄⋊C₄	C₂×Dic₃₁	semidirect
150. C₂²⋊C₄	C₂×Dic₃₁	semidirect
152. C₄²	C₂×Dic₃₁	semidirect
159. D₄	C₄⋊Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₃₁	2
#14520948	2
#14520924	2
#14520915	2
#14520909	2
C₄×C₄⋊Dic₃₁	2
C₂×C₃₁⋊#95	2
C₂×C₄×D₄×C₃₁	2
#2349	31