Subgroups — #16854377

Order 3968.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521110	C₂	1	4	yes	maximal, complemented
2	#14521106	C₂	1	2	yes	maximal, complemented
3	#14521102	C₂	1	2	yes	maximal, complemented
4	#14521073	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521016	C₂	1	2	yes	maximal
7	#14521005	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
12	C₄×D₄×C₃₁	C₄	1	2	yes	—
13	C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
14	C₄×D₄×C₃₁	C₄	1	2	yes	—
15	C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
16	C₃₁×#117	C₂²	1	2	yes	—
17	C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×#92474	C₂²	1	2	yes	—
21	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
22	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
23	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
24	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
28	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
29	#11758391	C₂²	1	4	yes	—
30	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
31	#41×C₃₁	—	2	2	no	—
32	#41×C₃₁	—	2	2	no	—
33	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₄²×C₃₁	C₂×C₄	1	2	yes	—
50	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₃₁	C₂×C₄	1	2	yes	—
52	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₃₁	C₂³	1	2	yes	—
54	C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	#92474	—	2	4	no	—
57	C₂×C₃₁⋊C₈	—	2	4	no	—
58	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
59	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
60	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
61	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	Q₈×C₃₁	—	2	2	no	—
64	D₄×C₃₁	—	2	4	no	—
65	D₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	4	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	4	1	no	—
71	C₂×C₄×C₃₁	—	4	2	no	—
72	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	#41	1	2	yes	—
81	C₂×C₄×C₃₁	#41	1	2	yes	—
82	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₃₁⋊C₈	—	2	2	no	—
84	C₃₁⋊C₈	—	2	2	no	—
85	C₃₁⋊C₈	—	2	4	no	—
86	#2925	—	31	1	no	maximal, Sylow
87	C₂²×C₃₁	—	2	1	no	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	—	2	2	no	—
90	C₂²×C₃₁	—	2	2	no	—
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	#117	1	2	yes	—
93	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	—	2	2	no	—
97	C₄×C₃₁	—	2	1	no	—
98	C₄×C₃₁	—	2	1	no	—
99	C₄×C₃₁	—	2	2	no	—
100	C₄×C₃₁	#117	1	2	yes	—
101	C₄×C₃₁	#131	1	2	yes	—
102	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
103	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
104	C₄×#41	D₃₁	1	1	yes	characteristic
105	#433	—	31	4	no	—
106	#432	—	31	2	no	—
107	#431	—	31	2	no	—
108	#423	—	31	1	no	—
109	C₂×C₄⋊C₈	—	31	1	no	—
110	#408	—	31	2	no	—
111	#405	—	31	2	no	—
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	—	4	2	no	—
114	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
115	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
116	C₂×C₃₁	#568	1	1	yes	characteristic
117	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
118	C₂×M₁₆	—	31	2	no	—
119	C₂²×C₈	—	31	2	no	—
120	C₄×Q₈	Dic₃₁	1	1	yes	characteristic
121	C₄×Q₈	Dic₃₁	1	1	yes	characteristic
122	C₄×D₄	Dic₃₁	1	2	yes	—
123	C₄×D₄	Dic₃₁	1	1	yes	characteristic
124	C₄×D₄	Dic₃₁	1	2	yes	—
125	C₄×D₄	Dic₃₁	1	1	yes	characteristic
126	#117	C₂×D₃₁	1	1	yes	characteristic
127	#117	C₂×D₃₁	1	2	yes	—
128	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₃₁	1	2	yes	—
130	C₄⋊C₈	—	31	1	no	—
131	C₄⋊C₈	—	31	1	no	—
132	C₄⋊C₈	—	31	2	no	—
133	C₄⋊C₈	—	31	2	no	—
134	C₄⋊C₈	—	31	2	no	—
135	C₂²⋊C₈	—	31	4	no	—
136	C₂²⋊C₈	—	31	4	no	—
137	#97	—	31	4	no	—
138	C₄×C₈	—	31	4	no	—
139	C₃₁	#2925	1	1	yes	characteristic, complemented, Sylow
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₃₁	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
144	C₂×D₄	C₂×Dic₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
146	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
148	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
150	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
151	M₁₆	—	62	4	no	—
152	C₂×C₈	—	31	2	no	—
153	C₂×C₈	—	31	4	no	—
154	C₂×C₈	—	31	2	no	—
155	C₂×C₈	—	62	4	no	—
156	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₃₁	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₃₁	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
163	C₄²	C₂×Dic₃₁	1	2	yes	—
164	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
165	C₄²	C₂×Dic₃₁	1	2	yes	—
166	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
167	C₄²	C₂²×D₃₁	1	2	yes	—
168	C₄²	C₂²×D₃₁	1	1	yes	characteristic
169	C₄²	C₂²×D₃₁	1	1	yes	characteristic
170	C₂³	C₂²×Dic₃₁	1	2	yes	—
171	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#92495	1	2	yes	—
182	C₂×C₄	#92495	1	2	yes	—
183	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
185	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
186	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
188	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
191	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
192	C₈	—	62	2	no	—
193	C₈	—	62	2	no	—
194	C₈	—	62	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#11758503	1	2	yes	—
201	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#11758503	1	2	yes	—
209	C₄	#11758525	1	2	yes	—
210	C₄	C₂×#92495	1	1	yes	characteristic
211	C₄	C₂×#92495	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#11758503	1	1	yes	characteristic
215	C₂	C₂×#11758525	1	1	yes	characteristic
216	C₂	#14521816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₄×#41×C₃₁	C₂
prime radical	C₂×#41×C₃₁	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×#41	D₃₁
hypocenter	C₃₁	#2925
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14521110	C₂	semidirect
2. #14521106	C₂	semidirect
3. #14521102	C₂	semidirect
4. #14521073	C₂	semidirect
5. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
7. #14521005	C₂	semidirect
139. C₃₁	#2925	semidirect

Maximal subgroups

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

subgroup	index
#14521110	2
#14521106	2
#14521102	2
#14521073	2
C₂×C₄⋊(C₃₁⋊C₈)	2
#14521016	2
#14521005	2
C₄×#41×C₃₁	2
#2925	31