Subgroups — #16854376

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	C₄×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521109	C₂	1	4	yes	maximal, complemented
3	#14521105	C₂	1	2	yes	maximal, complemented
4	#14521101	C₂	1	2	yes	maximal, complemented
5	#14521073	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521015	C₂	1	2	yes	maximal
7	#14521004	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	C₄×D₁₂₄	C₄	1	1	yes	characteristic
17	C₄×D₁₂₄	C₄	1	1	yes	characteristic
18	#11758449	C₂²	1	2	yes	—
19	C₄²×D₃₁	C₂²	1	2	yes	—
20	C₄×Dic₆₂	C₄	1	1	yes	characteristic
21	C₄×Dic₆₂	C₄	1	1	yes	characteristic
22	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
26	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
27	#11758403	C₂²	1	4	yes	—
28	C₈×Dic₃₁	C₂²	1	4	yes	—
29	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
30	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
31	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
35	#92495	—	2	2	no	—
36	#92495	—	2	2	no	—
37	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₃₁	C₂³	1	2	yes	—
41	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₃₁	—	2	4	no	—
43	C₂×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂×C₈×C₃₁	C₂³	1	2	yes	—
45	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
50	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
51	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
54	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
58	C₄×Dic₃₁	C₂³	1	2	yes	—
59	#92474	—	2	4	no	—
60	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
61	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	4	1	no	—
65	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#41	1	2	yes	—
71	C₂×C₄×C₃₁	#41	1	2	yes	—
72	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₃₁	—	2	4	no	—
74	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
77	D₁₂₄	—	2	2	no	—
78	C₄×D₃₁	—	2	4	no	—
79	C₄×D₃₁	—	2	2	no	—
80	C₄×D₃₁	—	2	2	no	—
81	C₄×D₃₁	—	4	2	no	—
82	Dic₆₂	—	2	2	no	—
83	C₈×C₃₁	—	2	2	no	—
84	C₈×C₃₁	—	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	#117	1	2	yes	—
90	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×D₃₁	—	2	2	no	—
92	C₂×D₃₁	—	2	2	no	—
93	C₂×D₃₁	—	2	2	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	—	2	1	no	—
97	C₄×C₃₁	#117	1	2	yes	—
98	C₄×C₃₁	#131	1	2	yes	—
99	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
100	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
101	Dic₃₁	—	2	2	no	—
102	Dic₃₁	—	2	2	no	—
103	Dic₃₁	—	2	2	no	—
104	C₄×#41	—	31	1	no	—
105	#433	—	31	4	no	—
106	#432	—	31	2	no	—
107	#431	—	31	2	no	—
108	#423	—	31	1	no	—
109	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic, complemented
110	#408	—	31	2	no	—
111	#405	—	31	2	no	—
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
114	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
115	C₂×C₃₁	#568	1	1	yes	characteristic
116	D₃₁	—	4	2	no	—
117	C₂×#41	—	31	1	no	—
118	C₂×M₁₆	—	31	2	no	—
119	C₂²×C₈	C₂×D₃₁	1	2	yes	—
120	C₄×Q₈	—	31	1	no	—
121	C₄×Q₈	—	31	1	no	—
122	C₄×D₄	—	31	2	no	—
123	C₄×D₄	—	31	2	no	—
124	C₄×D₄	—	31	1	no	—
125	C₄×D₄	—	31	1	no	—
126	#117	—	31	1	no	—
127	#117	—	31	2	no	—
128	C₂×C₄²	—	31	2	no	—
129	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
130	C₄⋊C₈	—	31	2	no	—
131	C₄⋊C₈	—	31	2	no	—
132	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
133	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
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	—	62	2	no	—
141	#41	—	62	2	no	—
142	C₂×Q₈	—	31	1	no	—
143	C₂×D₄	—	31	2	no	—
144	C₂×D₄	—	31	1	no	—
145	C₂²×C₄	—	31	2	no	—
146	C₂²×C₄	—	31	2	no	—
147	C₂²×C₄	—	31	2	no	—
148	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
151	M₁₆	—	62	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	31	4	no	—
154	C₂×C₈	C₂²×D₃₁	1	2	yes	—
155	C₂×C₈	C₂²×D₃₁	1	2	yes	—
156	C₄⋊C₄	—	31	2	no	—
157	C₄⋊C₄	—	31	2	no	—
158	C₄⋊C₄	—	31	1	no	—
159	C₄⋊C₄	—	31	1	no	—
160	C₂²⋊C₄	—	31	2	no	—
161	C₂²⋊C₄	—	31	2	no	—
162	C₂²⋊C₄	—	31	2	no	—
163	C₄²	—	31	2	no	—
164	C₄²	—	31	2	no	—
165	C₄²	—	31	2	no	—
166	C₄²	C₄×D₃₁	1	1	yes	characteristic
167	C₄²	C₄×D₃₁	1	1	yes	characteristic
168	C₄²	C₂²×D₃₁	1	1	yes	characteristic
169	C₄²	C₂²×D₃₁	1	1	yes	characteristic
170	C₂³	—	31	2	no	—
171	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
172	Q₈	—	62	2	no	—
173	D₄	—	62	4	no	—
174	D₄	—	62	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	31	2	no	—
178	C₂×C₄	—	31	2	no	—
179	C₂×C₄	—	31	2	no	—
180	C₂×C₄	—	62	4	no	—
181	C₂×C₄	—	62	2	no	—
182	C₂×C₄	—	62	2	no	—
183	C₂×C₄	—	124	2	no	—
184	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
189	C₂×C₄	#92497	1	2	yes	—
190	C₂×C₄	#92499	1	2	yes	—
191	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	62	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	62	2	no	—
198	C₂²	—	62	2	no	—
199	C₂²	—	62	2	no	—
200	C₂²	#11758469	1	2	yes	—
201	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	62	2	no	—
206	C₄	—	62	2	no	—
207	C₄	—	62	2	no	—
208	C₄	#11758469	1	2	yes	—
209	C₄	#11758478	1	2	yes	—
210	C₄	C₂×#92497	1	1	yes	characteristic
211	C₄	C₂×#92499	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	124	2	no	—
214	C₂	C₂×#11758469	1	1	yes	characteristic
215	C₂	C₂×#11758478	1	1	yes	characteristic
216	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	#2925
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
2. #14521109	C₂	semidirect
3. #14521105	C₂	semidirect
4. #14521101	C₂	semidirect
5. #14521073	C₂	semidirect
7. #14521004	C₂	semidirect
8. C₂×C₃₁×C₄⋊C₈	C₂	semidirect
109. C₂×C₄⋊C₈	D₃₁	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
C₄×#92495	2
#14521109	2
#14521105	2
#14521101	2
#14521073	2
#14521015	2
#14521004	2
C₂×C₃₁×C₄⋊C₈	2
#2925	31