Subgroups — C₂×#14521109 #16854336

Order 3968.

412 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758449	C₂	1	1	yes	maximal, characteristic
2	#14521109	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×(C₂×D₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758403	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₃₁)⋊C₄	C₄	1	1	yes	characteristic
14	C₂×(C₂×D₃₁)⋊C₄	C₄	1	1	yes	characteristic
15	C₂×Dic₃₁⋊C₄	C₄	1	1	yes	characteristic
16	C₂×Dic₃₁⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	#11758449	C₄	1	4	yes	—
21	#11758449	C₂²	1	4	yes	—
22	C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
23	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
24	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
25	#11758403	C₂²	1	4	yes	complemented
26	C₈×Dic₃₁	C₂²	1	4	yes	complemented
27	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
28	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃₁	—	2	2	no	—
36	C₂×C₄×D₃₁	—	2	2	no	—
37	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₃₁	C₂³	1	2	yes	—
39	C₂×C₈×C₃₁	—	2	2	no	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂×C₈×C₃₁	C₂³	1	2	yes	—
43	C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₄²×C₃₁	C₂³	1	2	yes	—
45	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
46	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
47	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
48	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
49	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
50	C₄×Dic₃₁	C₂³	1	2	yes	—
51	C₂×C₃₁⋊C₈	—	2	2	no	—
52	C₂×C₃₁⋊C₈	—	2	2	no	—
53	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
54	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
55	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₃₁	—	2	2	no	—
57	C₂²×D₃₁	—	2	2	no	—
58	C₂²×D₃₁	—	2	2	no	—
59	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	2	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	1	yes	characteristic
65	C₂×C₄×C₃₁	#41	1	2	yes	—
66	C₂×C₄×C₃₁	#41	1	2	yes	—
67	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₃₁	—	2	2	no	—
69	C₂×Dic₃₁	—	2	2	no	—
70	C₂×Dic₃₁	—	2	2	no	—
71	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
72	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
74	C₄×D₃₁	—	2	4	no	—
75	C₄×D₃₁	—	2	2	no	—
76	C₄×D₃₁	—	2	2	no	—
77	C₈×C₃₁	—	2	2	no	—
78	C₈×C₃₁	—	2	2	no	—
79	C₃₁⋊C₈	—	2	2	no	—
80	C₃₁⋊C₈	—	2	2	no	—
81	C₂×#433	—	31	1	no	maximal, Sylow
82	C₂²×C₃₁	#117	1	2	yes	—
83	C₂²×C₃₁	#131	1	2	yes	—
84	C₂²×C₃₁	#131	1	2	yes	—
85	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
86	C₂×D₃₁	—	2	2	no	—
87	C₂×D₃₁	—	2	2	no	—
88	C₂×D₃₁	—	2	2	no	—
89	C₂×D₃₁	—	4	4	no	—
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	2	2	no	—
92	C₄×C₃₁	#117	1	2	yes	—
93	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
94	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
95	Dic₃₁	—	2	2	no	—
96	Dic₃₁	—	2	2	no	—
97	Dic₃₁	—	2	2	no	—
98	C₂×#117	—	31	1	no	—
99	#433	—	31	8	no	—
100	C₂×C₄⋊C₈	—	31	1	no	—
101	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic, complemented
102	C₂×C₂²⋊C₈	—	31	1	no	—
103	C₂×C₂²⋊C₈	—	31	1	no	—
104	C₂×#97	—	31	1	no	—
105	C₂×C₄×C₈	—	31	1	no	—
106	C₂×C₃₁	#433	1	4	yes	complemented
107	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
108	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
109	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
110	D₃₁	—	4	2	no	—
111	C₂³×C₄	—	31	1	no	—
112	C₂²×C₈	—	31	1	no	—
113	C₂²×C₈	—	31	1	no	—
114	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
116	#117	—	31	4	no	—
117	#117	—	31	4	no	—
118	C₂×C₄⋊C₄	—	31	1	no	—
119	C₂×C₄⋊C₄	—	31	1	no	—
120	C₂×C₂²⋊C₄	—	31	1	no	—
121	C₂×C₂²⋊C₄	—	31	1	no	—
122	C₂×C₄²	—	31	1	no	—
123	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
124	C₄⋊C₈	—	31	4	no	—
125	C₄⋊C₈	C₂×D₃₁	1	4	yes	complemented
126	C₂²⋊C₈	—	31	4	no	—
127	C₂²⋊C₈	—	31	4	no	—
128	#97	—	31	4	no	—
129	C₄×C₈	—	31	4	no	—
130	C₃₁	C₂×#433	1	1	yes	characteristic, complemented, Sylow
131	C₂⁴	—	31	1	no	—
132	C₂²×C₄	—	31	1	no	—
133	C₂²×C₄	—	31	1	no	—
134	C₂²×C₄	—	31	1	no	—
135	C₂²×C₄	—	31	2	no	—
136	C₂²×C₄	—	31	2	no	—
137	C₂²×C₄	—	62	2	no	—
138	C₂²×C₄	—	62	2	no	—
139	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	31	2	no	—
145	C₂×C₈	—	31	2	no	—
146	C₂×C₈	—	62	2	no	—
147	C₂×C₈	—	62	2	no	—
148	C₂×C₈	C₂²×D₃₁	1	2	yes	—
149	C₂×C₈	C₂²×D₃₁	1	2	yes	—
150	C₄⋊C₄	—	31	4	no	—
151	C₄⋊C₄	—	31	4	no	—
152	C₂²⋊C₄	—	31	4	no	—
153	C₂²⋊C₄	—	31	4	no	—
154	C₄²	—	31	2	no	—
155	C₄²	—	31	2	no	—
156	C₄²	C₄×D₃₁	1	2	yes	—
157	C₄²	C₂²×D₃₁	1	2	yes	—
158	C₂³	—	31	2	no	—
159	C₂³	—	62	2	no	—
160	C₂³	—	62	2	no	—
161	C₂³	—	62	2	no	—
162	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	31	2	no	—
166	C₂×C₄	—	31	2	no	—
167	C₂×C₄	—	31	2	no	—
168	C₂×C₄	—	62	2	no	—
169	C₂×C₄	—	62	2	no	—
170	C₂×C₄	—	62	2	no	—
171	C₂×C₄	—	62	4	no	—
172	C₂×C₄	—	62	2	no	—
173	C₂×C₄	—	62	2	no	—
174	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
175	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
176	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
177	C₂×C₄	#92497	1	2	yes	—
178	C₂×C₄	#92499	1	2	yes	—
179	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	62	2	no	—
183	C₈	—	62	2	no	—
184	C₂²	—	62	2	no	—
185	C₂²	—	62	2	no	—
186	C₂²	—	62	2	no	—
187	C₂²	—	124	4	no	—
188	C₂²	#11758469	1	2	yes	—
189	C₂²	#11758478	1	2	yes	—
190	C₂²	#11758484	1	2	yes	—
191	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	62	2	no	—
195	C₄	—	62	2	no	—
196	C₄	—	62	2	no	—
197	C₄	#11758469	1	2	yes	—
198	C₄	C₂×#92497	1	1	yes	characteristic
199	C₄	C₂×#92499	1	1	yes	characteristic
200	C₂	—	124	2	no	—
201	C₂	#14521109	1	4	yes	direct product factor
202	C₂	C₂×#11758469	1	1	yes	characteristic
203	C₂	C₂×#11758478	1	1	yes	characteristic
204	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	C₂×#433
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14521109	C₂	direct (N × H)
3. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
6. C₂×#11758403	C₂	semidirect
7. C₂×C₈×Dic₃₁	C₂	semidirect
8. C₂×C₃₁×C₄⋊C₈	C₂	semidirect
22. C₃₁×C₄⋊C₈	C₂²	semidirect
25. #11758403	C₂²	semidirect
26. C₈×Dic₃₁	C₂²	semidirect
27. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
101. C₂×C₄⋊C₈	D₃₁	semidirect
106. C₂×C₃₁	#433	semidirect
125. C₄⋊C₈	C₂×D₃₁	semidirect
130. C₃₁	C₂×#433	semidirect
201. C₂	#14521109	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11758449	2
#14521109	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×(C₂×D₃₁)⋊C₈	2
C₂×#11758403	2
C₂×C₈×Dic₃₁	2
C₂×C₃₁×C₄⋊C₈	2
C₂×#433	31

Subgroups — C2×#14521109 #16854336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521109 #16854336