Subgroups — C₂×C₄×C₄⋊#3866 #18011026

Order 4736.

928 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#3866	C₂	1	2	yes	maximal
2	C₂²×C₄×#3866	C₂	1	2	yes	maximal
3	C₄×C₄⋊#3866	C₂	1	8	yes	maximal, direct product factor
4	C₂×(C₂×#3866)⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×#3866	C₂²	1	4	yes	—
7	C₂×C₄⋊#3866	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊#3866	C₂²	1	8	yes	—
9	C₂×C₄×#3866	—	2	8	no	—
10	C₂×C₄×#3866	C₂²	1	8	yes	—
11	(C₂×#3866)⋊C₄	C₂²	1	8	yes	—
12	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
18	C₄²×D₃₇	C₄	1	4	yes	complemented
19	C₄²×D₃₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂³×D₃₇	C₂³	1	1	yes	characteristic
22	C₂²×#3866	—	2	8	no	—
23	C₂²×#3866	—	2	8	no	—
24	C₂²×#3866	C₂×C₄	1	16	yes	—
25	C₂²×#3866	C₂³	1	8	yes	—
26	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₃₇	—	2	4	no	—
33	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₃₇	C₂×C₄	1	8	yes	—
36	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₃₇	D₄	1	4	yes	—
38	C₂×C₄×D₃₇	Q₈	1	4	yes	—
39	C₂×C₄×D₃₇	C₂³	1	2	yes	—
40	C₂×C₄×D₃₇	C₂³	1	2	yes	—
41	C₂×C₄×D₃₇	C₂³	1	2	yes	—
42	C₄⋊#3866	C₂×C₄	1	32	yes	direct product factor
43	C₄×#3866	—	2	16	no	—
44	C₄²×C₃₇	C₂×C₄	1	4	yes	complemented
45	C₄×Dic₃₇	C₂×C₄	1	4	yes	complemented
46	C₄×Dic₃₇	C₂×C₄	1	4	yes	complemented
47	C₄×Dic₃₇	C₂×C₄	1	4	yes	complemented
48	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
51	C₂²×D₃₇	C₂×D₄	1	2	yes	—
52	C₂²×D₃₇	C₂×Q₈	1	2	yes	—
53	C₂²×D₃₇	#41	1	4	yes	—
54	C₂²×D₃₇	C₂⁴	1	1	yes	characteristic
55	C₂×#3866	—	2	32	no	—
56	C₂×#3866	—	2	8	no	—
57	C₂×#3866	—	2	8	no	—
58	C₂×#3866	C₂²×C₄	1	16	yes	—
59	C₂×C₄×C₃₇	—	2	2	no	—
60	C₂×C₄×C₃₇	C₄²	1	4	yes	complemented
61	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	complemented
62	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
65	C₂×Dic₃₇	—	2	2	no	—
66	C₂×Dic₃₇	C₄²	1	4	yes	complemented
67	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	complemented
68	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
70	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
71	C₄×D₃₇	—	2	4	no	—
72	C₄×D₃₇	—	2	4	no	—
73	C₄×D₃₇	C₄²	1	8	yes	complemented
74	C₄×D₃₇	C₄⋊C₄	1	8	yes	complemented
75	C₄×D₃₇	C₂²×C₄	1	8	yes	—
76	C₄×D₃₇	C₂×D₄	1	4	yes	—
77	C₄×D₃₇	C₂×Q₈	1	4	yes	—
78	C₂²×C₃₇	C₂×C₄²	1	2	yes	—
79	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂²×C₃₇	#117	1	2	yes	—
81	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×D₃₇	C₂×C₄²	1	2	yes	—
83	C₂×D₃₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×D₃₇	#117	1	2	yes	—
85	C₂×D₃₇	C₄×D₄	1	8	yes	—
86	C₂×D₃₇	C₄×Q₈	1	8	yes	—
87	C₂×D₃₇	C₂³×C₄	1	2	yes	—
88	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₃₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×D₃₇	C₂×#41	1	2	yes	—
91	#3866	—	2	16	no	—
92	C₄×C₃₇	—	2	2	no	—
93	C₄×C₃₇	C₂×C₄²	1	4	yes	complemented
94	C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
95	Dic₃₇	—	2	2	no	—
96	Dic₃₇	C₂×C₄²	1	4	yes	complemented
97	Dic₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
98	C₂×C₄×C₄⋊C₄	—	37	1	no	maximal, Sylow
99	C₂×C₃₇	C₄×C₄⋊C₄	1	4	yes	complemented
100	C₂×C₃₇	C₂²×C₄²	1	1	yes	characteristic
101	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
103	D₃₇	C₄×C₄⋊C₄	1	4	yes	complemented
104	D₃₇	C₂×C₄×D₄	1	2	yes	—
105	D₃₇	C₂×C₄×Q₈	1	2	yes	—
106	C₂²×C₄⋊C₄	—	37	2	no	—
107	C₂²×C₄²	—	37	1	no	—
108	C₂²×C₄²	—	37	2	no	—
109	C₄×C₄⋊C₄	—	37	8	no	—
110	C₂×#95	—	37	2	no	—
111	C₃₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	—	37	1	no	—
113	C₂³×C₄	—	37	1	no	—
114	C₂³×C₄	—	37	4	no	—
115	C₂³×C₄	—	37	1	no	—
116	C₂×C₄⋊C₄	—	37	8	no	—
117	C₂×C₄⋊C₄	—	37	16	no	—
118	C₂×C₄²	—	37	1	no	—
119	C₂×C₄²	—	37	1	no	—
120	C₂×C₄²	—	37	1	no	—
121	C₂×C₄²	—	37	4	no	—
122	C₂×C₄²	—	37	4	no	—
123	C₂×C₄²	—	37	8	no	—
124	C₂×C₄²	—	74	8	no	—
125	C₂×C₄²	#3866	1	1	yes	characteristic, complemented
126	#95	—	37	8	no	—
127	C₂⁴	—	37	1	no	—
128	C₂²×C₄	—	37	1	no	—
129	C₂²×C₄	—	37	1	no	—
130	C₂²×C₄	—	37	8	no	—
131	C₂²×C₄	—	37	4	no	—
132	C₂²×C₄	—	37	4	no	—
133	C₂²×C₄	—	37	2	no	—
134	C₂²×C₄	—	37	2	no	—
135	C₂²×C₄	—	37	2	no	—
136	C₂²×C₄	—	37	2	no	—
137	C₂²×C₄	—	37	8	no	—
138	C₂²×C₄	—	37	16	no	—
139	C₂²×C₄	—	37	1	no	—
140	C₂²×C₄	—	37	2	no	—
141	C₂²×C₄	—	37	2	no	—
142	C₂²×C₄	—	74	4	no	—
143	C₂²×C₄	—	74	8	no	—
144	C₂²×C₄	—	74	8	no	—
145	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
146	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
147	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
148	C₄⋊C₄	—	37	32	no	—
149	C₄²	—	37	4	no	—
150	C₄²	—	37	4	no	—
151	C₄²	—	37	4	no	—
152	C₄²	—	74	16	no	—
153	C₄²	C₂×#3866	1	4	yes	complemented
154	C₂³	—	37	4	no	—
155	C₂³	—	37	2	no	—
156	C₂³	—	37	2	no	—
157	C₂³	—	37	4	no	—
158	C₂³	—	37	1	no	—
159	C₂³	—	37	1	no	—
160	C₂³	C₂²×#3866	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	37	4	no	—
163	C₂×C₄	—	37	4	no	—
164	C₂×C₄	—	37	2	no	—
165	C₂×C₄	—	37	2	no	—
166	C₂×C₄	—	37	8	no	—
167	C₂×C₄	—	37	8	no	—
168	C₂×C₄	—	37	4	no	—
169	C₂×C₄	—	37	8	no	—
170	C₂×C₄	—	37	4	no	—
171	C₂×C₄	—	37	16	no	—
172	C₂×C₄	—	37	2	no	—
173	C₂×C₄	—	74	4	no	—
174	C₂×C₄	—	74	4	no	—
175	C₂×C₄	—	74	8	no	—
176	C₂×C₄	—	74	32	no	—
177	C₂×C₄	—	74	8	no	—
178	C₂×C₄	—	74	2	no	—
179	C₂×C₄	C₄×#3866	1	4	yes	complemented
180	C₂×C₄	C₄⋊#3866	1	4	yes	direct product factor
181	C₂×C₄	C₂²×#3866	1	2	yes	—
182	C₂×C₄	C₂²×#3866	1	2	yes	—
183	C₂×C₄	C₂²×#3866	1	2	yes	—
184	C₂²	—	37	8	no	—
185	C₂²	—	37	8	no	—
186	C₂²	—	37	2	no	—
187	C₂²	—	37	2	no	—
188	C₂²	—	37	1	no	—
189	C₂²	—	37	2	no	—
190	C₂²	—	37	1	no	—
191	C₂²	—	37	2	no	—
192	C₂²	—	37	2	no	—
193	C₂²	C₂×C₄×#3866	1	2	yes	—
194	C₂²	C₂×C₄⋊#3866	1	2	yes	—
195	C₂²	#11924615	1	2	yes	—
196	C₂²	C₂³×#3866	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	37	4	no	—
199	C₄	—	37	4	no	—
200	C₄	—	74	16	no	—
201	C₄	—	74	2	no	—
202	C₄	C₂×C₄×#3866	1	4	yes	complemented
203	C₄	C₂×C₄⋊#3866	1	4	yes	direct product factor
204	C₂	—	37	2	no	—
205	C₂	—	37	2	no	—
206	C₂	—	37	4	no	—
207	C₂	C₄×C₄⋊#3866	1	4	yes	direct product factor
208	C₂	C₂²×C₄×#3866	1	1	yes	characteristic
209	C₂	C₂²×C₄⋊#3866	1	1	yes	characteristic
210	C₂	C₂×#11924615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#3866
commutator	C₂×C₃₇	C₂²×C₄²
Frattini	C₂²	C₂³×#3866
Fitting	C₂×C₄²×C₃₇	C₄
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂³×D₃₇	C₂³
cosocle	C₂²×D₃₇	C₂⁴
hypercenter	C₂×C₄²	#3866
hypocenter	C₃₇	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊#3866	C₂	direct (N × H)
7. C₂×C₄⋊#3866	C₄	direct (N × H)
15. C₂×C₄×Dic₃₇	C₄	semidirect
16. C₂×C₄×Dic₃₇	C₄	semidirect
17. C₂×C₄×Dic₃₇	C₄	semidirect
18. C₄²×D₃₇	C₄	semidirect
20. C₂×C₄²×C₃₇	C₄	semidirect
42. C₄⋊#3866	C₂×C₄	direct (N × H)
44. C₄²×C₃₇	C₂×C₄	semidirect
45. C₄×Dic₃₇	C₂×C₄	semidirect
46. C₄×Dic₃₇	C₂×C₄	semidirect
47. C₄×Dic₃₇	C₂×C₄	semidirect
60. C₂×C₄×C₃₇	C₄²	semidirect
61. C₂×C₄×C₃₇	C₄⋊C₄	semidirect
66. C₂×Dic₃₇	C₄²	semidirect
67. C₂×Dic₃₇	C₄⋊C₄	semidirect
73. C₄×D₃₇	C₄²	semidirect
74. C₄×D₃₇	C₄⋊C₄	semidirect
93. C₄×C₃₇	C₂×C₄²	semidirect
94. C₄×C₃₇	C₂×C₄⋊C₄	semidirect
96. Dic₃₇	C₂×C₄²	semidirect
97. Dic₃₇	C₂×C₄⋊C₄	semidirect
99. C₂×C₃₇	C₄×C₄⋊C₄	semidirect
103. D₃₇	C₄×C₄⋊C₄	semidirect
111. C₃₇	C₂×C₄×C₄⋊C₄	semidirect
125. C₂×C₄²	#3866	semidirect
153. C₄²	C₂×#3866	semidirect
179. C₂×C₄	C₄×#3866	semidirect
180. C₂×C₄	C₄⋊#3866	direct (N × H)
202. C₄	C₂×C₄×#3866	semidirect
203. C₄	C₂×C₄⋊#3866	direct (N × H)
207. C₂	C₄×C₄⋊#3866	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#3866	2
C₂²×C₄×#3866	2
C₄×C₄⋊#3866	2
C₂×(C₂×#3866)⋊C₄	2
C₂×C₄²×D₃₇	2
C₂×C₄×C₄⋊C₄	37

Subgroups — C2×C4×C4⋊#3866 #18011026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊#3866 #18011026