Subgroups — #18011045

Order 4736.

656 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924615	C₂	1	2	yes	maximal
2	#14558479	C₂	1	2	yes	maximal, complemented
3	#14558477	C₂	1	2	yes	maximal, complemented
4	#14558475	C₂	1	4	yes	maximal
5	C₄×C₄⋊#3866	C₂	1	4	yes	maximal, complemented
6	C₂×C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
7	#11924615	—	2	8	no	—
8	C₂×C₄⋊#3866	C₂²	1	8	yes	—
9	C₂×C₄⋊#3866	C₂²	1	4	yes	—
10	C₂×C₂²⋊#3866	C₂²	1	4	yes	—
11	C₂×C₄×#3866	C₂²	1	4	yes	—
12	(C₂×#3866)⋊C₄	C₂²	1	8	yes	—
13	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
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	2	yes	complemented
17	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
18	C₄²×D₃₇	C₄	1	2	yes	complemented
19	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
20	C₄²×D₃₇	C₂²	1	2	yes	—
21	C₄²×D₃₇	C₂²	1	1	yes	characteristic
22	C₄²×D₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₃₇	C₄	1	1	yes	characteristic, complemented
24	C₂³×D₃₇	C₂³	1	1	yes	characteristic
25	C₂²×#3866	C₂³	1	8	yes	—
26	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
29	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₃₇	—	2	8	no	—
31	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃₇	D₄	1	4	yes	—
36	C₂×C₄×D₃₇	Q₈	1	4	yes	—
37	C₂×C₄×D₃₇	C₂³	1	2	yes	—
38	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×D₃₇	C₂³	1	2	yes	—
41	C₂²⋊#3866	—	2	8	no	—
42	C₄⋊#3866	—	2	8	no	—
43	C₄⋊#3866	—	2	16	no	—
44	C₄×#3866	—	2	8	no	—
45	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₃₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
51	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
52	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
55	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₃₇	—	2	4	no	—
57	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×D₃₇	C₂×D₄	1	2	yes	—
59	C₂²×D₃₇	C₂×Q₈	1	2	yes	—
60	C₂²×D₃₇	C₂⁴	1	1	yes	characteristic
61	C₂×#3866	—	2	8	no	—
62	C₂×#3866	—	2	16	no	—
63	C₂×C₄×C₃₇	—	2	4	no	—
64	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	complemented
65	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₃₇	—	2	4	no	—
70	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	complemented
71	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
75	C₄×D₃₇	—	2	4	no	—
76	C₄×D₃₇	—	2	4	no	—
77	C₄×D₃₇	—	2	4	no	—
78	C₄×D₃₇	C₄⋊C₄	1	4	yes	complemented
79	C₄×D₃₇	C₄⋊C₄	1	4	yes	complemented
80	C₄×D₃₇	C₂×D₄	1	2	yes	—
81	C₄×D₃₇	C₂×D₄	1	2	yes	—
82	C₄×D₃₇	C₂×Q₈	1	2	yes	—
83	C₄×D₃₇	C₂×Q₈	1	2	yes	—
84	C₄×D₃₇	#41	1	8	yes	—
85	C₂²×C₃₇	—	2	2	no	—
86	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₃₇	—	2	2	no	—
89	C₂×D₃₇	—	2	4	no	—
90	C₂×D₃₇	—	2	4	no	—
91	C₂×D₃₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×D₃₇	C₂²×Q₈	1	1	yes	characteristic
94	C₂×D₃₇	C₂×#41	1	4	yes	—
95	#3866	—	4	8	no	—
96	C₄×C₃₇	—	2	2	no	—
97	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₃₇	#117	1	4	yes	complemented
100	Dic₃₇	—	2	2	no	—
101	Dic₃₇	C₂×C₄⋊C₄	1	2	yes	—
102	Dic₃₇	C₂×C₄⋊C₄	1	2	yes	—
103	Dic₃₇	#117	1	4	yes	complemented
104	#2285	—	37	1	no	maximal, Sylow
105	C₂×C₃₇	—	2	2	no	—
106	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃₇	C₂×#117	1	2	yes	—
108	D₃₇	—	2	2	no	—
109	D₃₇	#532	1	2	yes	—
110	D₃₇	#533	1	2	yes	—
111	C₂×#117	—	37	2	no	—
112	C₂²×C₄²	—	37	1	no	—
113	#384	—	37	2	no	—
114	#382	—	37	2	no	—
115	#380	—	37	4	no	—
116	C₄×C₄⋊C₄	—	37	4	no	—
117	C₃₇	#2285	1	1	yes	characteristic, complemented, Sylow
118	C₂³×C₄	—	37	1	no	—
119	C₂³×C₄	—	37	2	no	—
120	#117	—	74	8	no	—
121	C₂×C₄⋊C₄	—	37	4	no	—
122	C₂×C₄⋊C₄	—	37	8	no	—
123	C₂×C₂²⋊C₄	—	37	4	no	—
124	C₂×C₄²	—	37	2	no	—
125	C₂×C₄²	—	37	1	no	—
126	C₂×C₄²	—	37	1	no	—
127	C₂×C₄²	—	37	1	no	—
128	C₂×C₄²	—	37	2	no	—
129	C₂×C₄²	—	37	2	no	—
130	C₂×C₄²	—	37	1	no	—
131	C₂×C₄²	—	37	1	no	—
132	C₂×C₄²	—	37	4	no	—
133	C₂×C₄²	#3866	1	1	yes	characteristic, complemented
134	#95	—	37	8	no	—
135	C₂⁴	—	37	1	no	—
136	C₂²×C₄	—	37	1	no	—
137	C₂²×C₄	—	37	4	no	—
138	C₂²×C₄	—	37	4	no	—
139	C₂²×C₄	—	37	1	no	—
140	C₂²×C₄	—	37	1	no	—
141	C₂²×C₄	—	37	2	no	—
142	C₂²×C₄	—	37	1	no	—
143	C₂²×C₄	—	37	1	no	—
144	C₂²×C₄	—	37	2	no	—
145	C₂²×C₄	—	37	2	no	—
146	C₂²×C₄	—	37	2	no	—
147	C₂²×C₄	—	37	2	no	—
148	C₂²×C₄	—	37	8	no	—
149	C₂²×C₄	—	74	8	no	—
150	C₂²×C₄	C₂×#3866	1	2	yes	—
151	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
152	C₄⋊C₄	—	74	16	no	—
153	C₄⋊C₄	—	74	8	no	—
154	C₂²⋊C₄	—	74	8	no	—
155	C₄²	—	37	2	no	—
156	C₄²	—	37	1	no	—
157	C₄²	—	37	2	no	—
158	C₄²	—	37	2	no	—
159	C₄²	—	37	2	no	—
160	C₄²	—	37	1	no	—
161	C₄²	—	37	2	no	—
162	C₄²	—	74	8	no	—
163	C₄²	C₂×#3866	1	1	yes	characteristic
164	C₄²	C₂×#3866	1	1	yes	characteristic
165	C₄²	C₂×#3866	1	2	yes	—
166	C₂³	—	37	1	no	—
167	C₂³	—	37	2	no	—
168	C₂³	—	37	2	no	—
169	C₂³	—	37	1	no	—
170	C₂³	—	74	4	no	—
171	C₂³	C₂²×#3866	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	37	4	no	—
174	C₂×C₄	—	37	1	no	—
175	C₂×C₄	—	37	1	no	—
176	C₂×C₄	—	37	2	no	—
177	C₂×C₄	—	37	2	no	—
178	C₂×C₄	—	37	2	no	—
179	C₂×C₄	—	37	4	no	—
180	C₂×C₄	—	37	4	no	—
181	C₂×C₄	—	37	2	no	—
182	C₂×C₄	—	37	2	no	—
183	C₂×C₄	—	37	8	no	—
184	C₂×C₄	—	37	2	no	—
185	C₂×C₄	—	74	4	no	—
186	C₂×C₄	—	74	4	no	—
187	C₂×C₄	—	74	4	no	—
188	C₂×C₄	—	74	4	no	—
189	C₂×C₄	—	74	8	no	—
190	C₂×C₄	—	74	16	no	—
191	C₂×C₄	C₄⋊#3866	1	4	yes	complemented
192	C₂×C₄	C₂²×#3866	1	2	yes	—
193	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
194	C₂×C₄	C₂²×#3866	1	2	yes	—
195	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	37	1	no	—
198	C₂²	—	37	1	no	—
199	C₂²	—	37	4	no	—
200	C₂²	—	37	2	no	—
201	C₂²	—	74	4	no	—
202	C₂²	—	74	4	no	—
203	C₂²	—	74	2	no	—
204	C₂²	C₂×C₄⋊#3866	1	2	yes	—
205	C₂²	C₂³×#3866	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	37	2	no	—
208	C₄	—	37	2	no	—
209	C₄	—	37	4	no	—
210	C₄	—	74	2	no	—
211	C₄	—	148	8	no	—
212	C₄	C₂×C₄⋊#3866	1	2	yes	—
213	C₄	C₂×C₄⋊#3866	1	2	yes	—
214	C₄	#11924615	1	4	yes	complemented
215	C₂	—	2	2	no	—
216	C₂	—	37	2	no	—
217	C₂	—	37	2	no	—
218	C₂	—	74	2	no	—
219	C₂	C₂²×C₄⋊#3866	1	1	yes	characteristic
220	C₂	C₂×#11924615	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#3866
Fitting	C₂×C₄²×C₃₇	C₄
prime radical	C₂³×D₃₇	C₂³
cosocle	C₂²×D₃₇	C₂⁴
hypercenter	C₂×C₄²	#3866
hypocenter	C₃₇	#2285
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #14558479	C₂	semidirect
3. #14558477	C₂	semidirect
5. C₄×C₄⋊#3866	C₂	semidirect
15. C₂×C₄×Dic₃₇	C₄	semidirect
16. C₂×C₄×Dic₃₇	C₄	semidirect
17. C₄²×D₃₇	C₄	semidirect
18. C₄²×D₃₇	C₄	semidirect
19. C₄²×D₃₇	C₄	semidirect
23. C₂×C₄²×C₃₇	C₄	semidirect
64. C₂×C₄×C₃₇	C₄⋊C₄	semidirect
70. C₂×Dic₃₇	C₄⋊C₄	semidirect
78. C₄×D₃₇	C₄⋊C₄	semidirect
79. C₄×D₃₇	C₄⋊C₄	semidirect
99. C₄×C₃₇	#117	semidirect
103. Dic₃₇	#117	semidirect
117. C₃₇	#2285	semidirect
133. C₂×C₄²	#3866	semidirect
191. C₂×C₄	C₄⋊#3866	semidirect
214. C₄	#11924615	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924615	2
#14558479	2
#14558477	2
#14558475	2
C₄×C₄⋊#3866	2
C₂×C₄²×D₃₇	2
#2285	37