Subgroups — #18004447

Order 4736.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558161	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557840	C₂	1	2	yes	maximal, complemented
3	#14557833	C₂	1	2	yes	maximal, complemented
4	#14557828	C₂	1	2	yes	maximal, complemented
5	#14557813	C₂	1	2	yes	maximal, complemented
6	#14557573	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11924469	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557560	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557534	C₂	1	1	yes	maximal, characteristic
10	#14557529	C₂	1	1	yes	maximal, characteristic
11	C₃₇×#520	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596934	C₂²	1	1	yes	characteristic
13	C₂²×#64436	C₂²	1	1	yes	characteristic
14	#11924536	C₂²	1	2	yes	—
15	#11924535	—	2	1	no	—
16	#11924528	C₂²	1	2	yes	—
17	#11924527	—	2	1	no	—
18	#11924513	—	2	2	no	—
19	#11924495	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
21	#11924469	C₂²	1	2	yes	—
22	#11924469	C₂²	1	2	yes	—
23	#11924469	C₂²	1	2	yes	—
24	#11924467	C₂²	1	2	yes	—
25	C₄⋊#64436	C₂²	1	2	yes	—
26	C₄⋊#64436	C₂²	1	2	yes	—
27	#11924460	C₂²	1	1	yes	characteristic
28	#11924460	C₂²	1	1	yes	characteristic
29	#11924460	C₂²	1	2	yes	—
30	#11924457	C₂²	1	1	yes	characteristic
31	#11924457	C₂²	1	1	yes	characteristic
32	#11924457	C₂²	1	2	yes	—
33	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
35	C₄×D₄×C₃₇	—	2	2	no	—
36	C₄×D₄×C₃₇	C₂²	1	2	yes	—
37	#117×C₃₇	—	2	1	no	—
38	#117×C₃₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
41	#41×C₃₇	—	2	2	no	—
42	#41×C₃₇	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₃₇	—	2	1	no	—
45	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₇	—	2	1	no	—
47	C₂²×C₄×C₃₇	—	2	1	no	—
48	C₂²×C₄×C₃₇	—	2	2	no	—
49	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
50	C₂²×Dic₃₇	—	2	1	no	—
51	C₂×Dic₇₄	—	2	2	no	—
52	C₄⋊C₄×C₃₇	—	2	1	no	—
53	C₄⋊C₄×C₃₇	—	2	2	no	—
54	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₃₇	—	2	2	no	—
58	C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
59	C₄²×C₃₇	—	2	2	no	—
60	C₄²×C₃₇	C₂³	1	2	yes	—
61	C₂²⋊Dic₃₇	—	2	1	no	—
62	C₄⋊Dic₃₇	—	2	1	no	—
63	C₄⋊Dic₃₇	—	2	2	no	—
64	C₄⋊Dic₃₇	C₂³	1	2	yes	—
65	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
67	#10596936	—	2	2	no	—
68	#10596936	—	2	2	no	—
69	C₄×Dic₃₇	—	2	1	no	—
70	#10596934	—	2	2	no	—
71	C₂×#64436	—	2	2	no	—
72	C₂×#64436	C₂³	1	1	yes	characteristic
73	C₂×#64436	C₂³	1	1	yes	characteristic
74	C₂×#64436	C₂³	1	2	yes	—
75	C₂³×C₃₇	—	2	1	no	—
76	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₃₇	—	2	1	no	—
78	Q₈×C₃₇	C₂×Q₈	1	2	yes	—
79	D₄×C₃₇	—	2	4	no	—
80	D₄×C₃₇	—	2	1	no	—
81	D₄×C₃₇	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₃₇	—	2	1	no	—
83	C₂×C₄×C₃₇	—	2	4	no	—
84	C₂×C₄×C₃₇	—	2	2	no	—
85	C₂×C₄×C₃₇	—	8	2	no	—
86	C₂×C₄×C₃₇	—	8	1	no	—
87	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₃₇	#41	1	2	yes	—
92	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₃₇	—	2	1	no	—
94	C₂×Dic₃₇	—	2	1	no	—
95	C₂×Dic₃₇	—	2	2	no	—
96	C₂×Dic₃₇	—	8	1	no	—
97	Dic₇₄	—	8	2	no	—
98	#64436	—	2	1	no	—
99	#64436	—	2	1	no	—
100	#64436	—	2	2	no	—
101	C₂²×C₃₇	—	2	1	no	—
102	C₂²×C₃₇	—	2	1	no	—
103	C₂²×C₃₇	—	2	2	no	—
104	C₂²×C₃₇	—	4	1	no	—
105	C₂²×C₃₇	—	4	1	no	—
106	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₃₇	—	2	2	no	—
109	C₄×C₃₇	—	2	2	no	—
110	C₄×C₃₇	—	2	2	no	—
111	C₄×C₃₇	—	4	2	no	—
112	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
113	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
114	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
115	Dic₃₇	—	4	1	no	—
116	Dic₃₇	—	4	2	no	—
117	Dic₃₇	—	4	1	no	—
118	#3083	—	37	1	no	maximal, Sylow
119	C₂×C₃₇	—	2	2	no	—
120	C₂×C₃₇	—	4	2	no	—
121	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₃₇	#576	1	1	yes	characteristic
123	C₂×C₃₇	#578	1	1	yes	characteristic
124	#544	—	37	1	no	—
125	#520	D₃₇	1	1	yes	characteristic
126	#479	—	37	2	no	—
127	#478	—	37	2	no	—
128	#477	—	37	2	no	—
129	#474	—	37	2	no	—
130	#428	—	37	1	no	—
131	C₂×#107	—	37	1	no	—
132	#424	—	37	1	no	—
133	#417	—	37	1	no	—
134	#416	—	37	1	no	—
135	C₃₇	#3083	1	1	yes	characteristic, complemented, Sylow
136	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
137	C₂×M₁₆	—	37	1	no	—
138	C₂²×C₈	—	37	1	no	—
139	C₄⋊Q₈	—	37	2	no	—
140	C₄⋊Q₈	—	74	1	no	—
141	#125	—	37	2	no	—
142	#125	—	74	1	no	—
143	C₂²⋊Q₈	—	74	2	no	—
144	C₄×Q₈	C₂×D₃₇	1	2	yes	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂×D₃₇	1	2	yes	—
147	#117	—	2	1	no	—
148	#117	—	37	1	no	—
149	#117	C₂×D₃₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	37	1	no	—
152	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
153	#107	—	37	2	no	—
154	#107	—	37	2	no	—
155	#107	—	37	2	no	—
156	#106	—	37	2	no	—
157	C₄⋊C₈	—	37	2	no	—
158	C₄⋊C₈	—	37	2	no	—
159	#103	—	37	1	no	—
160	#103	—	37	1	no	—
161	#103	—	37	2	no	—
162	#102	—	37	1	no	—
163	#102	—	37	1	no	—
164	#102	—	37	2	no	—
165	#41	—	2	2	no	—
166	#41	Dic₇₄	1	4	yes	complemented
167	C₂×Q₈	—	74	2	no	—
168	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	74	1	no	—
175	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
176	M₁₆	—	74	2	no	—
177	C₂×C₈	—	37	2	no	—
178	C₂×C₈	—	37	1	no	—
179	C₂×C₈	—	37	1	no	—
180	C₂×C₈	—	74	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	37	1	no	—
184	C₄⋊C₄	—	37	2	no	—
185	C₄⋊C₄	—	37	1	no	—
186	C₄⋊C₄	—	74	2	no	—
187	C₄⋊C₄	—	74	2	no	—
188	C₄⋊C₄	—	74	1	no	—
189	C₄⋊C₄	—	74	2	no	—
190	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
191	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	74	1	no	—
195	C₂²⋊C₄	C₂²⋊D₃₇	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	74	1	no	—
198	C₄²	C₂²×D₃₇	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	296	2	no	—
203	Q₈	C₂×Dic₇₄	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₇₄	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	296	1	no	—
213	C₂×C₄	—	74	1	no	—
214	C₂×C₄	—	74	1	no	—
215	C₂×C₄	—	74	2	no	—
216	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
217	C₂×C₄	#10596962	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
221	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
222	C₈	—	74	1	no	—
223	C₈	—	74	1	no	—
224	C₈	—	74	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#11924515	1	2	yes	—
231	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	148	2	no	—
237	C₄	—	148	1	no	—
238	C₄	—	148	1	no	—
239	C₄	#11924515	1	2	yes	—
240	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
241	C₄	C₂×#10596962	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#11924515	1	1	yes	characteristic
245	C₂	#14558336	1	1	yes	characteristic
246	C₂	#14558348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#520	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#41×C₃₇	C₂²
hypercenter	#520	D₃₇
hypocenter	C₃₇	#3083
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #14558161	C₂	semidirect
2. #14557840	C₂	semidirect
3. #14557833	C₂	semidirect
4. #14557828	C₂	semidirect
5. #14557813	C₂	semidirect
6. #14557573	C₂	semidirect
7. C₂×#11924469	C₂	semidirect
8. #14557560	C₂	semidirect
42. #41×C₃₇	Q₈	semidirect
135. C₃₇	#3083	semidirect
166. #41	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#14558161	2
#14557840	2
#14557833	2
#14557828	2
#14557813	2
#14557573	2
C₂×#11924469	2
#14557560	2
#14557534	2
#14557529	2
C₃₇×#520	2
#3083	37