Subgroups — #18003538

Order 4736.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558087	C₂	1	1	yes	maximal, characteristic
2	#14557664	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557663	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557663	C₂	1	2	yes	maximal, complemented
5	#14557662	C₂	1	2	yes	maximal, complemented
6	#14557657	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557558	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
9	#14557487	C₂	1	2	yes	maximal
10	C₄×#10596929	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596929	C₂²	1	1	yes	characteristic
13	C₂×#10596929	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₃₇	C₂²	1	2	yes	—
15	#11924535	C₄	1	1	yes	characteristic
16	#11924534	C₄	1	1	yes	characteristic
17	#11924526	C₄	1	2	yes	—
18	#11924516	C₄	1	1	yes	characteristic
19	#11924514	C₄	1	2	yes	—
20	#11924512	C₄	1	1	yes	characteristic
21	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₇₄	C₂²	1	2	yes	—
23	#11924493	C₂²	1	2	yes	—
24	C₄²×D₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊#64436	C₂²	1	1	yes	characteristic
26	C₄⋊#64436	C₂²	1	2	yes	—
27	#11924465	C₂²	1	2	yes	—
28	#11924465	C₂²	1	2	yes	—
29	#11924465	C₂²	1	2	yes	—
30	#11924447	C₂²	1	2	yes	—
31	#11924446	C₂²	1	1	yes	characteristic
32	#11924446	C₂²	1	1	yes	characteristic
33	#11924446	C₂²	1	2	yes	—
34	C₄×#64436	C₂²	1	1	yes	characteristic
35	C₈×Dic₃₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
39	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₃₇	C₂²	1	2	yes	—
41	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₃₇	C₂³	1	2	yes	—
48	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₄²×C₃₇	C₂³	1	2	yes	—
54	#10596938	C₂×C₄	1	2	yes	—
55	#10596938	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
58	#10596936	C₂×C₄	1	2	yes	—
59	#10596936	C₂×C₄	1	2	yes	—
60	#10596936	C₂×C₄	1	2	yes	—
61	#10596936	C₂×C₄	1	2	yes	—
62	#10596936	C₂×C₄	1	2	yes	—
63	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
66	C₄×Dic₃₇	C₂³	1	2	yes	—
67	C₂×#64436	C₂³	1	2	yes	—
68	C₂×#64436	C₂³	1	1	yes	characteristic
69	C₂×#64436	C₂³	1	1	yes	characteristic
70	#10596929	—	2	2	no	—
71	#10596929	Q₈	1	4	yes	complemented
72	C₈×D₃₇	—	2	4	no	—
73	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₃₇	—	2	2	no	—
75	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₃₇	—	2	2	no	—
86	C₄×D₃₇	—	4	1	no	—
87	C₄×D₃₇	C₂×Q₈	1	2	yes	—
88	C₄×D₃₇	#41	1	2	yes	—
89	Dic₇₄	—	2	2	no	—
90	Dic₇₄	—	4	2	no	—
91	C₈×C₃₇	—	2	2	no	—
92	C₈×C₃₇	—	2	1	no	—
93	C₈×C₃₇	C₂×Q₈	1	2	yes	—
94	#64436	—	2	2	no	—
95	#64436	—	2	1	no	—
96	#64436	C₂×Q₈	1	2	yes	—
97	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₃₇	—	2	1	no	—
99	C₂×D₃₇	—	2	1	no	—
100	C₂×D₃₇	C₄×Q₈	1	2	yes	—
101	C₄×C₃₇	—	2	2	no	—
102	C₄×C₃₇	—	2	2	no	—
103	C₄×C₃₇	—	2	1	no	—
104	C₄×C₃₇	#131	1	2	yes	—
105	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
107	Dic₃₇	—	2	2	no	—
108	Dic₃₇	—	2	1	no	—
109	Dic₃₇	—	2	1	no	—
110	Dic₃₇	—	2	2	no	—
111	Dic₃₇	C₄×Q₈	1	2	yes	—
112	#2962	—	37	1	no	maximal, Sylow
113	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
115	C₂×C₃₇	#568	1	1	yes	characteristic
116	D₃₇	—	2	2	no	—
117	#533	—	37	1	no	—
118	#446	—	37	1	no	—
119	#446	—	37	2	no	—
120	#446	—	37	1	no	—
121	#446	—	37	2	no	—
122	C₈×Q₈	—	37	1	no	—
123	C₈×Q₈	D₃₇	1	1	yes	characteristic, complemented
124	#424	—	37	2	no	—
125	C₂×C₄⋊C₈	—	37	1	no	—
126	#405	—	37	2	no	—
127	C₄×M₁₆	—	37	1	no	—
128	C₃₇	#2962	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	37	1	no	—
130	C₂×M₁₆	—	37	1	no	—
131	C₂²×C₈	—	37	2	no	—
132	C₄⋊Q₈	—	37	1	no	—
133	C₄⋊Q₈	—	37	1	no	—
134	#125	—	37	2	no	—
135	C₂²⋊Q₈	—	37	1	no	—
136	C₂²⋊Q₈	—	37	2	no	—
137	C₂²⋊Q₈	—	37	1	no	—
138	C₄×Q₈	—	37	1	no	—
139	C₄×Q₈	—	37	2	no	—
140	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
141	#117	—	37	2	no	—
142	C₂×C₄²	—	37	1	no	—
143	C₄⋊C₈	—	37	2	no	—
144	C₄⋊C₈	—	37	2	no	—
145	C₄⋊C₈	—	37	2	no	—
146	C₄⋊C₈	—	37	1	no	—
147	C₄⋊C₈	—	37	2	no	—
148	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
150	#97	—	37	1	no	—
151	#97	—	37	1	no	—
152	#97	—	37	2	no	—
153	#97	—	37	2	no	—
154	C₄×C₈	—	37	2	no	—
155	C₄×C₈	—	37	1	no	—
156	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₃₇	1	2	yes	—
158	C₂×Q₈	—	37	1	no	—
159	C₂×Q₈	—	37	2	no	—
160	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
161	C₂²×C₄	—	37	1	no	—
162	C₂²×C₄	—	37	1	no	—
163	C₂²×C₄	—	37	1	no	—
164	M₁₆	—	37	4	no	—
165	M₁₆	—	74	2	no	—
166	C₂×C₈	—	37	1	no	—
167	C₂×C₈	—	37	1	no	—
168	C₂×C₈	—	37	2	no	—
169	C₂×C₈	—	74	4	no	—
170	C₂×C₈	C₂²×D₃₇	1	2	yes	—
171	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
173	C₄⋊C₄	—	37	2	no	—
174	C₄⋊C₄	—	37	2	no	—
175	C₄⋊C₄	—	37	2	no	—
176	C₄⋊C₄	—	37	2	no	—
177	C₄⋊C₄	—	37	2	no	—
178	C₄⋊C₄	—	37	2	no	—
179	C₄⋊C₄	—	37	1	no	—
180	C₄⋊C₄	C₄×D₃₇	1	2	yes	—
181	C₄⋊C₄	C₄×D₃₇	1	1	yes	characteristic
182	C₂²⋊C₄	—	37	2	no	—
183	C₂²⋊C₄	—	37	2	no	—
184	C₄²	—	37	1	no	—
185	C₄²	—	37	1	no	—
186	C₄²	—	37	2	no	—
187	C₄²	—	37	1	no	—
188	C₄²	C₂²×D₃₇	1	1	yes	characteristic
189	C₄²	C₂²×D₃₇	1	2	yes	—
190	C₂³	—	37	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	74	2	no	—
193	Q₈	—	148	2	no	—
194	C₂×C₄	—	37	1	no	—
195	C₂×C₄	—	37	2	no	—
196	C₂×C₄	—	37	2	no	—
197	C₂×C₄	—	37	1	no	—
198	C₂×C₄	—	37	1	no	—
199	C₂×C₄	—	37	2	no	—
200	C₂×C₄	—	37	2	no	—
201	C₂×C₄	—	74	2	no	—
202	C₂×C₄	—	148	1	no	—
203	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	37	2	no	—
211	C₈	—	74	2	no	—
212	C₈	—	74	1	no	—
213	C₈	Q₈×D₃₇	1	2	yes	—
214	C₂²	—	37	2	no	—
215	C₂²	—	74	1	no	—
216	C₂²	—	74	1	no	—
217	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	37	2	no	—
222	C₄	—	74	1	no	—
223	C₄	—	74	2	no	—
224	C₄	—	74	1	no	—
225	C₄	—	74	2	no	—
226	C₄	#11924542	1	2	yes	—
227	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
228	C₄	#41×D₃₇	1	1	yes	characteristic
229	C₂	—	74	2	no	—
230	C₂	C₄×Q₈×D₃₇	1	1	yes	characteristic
231	C₂	C₂×#11924542	1	1	yes	characteristic
232	C₂	#14558298	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₈×Q₈×C₃₇	C₂
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₈×Q₈	D₃₇
hypocenter	C₃₇	#2962
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #14557664	C₂	semidirect
3. #14557663	C₂	semidirect
4. #14557663	C₂	semidirect
5. #14557662	C₂	semidirect
6. #14557657	C₂	semidirect
11. C₈×Q₈×C₃₇	C₂	semidirect
71. #10596929	Q₈	semidirect
123. C₈×Q₈	D₃₇	semidirect
128. C₃₇	#2962	semidirect

Maximal subgroups

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

subgroup	index
#14558087	2
#14557664	2
#14557663	2
#14557662	2
#14557657	2
#14557558	2
C₄⋊C₈×D₃₇	2
#14557487	2
C₄×#10596929	2
C₈×Q₈×C₃₇	2
#2962	37