Subgroups — #18003542

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	#14558086	C₂	1	1	yes	maximal, characteristic
2	#14557662	C₂	1	2	yes	maximal, complemented
3	#14557662	C₂	1	2	yes	maximal, complemented
4	#14557660	C₂	1	2	yes	maximal, complemented
5	#14557657	C₂	1	2	yes	maximal, complemented
6	#14557561	C₂	1	1	yes	maximal, characteristic
7	C₂×#11924465	C₂	1	1	yes	maximal, characteristic
8	#14557489	C₂	1	1	yes	maximal, characteristic
9	#14557488	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
11	C₃₇×#424	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596934	C₂²	1	1	yes	characteristic
13	C₂²×#64436	C₂²	1	1	yes	characteristic
14	#11924535	C₄	1	1	yes	characteristic
15	#11924535	C₄	1	1	yes	characteristic
16	#11924527	C₄	1	1	yes	characteristic
17	#11924527	C₄	1	1	yes	characteristic
18	#11924513	C₄	1	2	yes	—
19	#11924513	C₄	1	2	yes	—
20	C₄×Dic₇₄	C₂²	1	2	yes	—
21	C₄×Dic₇₄	C₂²	1	2	yes	—
22	#11924495	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
24	#11924465	C₂²	1	1	yes	characteristic
25	#11924465	C₂²	1	1	yes	characteristic
26	#11924465	C₂²	1	1	yes	characteristic
27	#11924465	C₂²	1	1	yes	characteristic
28	#11924465	C₂²	1	2	yes	—
29	#11924465	C₂²	1	2	yes	—
30	#11924447	C₂²	1	2	yes	—
31	#11924446	C₂²	1	2	yes	—
32	#11924446	C₂²	1	2	yes	—
33	C₄×#64436	C₂²	1	2	yes	—
34	C₈×Dic₃₇	C₂²	1	2	yes	—
35	C₈×Dic₃₇	C₂²	1	2	yes	—
36	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
38	#117×C₃₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
40	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
41	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
45	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
46	M₁₆×C₃₇	—	2	2	no	—
47	C₂×C₈×C₃₇	—	2	2	no	—
48	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₇	C₂³	1	2	yes	—
51	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
54	C₄²×C₃₇	C₂³	1	2	yes	—
55	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
59	C₄⋊Dic₃₇	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	#10596936	C₂×C₄	1	2	yes	—
64	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
67	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
68	C₄×Dic₃₇	C₂³	1	2	yes	—
69	#10596934	Q₈	1	4	yes	complemented
70	C₂×#64436	—	2	2	no	—
71	C₂×#64436	C₂³	1	1	yes	characteristic
72	C₂×#64436	C₂³	1	1	yes	characteristic
73	C₂×#64436	C₂³	1	2	yes	—
74	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₃₇	#41	1	2	yes	—
80	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₃₇	—	2	2	no	—
82	C₂×Dic₃₇	—	4	1	no	—
83	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
89	Dic₇₄	—	2	4	no	—
90	Dic₇₄	—	4	2	no	—
91	C₈×C₃₇	—	2	1	no	—
92	C₈×C₃₇	—	2	1	no	—
93	C₈×C₃₇	—	2	2	no	—
94	#64436	—	2	1	no	—
95	#64436	—	2	1	no	—
96	#64436	C₂×Q₈	1	4	yes	—
97	C₂²×C₃₇	—	2	1	no	—
98	C₂²×C₃₇	—	2	1	no	—
99	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
100	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₃₇	—	2	2	no	—
102	C₄×C₃₇	—	2	2	no	—
103	C₄×C₃₇	C₄×Q₈	1	2	yes	—
104	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
106	Dic₃₇	—	2	2	no	—
107	Dic₃₇	—	2	1	no	—
108	Dic₃₇	—	2	1	no	—
109	Dic₃₇	—	2	2	no	—
110	Dic₃₇	—	2	1	no	—
111	Dic₃₇	#131	1	2	yes	—
112	#2962	—	37	1	no	maximal, Sylow
113	C₂×C₃₇	—	2	2	no	—
114	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
116	C₂×C₃₇	#568	1	1	yes	characteristic
117	#533	—	37	1	no	—
118	#446	—	37	2	no	—
119	#446	—	37	2	no	—
120	#446	—	37	2	no	—
121	C₈×Q₈	—	37	2	no	—
122	#424	—	37	1	no	—
123	#424	D₃₇	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	37	1	no	—
125	#405	—	37	1	no	—
126	#405	—	37	1	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	C₂×D₃₇	1	1	yes	characteristic
131	C₂²×C₈	—	37	1	no	—
132	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
133	C₄⋊Q₈	—	37	1	no	—
134	C₄⋊Q₈	—	37	1	no	—
135	#125	—	37	1	no	—
136	#125	—	37	1	no	—
137	C₂²⋊Q₈	—	37	2	no	—
138	C₂²⋊Q₈	—	37	2	no	—
139	C₄×Q₈	—	37	2	no	—
140	C₄×Q₈	—	37	2	no	—
141	#117	—	37	1	no	—
142	#117	C₂×D₃₇	1	1	yes	characteristic
143	C₂×C₄²	—	37	1	no	—
144	C₄⋊C₈	—	37	1	no	—
145	C₄⋊C₈	—	37	1	no	—
146	C₄⋊C₈	—	37	1	no	—
147	C₄⋊C₈	—	37	1	no	—
148	C₄⋊C₈	—	37	2	no	—
149	C₄⋊C₈	—	37	2	no	—
150	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
151	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
152	#97	—	37	2	no	—
153	#97	—	37	2	no	—
154	#97	—	37	2	no	—
155	C₄×C₈	—	37	2	no	—
156	C₄×C₈	—	37	2	no	—
157	C₄×C₈	—	37	2	no	—
158	C₂×Q₈	—	37	2	no	—
159	C₂×Q₈	—	37	2	no	—
160	C₂²×C₄	—	37	1	no	—
161	C₂²×C₄	—	37	1	no	—
162	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	37	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	37	2	no	—
167	C₂×C₈	—	37	1	no	—
168	C₂×C₈	—	37	1	no	—
169	C₂×C₈	—	74	2	no	—
170	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₃₇	1	2	yes	—
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	1	no	—
178	C₄⋊C₄	—	37	1	no	—
179	C₄⋊C₄	—	37	2	no	—
180	C₄⋊C₄	—	37	2	no	—
181	C₄⋊C₄	C₄×D₃₇	1	1	yes	characteristic
182	C₄⋊C₄	C₄×D₃₇	1	1	yes	characteristic
183	C₂²⋊C₄	—	37	2	no	—
184	C₂²⋊C₄	C₄×D₃₇	1	2	yes	—
185	C₄²	—	37	1	no	—
186	C₄²	—	37	1	no	—
187	C₄²	—	37	1	no	—
188	C₄²	—	37	2	no	—
189	C₄²	—	37	1	no	—
190	C₄²	C₂²×D₃₇	1	2	yes	—
191	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
192	Q₈	—	74	4	no	—
193	Q₈	—	148	2	no	—
194	C₂×C₄	—	37	1	no	—
195	C₂×C₄	—	37	1	no	—
196	C₂×C₄	—	37	1	no	—
197	C₂×C₄	—	37	1	no	—
198	C₂×C₄	—	37	2	no	—
199	C₂×C₄	—	37	2	no	—
200	C₂×C₄	—	74	2	no	—
201	C₂×C₄	—	148	1	no	—
202	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
205	C₂×C₄	#10596964	1	2	yes	—
206	C₂×C₄	Q₈×D₃₇	1	2	yes	—
207	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	37	4	no	—
212	C₈	—	74	1	no	—
213	C₈	—	74	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#11924501	1	2	yes	—
217	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	37	2	no	—
221	C₄	—	74	1	no	—
222	C₄	—	74	2	no	—
223	C₄	—	74	1	no	—
224	C₄	—	74	1	no	—
225	C₄	—	74	2	no	—
226	C₄	#11924501	1	2	yes	—
227	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
228	C₄	C₂×#10596964	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×#11924501	1	1	yes	characteristic
231	C₂	#131×D₃₇	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	C₃₇×#424	C₂
prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#424	D₃₇
hypocenter	C₃₇	#2962
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #14557662	C₂	semidirect
3. #14557662	C₂	semidirect
4. #14557660	C₂	semidirect
5. #14557657	C₂	semidirect
69. #10596934	Q₈	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
#14558086	2
#14557662	2
#14557660	2
#14557657	2
#14557561	2
C₂×#11924465	2
#14557489	2
#14557488	2
M₁₆×Dic₃₇	2
C₃₇×#424	2
#2962	37