Subgroups — #17997560

Order 4736.

464 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
2	#14557440	C₂	1	1	yes	maximal, characteristic
3	#14557418	C₂	1	2	yes	maximal
4	#14557416	C₂	1	2	yes	maximal
5	#14557400	C₂	1	2	yes	maximal
6	#14557398	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
8	C₄×#10596936	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
10	#14557378	C₂	1	2	yes	maximal
11	C₃₇×#391	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₃₇	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
16	C₂×#10596936	C₂²	1	1	yes	characteristic
17	C₂×#10596936	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
23	#11924443	C₂²	1	2	yes	—
24	#11924443	C₂²	1	2	yes	—
25	#11924443	C₂²	1	1	yes	characteristic
26	#11924443	C₂²	1	2	yes	—
27	#11924443	C₂²	1	2	yes	—
28	#11924443	C₂²	1	1	yes	characteristic
29	#11924443	C₂²	1	2	yes	—
30	#11924443	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
34	#95×C₃₇	C₂²	1	2	yes	—
35	#95×C₃₇	C₂²	1	2	yes	—
36	C₂×Q₈×C₃₇	—	2	2	no	—
37	C₂×Q₈×C₃₇	—	2	2	no	—
38	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₃₇	C₂³	1	2	yes	—
46	C₂²×Dic₃₇	C₂³	1	2	yes	—
47	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
48	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₃₇	—	2	2	no	—
51	C₄²×C₃₇	—	2	2	no	—
52	C₄⋊Dic₃₇	—	2	2	no	—
53	C₄⋊Dic₃₇	C₂×C₄	1	8	yes	—
54	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
55	#10596936	—	2	2	no	—
56	#10596936	—	2	2	no	—
57	C₄×Dic₃₇	—	2	2	no	—
58	C₄×Dic₃₇	—	2	4	no	—
59	C₄×Dic₃₇	—	2	4	no	—
60	C₄×Dic₃₇	C₂×C₄	1	8	yes	—
61	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
62	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
63	Q₈×C₃₇	—	2	8	no	—
64	C₂×C₄×C₃₇	—	2	1	no	—
65	C₂×C₄×C₃₇	—	2	2	no	—
66	C₂×C₄×C₃₇	—	2	2	no	—
67	C₂×C₄×C₃₇	—	2	1	no	—
68	C₂×C₄×C₃₇	—	2	2	no	—
69	C₂×C₄×C₃₇	—	2	1	no	—
70	C₂×C₄×C₃₇	—	2	1	no	—
71	C₂×C₄×C₃₇	—	2	2	no	—
72	C₂×C₄×C₃₇	—	2	2	no	—
73	C₂×C₄×C₃₇	—	2	1	no	—
74	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₇	#41	1	2	yes	—
77	C₂×C₄×C₃₇	#41	1	2	yes	—
78	C₂×C₄×C₃₇	#41	1	2	yes	—
79	C₂×Dic₃₇	—	2	2	no	—
80	C₂×Dic₃₇	—	2	1	no	—
81	C₂×Dic₃₇	—	2	2	no	—
82	C₂×Dic₃₇	—	2	1	no	—
83	C₂×Dic₃₇	—	2	1	no	—
84	C₂×Dic₃₇	—	2	1	no	—
85	C₂×Dic₃₇	—	2	1	no	—
86	C₂×Dic₃₇	—	2	1	no	—
87	C₂×Dic₃₇	—	2	1	no	—
88	C₂×Dic₃₇	—	2	2	no	—
89	C₂×Dic₃₇	—	2	1	no	—
90	C₂×Dic₃₇	—	2	2	no	—
91	C₂×Dic₃₇	—	2	1	no	—
92	C₂×Dic₃₇	—	2	2	no	—
93	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
95	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
96	C₂×Dic₃₇	#41	1	2	yes	—
97	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
101	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
102	C₂²×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂²×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₃₇	—	2	4	no	—
105	C₄×C₃₇	—	2	2	no	—
106	C₄×C₃₇	—	4	1	no	—
107	C₄×C₃₇	—	4	2	no	—
108	C₄×C₃₇	#117	1	4	yes	—
109	Dic₃₇	—	2	4	no	—
110	Dic₃₇	—	2	2	no	—
111	Dic₃₇	—	2	2	no	—
112	Dic₃₇	—	2	2	no	—
113	Dic₃₇	—	4	1	no	—
114	Dic₃₇	—	4	2	no	—
115	#2356	—	37	1	no	maximal, Sylow
116	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
117	C₂×C₃₇	C₄×#41	1	1	yes	characteristic
118	C₂×C₃₇	#519	1	1	yes	characteristic
119	C₂×C₃₇	#552	1	1	yes	characteristic
120	C₂×C₃₇	#552	1	1	yes	characteristic
121	C₂×C₃₇	#556	1	1	yes	characteristic
122	C₂×C₃₇	#556	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	37	1	no	—
124	#391	—	37	1	no	—
125	#391	D₃₇	1	1	yes	characteristic
126	#387	—	37	2	no	—
127	#387	—	37	2	no	—
128	#383	—	37	2	no	—
129	#382	—	37	1	no	—
130	C₄×C₄⋊C₄	—	37	1	no	—
131	C₄×C₄⋊C₄	—	37	1	no	—
132	C₄×C₄⋊C₄	—	37	1	no	—
133	#376	—	37	2	no	—
134	C₃₇	#2356	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
136	C₄×Q₈	—	37	8	no	—
137	C₂×C₄⋊C₄	—	37	1	no	—
138	C₂×C₄⋊C₄	—	37	2	no	—
139	C₂×C₄⋊C₄	—	37	1	no	—
140	C₂×C₄⋊C₄	—	37	1	no	—
141	C₂×C₄⋊C₄	—	37	1	no	—
142	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
143	C₂×C₄²	—	37	2	no	—
144	C₂×C₄²	—	37	1	no	—
145	C₂×C₄²	—	37	2	no	—
146	C₂×C₄²	—	37	2	no	—
147	C₂×C₄²	—	37	1	no	—
148	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
149	#95	—	37	1	no	—
150	#95	—	37	2	no	—
151	#95	—	37	2	no	—
152	#95	—	37	1	no	—
153	#95	—	37	2	no	—
154	#95	—	37	2	no	—
155	#95	—	37	2	no	—
156	#95	—	37	2	no	—
157	#95	C₂×D₃₇	1	2	yes	—
158	#95	C₂×D₃₇	1	2	yes	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₄×D₃₇	1	4	yes	—
162	C₂²×C₄	—	37	1	no	—
163	C₂²×C₄	—	37	1	no	—
164	C₂²×C₄	—	37	2	no	—
165	C₂²×C₄	—	37	1	no	—
166	C₂²×C₄	—	37	1	no	—
167	C₂²×C₄	—	37	2	no	—
168	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
171	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	37	8	no	—
175	C₄⋊C₄	—	37	4	no	—
176	C₄⋊C₄	—	74	2	no	—
177	C₄⋊C₄	—	74	2	no	—
178	C₄⋊C₄	—	74	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	37	8	no	—
181	C₄²	—	37	4	no	—
182	C₄²	—	74	4	no	—
183	C₄²	—	74	2	no	—
184	C₄²	—	74	4	no	—
185	C₂³	C₂³×D₃₇	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	37	2	no	—
198	C₂×C₄	—	37	4	no	—
199	C₂×C₄	—	37	2	no	—
200	C₂×C₄	—	37	2	no	—
201	C₂×C₄	—	74	2	no	—
202	C₂×C₄	—	74	1	no	—
203	C₂×C₄	—	74	1	no	—
204	C₂×C₄	—	74	2	no	—
205	C₂×C₄	—	74	1	no	—
206	C₂×C₄	—	74	1	no	—
207	C₂×C₄	—	74	1	no	—
208	C₂×C₄	—	74	1	no	—
209	C₂×C₄	—	74	1	no	—
210	C₂×C₄	—	74	2	no	—
211	C₂×C₄	—	74	2	no	—
212	C₂×C₄	—	74	1	no	—
213	C₂×C₄	—	74	1	no	—
214	C₂×C₄	—	74	2	no	—
215	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
216	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
217	C₂×C₄	#10596964	1	2	yes	—
218	C₂×C₄	#10596964	1	2	yes	—
219	C₂×C₄	#10596966	1	2	yes	—
220	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
221	C₂²	#41×D₃₇	1	1	yes	characteristic
222	C₂²	C₂×#10596964	1	1	yes	characteristic
223	C₂²	C₂×#10596964	1	1	yes	characteristic
224	C₂²	C₂×#10596966	1	1	yes	characteristic
225	C₂²	#11924583	1	1	yes	characteristic
226	C₂²	#11924584	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	74	2	no	—
232	C₄	—	74	4	no	—
233	C₄	—	74	2	no	—
234	C₄	—	74	2	no	—
235	C₄	—	148	1	no	—
236	C₄	—	148	2	no	—
237	C₄	#11924492	1	4	yes	—
238	C₂	C₂×#11924492	1	1	yes	characteristic
239	C₂	C₄×#10596966	1	1	yes	characteristic
240	C₂	#14558009	1	1	yes	characteristic
241	C₂	#14558217	1	1	yes	characteristic
242	C₂	#14558225	1	1	yes	characteristic
243	C₂	#14558252	1	1	yes	characteristic
244	C₂	#14558257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#391	C₂
socle, prime radical, cosocle	C₂³×C₃₇	C₂⁴
hypercenter	#391	D₃₇
hypocenter	C₃₇	#2356
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
12. Q₈×Dic₃₇	C₄	semidirect
134. C₃₇	#2356	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₃₇	2
#14557440	2
#14557418	2
#14557416	2
#14557400	2
#14557398	2
C₄⋊C₄×Dic₃₇	2
C₄×#10596936	2
C₄×C₄⋊Dic₃₇	2
#14557378	2
C₃₇×#391	2
#2356	37