Subgroups — #18003394

Order 4736.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924556	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596962	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557643	C₂	1	2	yes	maximal, complemented
4	#14557638	C₂	1	2	yes	maximal, complemented
5	#14557633	C₂	1	2	yes	maximal, complemented
6	#14557631	C₂	1	2	yes	maximal, complemented
7	#14557569	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557529	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557527	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924556	C₄	1	8	yes	complemented
14	C₂×#10596942	C₂²	1	1	yes	characteristic
15	C₂×#10596940	C₂²	1	1	yes	characteristic
16	C₂×#10596941	C₂²	1	1	yes	characteristic
17	C₂×#10596939	C₂²	1	1	yes	characteristic
18	C₂²×#64436	C₂²	1	1	yes	characteristic
19	C₂²×#64436	C₂²	1	1	yes	characteristic
20	C₄×Dic₇₄	C₂²	1	2	yes	—
21	C₄×D₁₄₈	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₃₇	—	2	2	no	—
23	#11924495	C₂²	1	1	yes	characteristic
24	#11924493	—	2	1	no	—
25	C₄²×D₃₇	—	2	1	no	—
26	#11924469	C₂²	1	2	yes	—
27	#11924467	C₂²	1	2	yes	—
28	#11924460	C₂²	1	2	yes	—
29	#11924459	C₂²	1	2	yes	—
30	#11924457	C₂²	1	2	yes	—
31	#11924456	C₂²	1	2	yes	—
32	#11924447	C₂²	1	2	yes	—
33	#11924447	C₂²	1	2	yes	—
34	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
36	C₄×D₄×C₃₇	—	2	2	no	—
37	C₄×D₄×C₃₇	C₂²	1	2	yes	—
38	#117×C₃₇	—	2	1	no	—
39	#117×C₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₃₇	—	2	1	no	—
41	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
42	#41×C₃₇	—	2	2	no	—
43	#41×C₃₇	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₇	—	2	1	no	—
46	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₇	—	2	1	no	—
48	C₂²×C₄×C₃₇	—	2	2	no	—
49	C₂²×C₄×C₃₇	D₄	1	2	yes	—
50	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₃₇	—	2	1	no	—
52	#10596962	—	2	2	no	—
53	#10596962	C₂×C₄	1	4	yes	complemented
54	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₃₇	—	2	1	no	—
56	C₂×C₄×D₃₇	—	2	2	no	—
57	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₃₇	—	2	2	no	—
59	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
60	C₂²⋊C₄×C₃₇	—	2	1	no	—
61	C₂²⋊C₄×C₃₇	—	2	2	no	—
62	C₄²×C₃₇	—	2	1	no	—
63	C₄²×C₃₇	—	2	2	no	—
64	C₄²×C₃₇	D₄	1	1	yes	characteristic
65	C₄²×C₃₇	D₄	1	1	yes	characteristic
66	C₄²×C₃₇	C₂³	1	2	yes	—
67	C₂²⋊Dic₃₇	—	2	1	no	—
68	#10596942	C₂×C₄	1	4	yes	complemented
69	#10596941	C₂×C₄	1	4	yes	complemented
70	#10596940	C₂×C₄	1	4	yes	complemented
71	#10596939	C₂×C₄	1	4	yes	complemented
72	#10596938	—	2	2	no	—
73	C₄⋊Dic₃₇	C₂³	1	2	yes	—
74	#10596936	—	2	2	no	—
75	C₄×Dic₃₇	—	2	1	no	—
76	C₄×Dic₃₇	—	2	2	no	—
77	C₂×#64436	—	2	2	no	—
78	C₂×#64436	C₂×C₄	1	4	yes	complemented
79	C₂×#64436	C₂³	1	2	yes	—
80	C₂×#64436	C₂³	1	1	yes	characteristic
81	C₂×#64436	C₂³	1	1	yes	characteristic
82	C₂³×C₃₇	—	2	1	no	—
83	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₃₇	—	2	1	no	—
85	Q₈×C₃₇	—	2	1	no	—
86	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
87	D₄×C₃₇	—	2	4	no	—
88	D₄×C₃₇	—	2	1	no	—
89	D₄×C₃₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₃₇	—	2	1	no	—
91	C₂×C₄×C₃₇	—	2	4	no	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	4	2	no	—
94	C₂×C₄×C₃₇	—	4	4	no	—
95	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₇	#41	1	2	yes	—
100	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₃₇	—	2	4	no	—
102	C₂×Dic₃₇	—	2	1	no	—
103	C₂×Dic₃₇	—	2	2	no	—
104	D₁₄₈	—	2	1	no	—
105	D₁₄₈	C₂²×C₄	1	2	yes	—
106	C₄×D₃₇	—	2	4	no	—
107	C₄×D₃₇	—	4	4	no	—
108	Dic₇₄	—	2	1	no	—
109	Dic₇₄	C₂²×C₄	1	2	yes	—
110	#64436	—	2	2	no	—
111	#64436	C₂²×C₄	1	2	yes	—
112	#64436	C₂²×C₄	1	2	yes	—
113	C₂²×C₃₇	—	2	2	no	—
114	C₂²×C₃₇	—	2	2	no	—
115	C₂²×C₃₇	—	4	2	no	—
116	C₂²×C₃₇	C₄×D₄	1	2	yes	—
117	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₃₇	—	2	2	no	—
119	C₂×D₃₇	—	4	2	no	—
120	C₄×C₃₇	—	2	2	no	—
121	C₄×C₃₇	—	2	2	no	—
122	C₄×C₃₇	—	2	2	no	—
123	C₄×C₃₇	—	4	2	no	—
124	C₄×C₃₇	C₄×D₄	1	2	yes	—
125	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
127	Dic₃₇	—	2	2	no	—
128	Dic₃₇	—	4	2	no	—
129	#2943	—	37	1	no	maximal, Sylow
130	C₂×C₃₇	—	2	2	no	—
131	C₂×C₃₇	—	4	2	no	—
132	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₃₇	#575	1	2	yes	—
134	D₃₇	—	4	2	no	—
135	C₂×#135	—	37	1	no	—
136	C₄×#41	—	37	1	no	—
137	C₄×#41	D₃₇	1	1	yes	characteristic, complemented
138	#442	—	37	2	no	—
139	#441	—	37	2	no	—
140	#440	—	37	2	no	—
141	#440	—	37	2	no	—
142	#427	—	37	1	no	—
143	#416	—	37	1	no	—
144	#416	—	37	1	no	—
145	C₂×#97	—	37	1	no	—
146	C₃₇	#2943	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	37	1	no	—
148	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
149	#135	—	37	8	no	—
150	C₂×Q₁₆	—	37	1	no	—
151	C₂×SD₁₆	—	37	1	no	—
152	C₂×SD₁₆	—	37	1	no	—
153	C₂×D₈	—	37	1	no	—
154	C₂²×C₈	—	37	1	no	—
155	C₂²×C₈	—	37	1	no	—
156	C₄×Q₈	—	37	2	no	—
157	C₄×Q₈	C₂×D₃₇	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	37	2	no	—
160	C₄×D₄	—	74	2	no	—
161	C₄×D₄	C₂×D₃₇	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	37	1	no	—
164	#117	—	74	1	no	—
165	#117	C₂×D₃₇	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	74	1	no	—
168	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
169	#107	—	37	2	no	—
170	#106	—	37	2	no	—
171	#103	—	37	2	no	—
172	#103	—	37	2	no	—
173	#102	—	37	2	no	—
174	#102	—	37	2	no	—
175	#97	—	37	2	no	—
176	#97	—	37	2	no	—
177	#41	—	2	2	no	—
178	#41	—	37	4	no	—
179	#41	—	74	2	no	—
180	#41	C₄×D₃₇	1	4	yes	complemented
181	C₂×Q₈	—	37	1	no	—
182	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	37	1	no	—
185	C₂×D₄	—	74	1	no	—
186	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	74	1	no	—
190	C₂²×C₄	—	74	2	no	—
191	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
192	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
193	Q₁₆	—	37	4	no	—
194	SD₁₆	—	37	4	no	—
195	SD₁₆	—	37	4	no	—
196	D₈	—	37	4	no	—
197	C₂×C₈	—	37	4	no	—
198	C₂×C₈	—	37	1	no	—
199	C₂×C₈	—	37	1	no	—
200	C₂×C₈	—	37	2	no	—
201	C₂×C₈	—	74	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	37	2	no	—
204	C₄⋊C₄	—	74	2	no	—
205	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	74	1	no	—
209	C₂²⋊C₄	—	74	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	74	2	no	—
213	C₄²	—	74	1	no	—
214	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
215	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
216	C₄²	C₂²×D₃₇	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	74	1	no	—
219	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	37	2	no	—
222	Q₈	—	74	1	no	—
223	Q₈	C₂×C₄×D₃₇	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	37	2	no	—
227	D₄	—	74	4	no	—
228	D₄	—	74	1	no	—
229	D₄	C₂×C₄×D₃₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	74	1	no	—
236	C₂×C₄	—	74	4	no	—
237	C₂×C₄	—	74	2	no	—
238	C₂×C₄	—	148	4	no	—
239	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
240	C₂×C₄	#10596962	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
244	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
245	C₈	—	37	2	no	—
246	C₈	—	37	2	no	—
247	C₈	—	74	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	74	2	no	—
252	C₂²	—	148	2	no	—
253	C₂²	C₄×C₂²⋊D₃₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	74	2	no	—
260	C₄	—	148	2	no	—
261	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
262	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
263	C₄	C₂×#10596962	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	148	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₃₇	1	1	yes	characteristic
268	C₂	#14558331	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×#41×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924556	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×#41	D₃₇
hypocenter	C₃₇	#2943
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#10596962	C₂	semidirect
3. #14557643	C₂	semidirect
4. #14557638	C₂	semidirect
5. #14557633	C₂	semidirect
6. #14557631	C₂	semidirect
7. #14557569	C₂	semidirect
8. #14557529	C₂	semidirect
9. #14557527	C₂	semidirect
10. C₂×#11924447	C₂	semidirect
11. C₄×#41×C₃₇	C₂	semidirect
13. #11924556	C₄	semidirect
43. #41×C₃₇	C₂×C₄	semidirect
53. #10596962	C₂×C₄	semidirect
68. #10596942	C₂×C₄	semidirect
69. #10596941	C₂×C₄	semidirect
70. #10596940	C₂×C₄	semidirect
71. #10596939	C₂×C₄	semidirect
78. C₂×#64436	C₂×C₄	semidirect
137. C₄×#41	D₃₇	semidirect
146. C₃₇	#2943	semidirect
180. #41	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924556	2
C₄×#10596962	2
#14557643	2
#14557638	2
#14557633	2
#14557631	2
#14557569	2
#14557529	2
#14557527	2
C₂×#11924447	2
C₄×#41×C₃₇	2
#2943	37