Subgroups — #18007986

Order 4736.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558154	C₂	1	4	yes	maximal, complemented
2	#14558145	C₂	1	2	yes	maximal, complemented
3	#14558138	C₂	1	2	yes	maximal, complemented
4	#14558130	C₂	1	4	yes	maximal, complemented
5	#14558121	C₂	1	4	yes	maximal, complemented
6	#14558090	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558083	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558061	C₂	1	8	yes	maximal, complemented
9	#14558044	C₂	1	2	yes	maximal, complemented
10	C₄×#10596962	C₂	1	2	yes	maximal, complemented
11	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	2	yes	—
13	#11924536	C₂²	1	1	yes	characteristic, complemented
14	#11924536	C₂²	1	2	yes	complemented
15	#11924533	C₂²	1	1	yes	characteristic, complemented
16	#11924531	C₂²	1	8	yes	complemented
17	#11924529	C₂²	1	8	yes	—
18	#11924528	C₂²	1	2	yes	complemented
19	#11924526	C₂²	1	8	yes	complemented
20	#11924524	C₂²	1	2	yes	complemented
21	#11924522	C₂²	1	8	yes	—
22	#11924521	C₂²	1	8	yes	complemented
23	#11924519	C₂²	1	4	yes	—
24	#11924518	C₂²	1	8	yes	—
25	#11924515	C₂²	1	4	yes	—
26	#11924515	C₂²	1	4	yes	—
27	#11924514	C₂²	1	8	yes	complemented
28	#11924510	C₂²	1	4	yes	—
29	#11924510	C₂²	1	4	yes	—
30	#11924509	C₂²	1	8	yes	—
31	#11924505	C₂²	1	4	yes	—
32	C₄×Dic₇₄	C₂²	1	4	yes	complemented
33	C₄×Dic₇₄	C₂²	1	4	yes	complemented
34	C₄×D₁₄₈	C₂²	1	4	yes	complemented
35	C₄×D₁₄₈	C₂²	1	4	yes	complemented
36	C₄×C₂²⋊D₃₇	C₂²	1	8	yes	—
37	C₄×C₂²⋊D₃₇	C₂²	1	8	yes	—
38	#11924495	C₂²	1	2	yes	—
39	#11924493	C₂²	1	4	yes	complemented
40	C₄²×D₃₇	C₂²	1	4	yes	complemented
41	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
42	#117×C₃₇	C₂²	1	4	yes	—
43	#117×C₃₇	C₂²	1	2	yes	—
44	#117×C₃₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
47	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
49	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₇	—	2	4	no	—
51	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₃₇	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₃₇	C₂³	1	8	yes	—
59	#10596962	—	2	8	no	—
60	C₂×D₁₄₈	C₂³	1	2	yes	—
61	C₂×D₁₄₈	C₂³	1	4	yes	—
62	C₂×C₄×D₃₇	C₂³	1	4	yes	—
63	C₂×C₄×D₃₇	C₂³	1	8	yes	—
64	C₂×Dic₇₄	C₂³	1	2	yes	—
65	C₂×Dic₇₄	C₂³	1	4	yes	—
66	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
67	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₃₇	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₃₇	C₂³	1	4	yes	—
70	C₄²×C₃₇	C₂³	1	2	yes	—
71	C₄²×C₃₇	C₂³	1	2	yes	—
72	C₄²×C₃₇	C₂³	1	2	yes	—
73	C₄²×C₃₇	C₂³	1	1	yes	characteristic
74	C₄²×C₃₇	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₃₇	C₂³	1	4	yes	—
76	C₂²⋊Dic₃₇	C₂³	1	8	yes	—
77	#10596938	C₂³	1	8	yes	—
78	#10596938	C₂³	1	8	yes	—
79	#10596938	C₂³	1	8	yes	—
80	C₄⋊Dic₃₇	C₂³	1	4	yes	—
81	C₄⋊Dic₃₇	C₂³	1	4	yes	—
82	C₄⋊Dic₃₇	C₂³	1	4	yes	—
83	#10596936	C₂³	1	8	yes	—
84	#10596936	C₂³	1	8	yes	—
85	#10596936	C₂³	1	8	yes	—
86	C₄×Dic₃₇	C₂³	1	4	yes	—
87	C₄×Dic₃₇	C₂³	1	8	yes	—
88	C₂³×C₃₇	—	2	2	no	—
89	C₂³×C₃₇	—	4	2	no	—
90	C₂³×C₃₇	C₂⁴	1	2	yes	—
91	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
92	C₂²×D₃₇	C₂⁴	1	4	yes	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	4	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	4	2	no	—
97	C₂×C₄×C₃₇	—	4	2	no	—
98	C₂×C₄×C₃₇	#41	1	4	yes	—
99	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
100	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
101	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
102	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
103	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₃₇	—	2	8	no	—
106	C₂²⋊D₃₇	—	4	8	no	—
107	C₂×Dic₃₇	C₂⁴	1	4	yes	—
108	C₂×Dic₃₇	C₂⁴	1	8	yes	—
109	D₁₄₈	—	2	4	no	—
110	D₁₄₈	—	4	4	no	—
111	C₄×D₃₇	—	2	8	no	—
112	C₄×D₃₇	—	4	8	no	—
113	Dic₇₄	—	2	4	no	—
114	Dic₇₄	—	4	4	no	—
115	C₂²×C₃₇	—	2	2	no	—
116	C₂²×C₃₇	—	2	2	no	—
117	C₂²×C₃₇	—	2	4	no	—
118	C₂²×C₃₇	—	4	4	no	—
119	C₂²×C₃₇	C₂×#41	1	2	yes	—
120	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
121	C₂×D₃₇	—	2	4	no	—
122	C₂×D₃₇	—	2	8	no	—
123	C₄×C₃₇	—	2	2	no	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	—	2	4	no	—
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	C₂×#41	1	2	yes	—
128	C₄×C₃₇	C₂×#41	1	2	yes	—
129	Dic₃₇	—	2	4	no	—
130	Dic₃₇	—	2	8	no	—
131	#3455	—	37	1	no	maximal, Sylow
132	C₂×C₃₇	—	2	2	no	—
133	C₂×C₃₇	—	4	2	no	—
134	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
135	C₂×C₃₇	#585	1	2	yes	—
136	D₃₇	—	4	4	no	—
137	#542	—	37	4	no	—
138	#541	—	37	2	no	—
139	#540	—	37	2	no	—
140	#539	—	37	4	no	—
141	#538	—	37	4	no	—
142	#533	—	37	1	no	—
143	#532	—	37	1	no	—
144	#529	—	37	8	no	—
145	#525	—	37	2	no	—
146	C₄×#41	—	37	2	no	—
147	C₂×#117	D₃₇	1	1	yes	characteristic, complemented
148	C₃₇	#3455	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	37	2	no	—
150	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
151	C₄⋊Q₈	—	37	1	no	—
152	C₄⋊Q₈	—	37	2	no	—
153	#127	—	37	1	no	—
154	#126	—	37	8	no	—
155	#126	—	37	8	no	—
156	#125	—	37	8	no	—
157	#125	—	37	2	no	—
158	#124	—	37	8	no	—
159	#124	—	37	2	no	—
160	#123	—	37	4	no	—
161	#123	—	37	8	no	—
162	#123	—	37	8	no	—
163	C₂²⋊Q₈	—	37	4	no	—
164	C₂²⋊Q₈	—	37	4	no	—
165	C₂²⋊Q₈	—	37	8	no	—
166	#121	—	37	4	no	—
167	#121	—	37	4	no	—
168	#121	—	37	8	no	—
169	#120	—	37	4	no	—
170	C₄×Q₈	—	37	4	no	—
171	C₄×Q₈	—	37	4	no	—
172	C₄×D₄	—	37	8	no	—
173	C₄×D₄	—	37	8	no	—
174	C₄×D₄	—	37	4	no	—
175	C₄×D₄	—	37	4	no	—
176	#117	—	37	2	no	—
177	#117	—	37	4	no	—
178	#117	C₂×D₃₇	1	4	yes	—
179	#117	C₂×D₃₇	1	2	yes	—
180	#117	C₂×D₃₇	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
183	C₂×C₄²	—	37	4	no	—
184	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
186	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
187	#41	—	74	8	no	—
188	C₂×Q₈	—	37	2	no	—
189	C₂×Q₈	—	37	4	no	—
190	C₂×D₄	—	37	4	no	—
191	C₂×D₄	—	37	8	no	—
192	C₂×D₄	—	37	2	no	—
193	C₂×D₄	—	37	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	37	4	no	—
196	C₂²×C₄	—	37	8	no	—
197	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
199	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
200	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
201	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
203	C₄⋊C₄	—	37	8	no	—
204	C₄⋊C₄	—	37	8	no	—
205	C₄⋊C₄	—	37	8	no	—
206	C₄⋊C₄	—	37	4	no	—
207	C₄⋊C₄	—	37	4	no	—
208	C₄⋊C₄	—	37	4	no	—
209	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
210	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
211	C₂²⋊C₄	—	37	4	no	—
212	C₂²⋊C₄	—	37	8	no	—
213	C₂²⋊C₄	—	37	8	no	—
214	C₂²⋊C₄	—	37	8	no	—
215	C₂²⋊C₄	—	37	8	no	—
216	C₂²⋊C₄	C₂²×D₃₇	1	4	yes	—
217	C₂²⋊C₄	C₂²×D₃₇	1	4	yes	—
218	C₄²	—	37	8	no	—
219	C₄²	—	37	4	no	—
220	C₄²	C₂²×D₃₇	1	2	yes	—
221	C₄²	C₂²×D₃₇	1	1	yes	characteristic
222	C₄²	C₂²×D₃₇	1	1	yes	characteristic
223	C₄²	C₂²×D₃₇	1	2	yes	—
224	C₄²	C₂²×D₃₇	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	37	4	no	—
228	C₂³	C₂³×D₃₇	1	2	yes	—
229	C₂³	C₂³×D₃₇	1	1	yes	characteristic
230	Q₈	—	74	4	no	—
231	Q₈	—	148	4	no	—
232	D₄	—	74	8	no	—
233	D₄	—	74	4	no	—
234	D₄	—	148	4	no	—
235	D₄	—	148	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	37	4	no	—
242	C₂×C₄	—	37	8	no	—
243	C₂×C₄	—	74	8	no	—
244	C₂×C₄	—	148	8	no	—
245	C₂×C₄	#10596962	1	4	yes	—
246	C₂×C₄	C₂³×D₃₇	1	4	yes	—
247	C₂×C₄	C₂³×D₃₇	1	2	yes	—
248	C₂×C₄	C₂³×D₃₇	1	2	yes	—
249	C₂×C₄	C₂³×D₃₇	1	2	yes	—
250	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	74	4	no	—
257	C₂²	—	74	8	no	—
258	C₂²	C₂×#10596962	1	2	yes	—
259	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	74	4	no	—
265	C₄	—	74	8	no	—
266	C₄	C₂×#10596962	1	2	yes	—
267	C₄	C₂×#10596962	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	148	4	no	—
271	C₂	C₂²×#10596962	1	1	yes	characteristic
272	C₂	#14558369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#117×C₃₇	C₂
hypercenter	C₂×#117	D₃₇
hypocenter	C₃₇	#3455
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #14558154	C₂	semidirect
2. #14558145	C₂	semidirect
3. #14558138	C₂	semidirect
4. #14558130	C₂	semidirect
5. #14558121	C₂	semidirect
6. #14558090	C₂	semidirect
7. #14558083	C₂	semidirect
8. #14558061	C₂	semidirect
9. #14558044	C₂	semidirect
10. C₄×#10596962	C₂	semidirect
11. C₂×#117×C₃₇	C₂	semidirect
13. #11924536	C₂²	semidirect
14. #11924536	C₂²	semidirect
15. #11924533	C₂²	semidirect
16. #11924531	C₂²	semidirect
18. #11924528	C₂²	semidirect
19. #11924526	C₂²	semidirect
20. #11924524	C₂²	semidirect
22. #11924521	C₂²	semidirect
27. #11924514	C₂²	semidirect
32. C₄×Dic₇₄	C₂²	semidirect
33. C₄×Dic₇₄	C₂²	semidirect
34. C₄×D₁₄₈	C₂²	semidirect
35. C₄×D₁₄₈	C₂²	semidirect
39. #11924493	C₂²	semidirect
40. C₄²×D₃₇	C₂²	semidirect
147. C₂×#117	D₃₇	semidirect
148. C₃₇	#3455	semidirect

Maximal subgroups

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

subgroup	index
#14558154	2
#14558145	2
#14558138	2
#14558130	2
#14558121	2
#14558090	2
#14558083	2
#14558061	2
#14558044	2
C₄×#10596962	2
C₂×#117×C₃₇	2
#3455	37