Subgroups — C₂×#14558120 #18007936

Order 4736.

1336 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558120	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924531	C₂	1	2	yes	maximal, complemented
3	C₂×#11924522	C₂	1	2	yes	maximal, complemented
4	C₂×#11924520	C₂	1	2	yes	maximal, complemented
5	C₂×#11924515	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924508	C₂	1	2	yes	maximal, complemented
7	C₂×#11924504	C₂	1	2	yes	maximal, complemented
8	C₂×D₄×Dic₃₇	C₂	1	2	yes	maximal, complemented
9	C₂²×#10596938	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#121×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₃₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₂²⋊D₃₇	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₃₇	C₂²	1	2	yes	—
15	C₂³×Dic₃₇	C₂²	1	1	yes	characteristic
16	C₂³×Dic₃₇	C₂²	1	2	yes	complemented
17	#11924531	C₂²	1	16	yes	complemented
18	#11924522	C₂²	1	16	yes	complemented
19	#11924520	C₂²	1	16	yes	complemented
20	#11924515	C₂²	1	8	yes	complemented
21	#11924508	C₂²	1	16	yes	complemented
22	#11924504	C₂²	1	16	yes	complemented
23	D₄×Dic₃₇	C₂²	1	16	yes	complemented
24	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	complemented
25	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×#10596936	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊Dic₃₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₃₇	C₂²	1	2	yes	complemented
29	C₂×#10596938	C₂²	1	2	yes	—
30	C₂×#10596938	C₂²	1	2	yes	—
31	C₂×#10596938	C₂²	1	2	yes	—
32	C₂×#10596938	C₂²	1	2	yes	—
33	C₂×#10596938	C₂²	1	8	yes	complemented
34	C₂×C₄×Dic₃₇	C₂²	1	2	yes	complemented
35	C₂²×D₄×C₃₇	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic, complemented
37	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
38	#121×C₃₇	C₂²	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	complemented
41	C₂⁴×C₃₇	C₂³	1	2	yes	—
42	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
43	C₂³×D₃₇	—	2	4	no	—
44	C₂³×D₃₇	—	4	2	no	—
45	C₂³×D₃₇	—	4	2	no	—
46	C₂³×D₃₇	C₂³	1	2	yes	—
47	C₂³×D₃₇	C₂³	1	4	yes	—
48	C₂×D₄×C₃₇	—	2	8	no	—
49	C₂×D₄×C₃₇	—	4	2	no	—
50	C₂×D₄×C₃₇	C₂³	1	8	yes	—
51	C₂×D₄×C₃₇	C₂³	1	4	yes	complemented
52	C₂²×C₄×C₃₇	—	4	2	no	—
53	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₃₇	—	4	4	no	—
58	C₂×C₂²⋊D₃₇	—	4	4	no	—
59	C₂×C₂²⋊D₃₇	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₃₇	C₂³	1	8	yes	—
61	C₂²×Dic₃₇	—	4	2	no	—
62	C₂²×Dic₃₇	—	4	4	no	—
63	C₂²×Dic₃₇	C₂³	1	2	yes	—
64	C₂²×Dic₃₇	C₂³	1	2	yes	—
65	C₂²×Dic₃₇	C₂³	1	2	yes	—
66	C₂²×Dic₃₇	C₂³	1	4	yes	—
67	C₂²×Dic₃₇	C₂³	1	8	yes	complemented
68	C₂×Dic₇₄	—	4	2	no	—
69	C₂×Dic₇₄	C₂³	1	4	yes	complemented
70	C₄⋊C₄×C₃₇	C₂³	1	4	yes	complemented
71	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	complemented
72	C₂²⋊Dic₃₇	C₂³	1	8	yes	—
73	C₂²⋊Dic₃₇	C₂³	1	8	yes	complemented
74	#10596938	C₂³	1	8	yes	—
75	#10596938	C₂³	1	8	yes	—
76	#10596938	C₂³	1	8	yes	—
77	#10596938	C₂³	1	8	yes	—
78	C₄⋊Dic₃₇	C₂³	1	8	yes	complemented
79	C₄⋊Dic₃₇	C₂³	1	4	yes	complemented
80	#10596936	C₂³	1	8	yes	complemented
81	C₄×Dic₃₇	C₂³	1	8	yes	complemented
82	C₂³×C₃₇	—	2	2	no	—
83	C₂³×C₃₇	—	2	2	no	—
84	C₂³×C₃₇	—	2	4	no	—
85	C₂³×C₃₇	—	2	4	no	—
86	C₂³×C₃₇	—	2	4	no	—
87	C₂³×C₃₇	#41	1	4	yes	—
88	C₂³×C₃₇	C₂⁴	1	4	yes	—
89	C₂³×C₃₇	C₂⁴	1	2	yes	—
90	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
91	C₂²×D₃₇	—	2	8	no	—
92	C₂²×D₃₇	—	2	4	no	—
93	C₂²×D₃₇	—	2	4	no	—
94	C₂²×D₃₇	—	2	4	no	—
95	C₂²×D₃₇	—	4	4	no	—
96	C₂²×D₃₇	—	4	4	no	—
97	C₂²×D₃₇	—	4	16	no	—
98	C₂²×D₃₇	C₂⁴	1	4	yes	—
99	D₄×C₃₇	—	2	16	no	—
100	D₄×C₃₇	—	4	4	no	—
101	C₂×C₄×C₃₇	—	2	4	no	—
102	C₂×C₄×C₃₇	—	2	2	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
106	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
107	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
108	C₂²⋊D₃₇	—	4	8	no	—
109	C₂²⋊D₃₇	—	4	8	no	—
110	C₂×Dic₃₇	—	2	4	no	—
111	C₂×Dic₃₇	—	2	4	no	—
112	C₂×Dic₃₇	—	2	4	no	—
113	C₂×Dic₃₇	—	4	4	no	—
114	C₂×Dic₃₇	—	4	8	no	—
115	C₂×Dic₃₇	C₂⁴	1	4	yes	—
116	C₂×Dic₃₇	C₂⁴	1	4	yes	—
117	C₂×Dic₃₇	C₂⁴	1	4	yes	—
118	Dic₇₄	—	4	4	no	—
119	C₂²×C₃₇	—	2	8	no	—
120	C₂²×C₃₇	—	2	2	no	—
121	C₂²×C₃₇	—	2	2	no	—
122	C₂²×C₃₇	—	2	4	no	—
123	C₂²×C₃₇	—	2	4	no	—
124	C₂²×C₃₇	—	2	4	no	—
125	C₂²×C₃₇	—	4	8	no	—
126	C₂²×C₃₇	C₂×#41	1	4	yes	—
127	C₂²×C₃₇	C₂×#41	1	2	yes	—
128	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
129	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
131	C₂×D₃₇	—	2	4	no	—
132	C₂×D₃₇	—	2	4	no	—
133	C₂×D₃₇	—	2	4	no	—
134	C₂×D₃₇	—	4	16	no	—
135	C₂×D₃₇	—	4	8	no	—
136	C₄×C₃₇	—	2	4	no	—
137	C₄×C₃₇	—	2	2	no	—
138	C₄×C₃₇	—	2	2	no	—
139	Dic₃₇	—	2	4	no	—
140	Dic₃₇	—	2	4	no	—
141	Dic₃₇	—	2	4	no	—
142	C₂×#538	—	37	1	no	maximal, Sylow
143	C₂×C₃₇	—	2	4	no	—
144	C₂×C₃₇	—	4	4	no	—
145	C₂×C₃₇	#538	1	4	yes	complemented
146	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
147	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
149	D₃₇	—	4	4	no	—
150	#538	—	37	16	no	—
151	C₂×#126	—	37	2	no	—
152	C₂×#124	—	37	2	no	—
153	C₂×#123	—	37	2	no	—
154	C₂×C₂²⋊Q₈	—	37	1	no	—
155	C₂×#121	—	37	2	no	—
156	C₂×#121	D₃₇	1	1	yes	characteristic, complemented
157	C₂×#120	—	37	2	no	—
158	C₂×C₄×D₄	—	37	2	no	—
159	C₂²×C₂²⋊C₄	—	37	1	no	—
160	C₃₇	C₂×#538	1	1	yes	characteristic, complemented, Sylow
161	C₂⁵	—	37	1	no	—
162	C₂²×Q₈	—	37	1	no	—
163	C₂²×D₄	—	37	2	no	—
164	C₂²×D₄	—	37	2	no	—
165	C₂²×D₄	C₂×D₃₇	1	2	yes	—
166	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic, complemented
167	C₂³×C₄	—	37	1	no	—
168	C₂³×C₄	—	37	2	no	—
169	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
170	#126	—	37	16	no	—
171	#124	—	37	16	no	—
172	#123	—	37	16	no	—
173	C₂²⋊Q₈	—	37	8	no	—
174	#121	—	37	16	no	—
175	#121	C₂×D₃₇	1	8	yes	complemented
176	#120	—	37	16	no	—
177	C₄×D₄	—	37	16	no	—
178	C₂×C₄⋊C₄	—	37	2	no	—
179	C₂×C₄⋊C₄	—	37	2	no	—
180	C₂×C₄⋊C₄	—	37	1	no	—
181	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic, complemented
182	C₂×C₂²⋊C₄	—	37	2	no	—
183	C₂×C₂²⋊C₄	—	37	2	no	—
184	C₂×C₂²⋊C₄	—	37	2	no	—
185	C₂×C₂²⋊C₄	—	37	2	no	—
186	C₂×C₂²⋊C₄	—	37	8	no	—
187	C₂×C₂²⋊C₄	—	37	2	no	—
188	C₂×C₂²⋊C₄	—	37	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	complemented
190	C₂×C₄²	—	37	2	no	—
191	C₂⁴	—	37	2	no	—
192	C₂⁴	—	37	4	no	—
193	C₂⁴	—	74	4	no	—
194	C₂⁴	—	148	2	no	—
195	C₂⁴	—	148	2	no	—
196	C₂⁴	C₂²×D₃₇	1	2	yes	—
197	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
198	C₂×Q₈	—	37	4	no	—
199	C₂×Q₈	—	148	2	no	—
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	37	8	no	—
203	C₂×D₄	—	37	8	no	—
204	C₂×D₄	—	148	4	no	—
205	C₂×D₄	—	148	4	no	—
206	C₂×D₄	C₂²×D₃₇	1	8	yes	—
207	C₂×D₄	C₂²×D₃₇	1	4	yes	complemented
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	37	2	no	—
210	C₂²×C₄	—	37	2	no	—
211	C₂²×C₄	—	37	2	no	—
212	C₂²×C₄	—	37	4	no	—
213	C₂²×C₄	—	37	8	no	—
214	C₂²×C₄	—	148	2	no	—
215	C₂²×C₄	—	148	4	no	—
216	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
217	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
220	C₄⋊C₄	—	37	8	no	—
221	C₄⋊C₄	—	37	8	no	—
222	C₄⋊C₄	—	37	4	no	—
223	C₄⋊C₄	C₂²×D₃₇	1	4	yes	complemented
224	C₂²⋊C₄	—	37	8	no	—
225	C₂²⋊C₄	—	37	8	no	—
226	C₂²⋊C₄	—	37	8	no	—
227	C₂²⋊C₄	—	37	8	no	—
228	C₂²⋊C₄	—	37	8	no	—
229	C₂²⋊C₄	—	37	8	no	—
230	C₂²⋊C₄	C₂²×D₃₇	1	8	yes	complemented
231	C₄²	—	37	8	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	37	4	no	—
238	C₂³	—	74	8	no	—
239	C₂³	—	74	4	no	—
240	C₂³	—	74	4	no	—
241	C₂³	—	74	4	no	—
242	C₂³	—	148	4	no	—
243	C₂³	—	148	4	no	—
244	C₂³	—	148	16	no	—
245	C₂³	#10596964	1	4	yes	—
246	C₂³	C₂³×D₃₇	1	4	yes	—
247	C₂³	C₂³×D₃₇	1	2	yes	—
248	C₂³	C₂³×D₃₇	1	1	yes	characteristic
249	Q₈	—	148	4	no	—
250	D₄	—	2	16	no	—
251	D₄	—	4	4	no	—
252	D₄	—	148	8	no	—
253	D₄	—	148	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	37	4	no	—
259	C₂×C₄	—	37	4	no	—
260	C₂×C₄	—	37	4	no	—
261	C₂×C₄	—	74	4	no	—
262	C₂×C₄	—	74	4	no	—
263	C₂×C₄	—	74	4	no	—
264	C₂×C₄	—	148	4	no	—
265	C₂×C₄	—	148	8	no	—
266	C₂×C₄	C₂³×D₃₇	1	4	yes	—
267	C₂×C₄	C₂³×D₃₇	1	2	yes	—
268	C₂×C₄	C₂³×D₃₇	1	2	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	74	4	no	—
277	C₂²	—	74	4	no	—
278	C₂²	—	74	4	no	—
279	C₂²	—	148	16	no	—
280	C₂²	—	148	8	no	—
281	C₂²	C₂×#10596964	1	4	yes	—
282	C₂²	C₂×#10596964	1	2	yes	—
283	C₂²	#11924581	1	2	yes	—
284	C₂²	#11924582	1	2	yes	—
285	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	74	4	no	—
290	C₄	—	74	4	no	—
291	C₄	—	74	4	no	—
292	C₂	—	2	4	no	—
293	C₂	—	4	4	no	—
294	C₂	—	148	4	no	—
295	C₂	#14558120	1	4	yes	direct product factor
296	C₂	C₂²×#10596964	1	1	yes	characteristic
297	C₂	C₂×#11924581	1	1	yes	characteristic
298	C₂	C₂×#11924582	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#121×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
hypercenter	C₂×#121	D₃₇
hypocenter	C₃₇	C₂×#538
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. #14558120	C₂	direct (N × H)
2. C₂×#11924531	C₂	semidirect
3. C₂×#11924522	C₂	semidirect
4. C₂×#11924520	C₂	semidirect
5. C₂×#11924515	C₂	semidirect
6. C₂×#11924508	C₂	semidirect
7. C₂×#11924504	C₂	semidirect
8. C₂×D₄×Dic₃₇	C₂	semidirect
9. C₂²×#10596938	C₂	semidirect
10. C₂×#121×C₃₇	C₂	semidirect
12. C₂²×Dic₇₄	C₂²	semidirect
16. C₂³×Dic₃₇	C₂²	semidirect
17. #11924531	C₂²	semidirect
18. #11924522	C₂²	semidirect
19. #11924520	C₂²	semidirect
20. #11924515	C₂²	semidirect
21. #11924508	C₂²	semidirect
22. #11924504	C₂²	semidirect
23. D₄×Dic₃₇	C₂²	semidirect
24. C₂×C₄⋊Dic₃₇	C₂²	semidirect
25. C₂×C₄⋊Dic₃₇	C₂²	semidirect
26. C₂×#10596936	C₂²	semidirect
28. C₂×C₂²⋊Dic₃₇	C₂²	semidirect
33. C₂×#10596938	C₂²	semidirect
34. C₂×C₄×Dic₃₇	C₂²	semidirect
36. C₂²×D₄×C₃₇	C₂²	semidirect
38. #121×C₃₇	C₂²	semidirect
39. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₃₇	C₂²	semidirect
51. C₂×D₄×C₃₇	C₂³	semidirect
67. C₂²×Dic₃₇	C₂³	semidirect
69. C₂×Dic₇₄	C₂³	semidirect
70. C₄⋊C₄×C₃₇	C₂³	semidirect
71. C₂²⋊C₄×C₃₇	C₂³	semidirect
73. C₂²⋊Dic₃₇	C₂³	semidirect
78. C₄⋊Dic₃₇	C₂³	semidirect
79. C₄⋊Dic₃₇	C₂³	semidirect
80. #10596936	C₂³	semidirect
81. C₄×Dic₃₇	C₂³	semidirect
145. C₂×C₃₇	#538	semidirect
156. C₂×#121	D₃₇	semidirect
160. C₃₇	C₂×#538	semidirect
166. C₂²×D₄	C₂×D₃₇	semidirect
175. #121	C₂×D₃₇	semidirect
181. C₂×C₄⋊C₄	C₂×D₃₇	semidirect
189. C₂×C₂²⋊C₄	C₂×D₃₇	semidirect
207. C₂×D₄	C₂²×D₃₇	semidirect
223. C₄⋊C₄	C₂²×D₃₇	semidirect
230. C₂²⋊C₄	C₂²×D₃₇	semidirect
295. C₂	#14558120	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558120	2
C₂×#11924531	2
C₂×#11924522	2
C₂×#11924520	2
C₂×#11924515	2
C₂×#11924508	2
C₂×#11924504	2
C₂×D₄×Dic₃₇	2
C₂²×#10596938	2
C₂×#121×C₃₇	2
C₂×#538	37

Subgroups — C2×#14558120 #18007936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558120 #18007936