Subgroups — C₂×#14558239 #18008126

Order 4736.

1200 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558239	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924535	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924527	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11924516	C₂	1	2	yes	maximal, complemented
5	C₂×#11924514	C₂	1	2	yes	maximal, complemented
6	C₂×#11924514	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₄₈	C₂	1	1	yes	maximal, characteristic
9	C₂×#11924498	C₂	1	1	yes	maximal, characteristic
10	C₂×#11924498	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₃₇	C₂	1	2	yes	maximal, complemented
12	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇₄	C₂²	1	2	yes	—
15	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
17	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
18	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
19	#11924535	C₂²	1	8	yes	complemented
20	#11924527	C₂²	1	8	yes	complemented
21	#11924516	C₂²	1	16	yes	complemented
22	#11924514	C₂²	1	16	yes	complemented
23	#11924514	C₂²	1	16	yes	complemented
24	C₄×Dic₇₄	C₂²	1	8	yes	complemented
25	C₄×D₁₄₈	C₂²	1	8	yes	—
26	#11924498	C₂²	1	8	yes	—
27	#11924498	C₂²	1	8	yes	—
28	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
31	C₂×#10596936	C₂²	1	1	yes	characteristic
32	C₂×#10596936	C₂²	1	2	yes	—
33	C₂×#10596936	C₂²	1	1	yes	characteristic
34	C₂×#10596936	C₂²	1	2	yes	—
35	C₄⋊C₄×D₃₇	C₂²	1	16	yes	complemented
36	C₂×#10596938	C₂²	1	2	yes	—
37	C₂×#10596938	C₂²	1	2	yes	—
38	C₂×#10596938	C₂²	1	2	yes	—
39	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
41	#125×C₃₇	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
47	C₂³×D₃₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₃₇	C₂³	1	2	yes	—
57	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
58	C₂×D₁₄₈	Q₈	1	8	yes	complemented
59	C₂×D₁₄₈	C₂³	1	4	yes	—
60	C₂×C₄×D₃₇	—	2	8	no	—
61	C₂×C₄×D₃₇	—	4	4	no	—
62	C₂×C₄×D₃₇	—	4	4	no	—
63	C₂×C₄×D₃₇	C₂³	1	8	yes	—
64	C₂×C₄×D₃₇	C₂³	1	8	yes	—
65	C₂×C₄×D₃₇	C₂³	1	8	yes	—
66	C₂×Dic₇₄	—	2	4	no	—
67	C₂×Dic₇₄	—	4	4	no	—
68	C₂×Dic₇₄	C₂³	1	4	yes	—
69	C₂×Dic₇₄	C₂³	1	8	yes	—
70	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
71	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
72	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
73	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
74	C₄²×C₃₇	C₂³	1	4	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	—
79	C₄⋊Dic₃₇	C₂³	1	4	yes	—
80	C₄⋊Dic₃₇	C₂³	1	4	yes	—
81	#10596936	C₂³	1	4	yes	—
82	#10596936	C₂³	1	8	yes	—
83	#10596936	C₂³	1	4	yes	—
84	#10596936	C₂³	1	8	yes	—
85	C₄×Dic₃₇	C₂³	1	4	yes	—
86	C₄×Dic₃₇	C₂³	1	4	yes	—
87	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
88	C₂²×D₃₇	—	2	4	no	—
89	C₂²×D₃₇	—	2	4	no	—
90	C₂²×D₃₇	C₂×Q₈	1	8	yes	—
91	C₂²×D₃₇	C₂⁴	1	4	yes	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	2	4	no	—
94	C₂×C₄×C₃₇	—	2	4	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
98	C₂×C₄×C₃₇	C₂⁴	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×Dic₃₇	—	2	2	no	—
103	C₂×Dic₃₇	—	2	2	no	—
104	C₂×Dic₃₇	—	2	4	no	—
105	C₂×Dic₃₇	—	2	2	no	—
106	C₂×Dic₃₇	#41	1	4	yes	—
107	C₂×Dic₃₇	C₂⁴	1	2	yes	—
108	C₂×Dic₃₇	C₂⁴	1	2	yes	—
109	C₂×Dic₃₇	C₂⁴	1	2	yes	—
110	C₂×Dic₃₇	C₂⁴	1	4	yes	—
111	C₂×Dic₃₇	C₂⁴	1	2	yes	—
112	D₁₄₈	C₂×Q₈	1	16	yes	complemented
113	C₄×D₃₇	—	2	16	no	—
114	C₄×D₃₇	—	4	8	no	—
115	C₄×D₃₇	—	4	8	no	—
116	Dic₇₄	—	2	8	no	—
117	Dic₇₄	—	4	8	no	—
118	C₂²×C₃₇	C₂²×Q₈	1	2	yes	—
119	C₂²×C₃₇	C₂×#41	1	2	yes	—
120	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
122	C₂×D₃₇	—	2	16	no	—
123	C₂×D₃₇	—	2	4	no	—
124	C₂×D₃₇	—	2	4	no	—
125	C₂×D₃₇	C₂²×Q₈	1	8	yes	—
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	—	2	4	no	—
128	C₄×C₃₇	—	2	4	no	—
129	C₄×C₃₇	—	2	2	no	—
130	C₄×C₃₇	C₂²×Q₈	1	4	yes	—
131	Dic₃₇	—	2	2	no	—
132	Dic₃₇	—	2	2	no	—
133	Dic₃₇	—	2	4	no	—
134	Dic₃₇	—	2	2	no	—
135	Dic₃₇	C₂×#41	1	4	yes	—
136	C₂×#554	—	37	1	no	maximal, Sylow
137	C₂×C₃₇	#554	1	4	yes	complemented
138	C₂×C₃₇	C₂³×Q₈	1	1	yes	characteristic
139	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
140	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	D₃₇	—	2	8	no	—
142	#554	—	37	16	no	—
143	C₂×C₄⋊Q₈	—	37	1	no	—
144	C₂×#125	—	37	1	no	—
145	C₂×#125	D₃₇	1	1	yes	characteristic, complemented
146	C₂×C₂²⋊Q₈	—	37	2	no	—
147	C₂×C₂²⋊Q₈	—	37	2	no	—
148	C₂×C₂²⋊Q₈	—	37	2	no	—
149	C₂×C₄×Q₈	—	37	1	no	—
150	C₂×C₄×D₄	—	37	1	no	—
151	C₂×C₄×D₄	—	37	1	no	—
152	C₂×C₄×D₄	—	37	1	no	—
153	C₂²×C₄⋊C₄	—	37	2	no	—
154	C₃₇	C₂×#554	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	37	1	no	—
156	C₂²×Q₈	—	37	2	no	—
157	C₂²×D₄	—	37	1	no	—
158	C₂³×C₄	—	37	2	no	—
159	C₂³×C₄	—	37	2	no	—
160	C₂³×C₄	—	37	2	no	—
161	C₄⋊Q₈	—	37	8	no	—
162	#125	—	37	8	no	—
163	#125	C₂×D₃₇	1	8	yes	complemented
164	C₂²⋊Q₈	—	37	16	no	—
165	C₂²⋊Q₈	—	37	16	no	—
166	C₂²⋊Q₈	—	37	16	no	—
167	C₄×Q₈	—	37	8	no	—
168	C₄×D₄	—	37	8	no	—
169	C₄×D₄	—	37	8	no	—
170	C₄×D₄	—	37	8	no	—
171	C₂×C₄⋊C₄	—	37	1	no	—
172	C₂×C₄⋊C₄	—	37	2	no	—
173	C₂×C₄⋊C₄	—	37	2	no	—
174	C₂×C₄⋊C₄	—	37	1	no	—
175	C₂×C₄⋊C₄	—	37	1	no	—
176	C₂×C₄⋊C₄	—	37	1	no	—
177	C₂×C₄⋊C₄	—	37	2	no	—
178	C₂×C₄⋊C₄	—	37	16	no	—
179	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
180	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
183	C₂×C₂²⋊C₄	—	37	2	no	—
184	C₂×C₂²⋊C₄	—	37	2	no	—
185	C₂×C₂²⋊C₄	—	37	2	no	—
186	C₂×C₄²	—	37	1	no	—
187	C₂×C₄²	—	37	1	no	—
188	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
189	C₂⁴	—	37	2	no	—
190	C₂×Q₈	—	37	4	no	—
191	C₂×Q₈	—	37	8	no	—
192	C₂×Q₈	—	74	4	no	—
193	C₂×Q₈	—	148	4	no	—
194	C₂×D₄	—	37	8	no	—
195	C₂×D₄	—	37	4	no	—
196	C₂²×C₄	—	37	1	no	—
197	C₂²×C₄	—	37	1	no	—
198	C₂²×C₄	—	37	1	no	—
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	37	2	no	—
201	C₂²×C₄	—	37	8	no	—
202	C₂²×C₄	—	37	8	no	—
203	C₂²×C₄	—	37	8	no	—
204	C₂²×C₄	—	74	8	no	—
205	C₂²×C₄	—	148	4	no	—
206	C₂²×C₄	—	148	4	no	—
207	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
210	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
211	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
212	C₄⋊C₄	—	37	4	no	—
213	C₄⋊C₄	—	37	8	no	—
214	C₄⋊C₄	—	37	4	no	—
215	C₄⋊C₄	—	37	8	no	—
216	C₄⋊C₄	—	37	8	no	—
217	C₄⋊C₄	—	37	4	no	—
218	C₄⋊C₄	—	37	4	no	—
219	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
220	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
221	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
222	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
223	C₂²⋊C₄	—	37	8	no	—
224	C₂²⋊C₄	—	37	8	no	—
225	C₂²⋊C₄	—	37	8	no	—
226	C₄²	—	37	4	no	—
227	C₄²	—	37	4	no	—
228	C₄²	C₂²×D₃₇	1	4	yes	—
229	C₂³	—	37	8	no	—
230	C₂³	—	37	4	no	—
231	C₂³	—	74	4	no	—
232	C₂³	—	74	4	no	—
233	C₂³	C₂³×D₃₇	1	1	yes	characteristic
234	Q₈	—	74	8	no	—
235	Q₈	—	148	8	no	—
236	D₄	—	37	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	37	4	no	—
242	C₂×C₄	—	37	2	no	—
243	C₂×C₄	—	37	2	no	—
244	C₂×C₄	—	37	2	no	—
245	C₂×C₄	—	37	2	no	—
246	C₂×C₄	—	37	4	no	—
247	C₂×C₄	—	74	16	no	—
248	C₂×C₄	—	74	2	no	—
249	C₂×C₄	—	74	2	no	—
250	C₂×C₄	—	74	4	no	—
251	C₂×C₄	—	74	2	no	—
252	C₂×C₄	—	148	8	no	—
253	C₂×C₄	—	148	8	no	—
254	C₂×C₄	Q₈×D₃₇	1	4	yes	—
255	C₂×C₄	C₂³×D₃₇	1	2	yes	—
256	C₂×C₄	C₂³×D₃₇	1	2	yes	—
257	C₂×C₄	C₂³×D₃₇	1	4	yes	—
258	C₂×C₄	C₂³×D₃₇	1	4	yes	—
259	C₂×C₄	C₂³×D₃₇	1	2	yes	—
260	C₂²	—	37	8	no	—
261	C₂²	—	74	16	no	—
262	C₂²	—	74	4	no	—
263	C₂²	—	74	4	no	—
264	C₂²	C₂×Q₈×D₃₇	1	2	yes	—
265	C₂²	#41×D₃₇	1	2	yes	—
266	C₂²	#11924582	1	2	yes	—
267	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	37	4	no	—
273	C₄	—	74	2	no	—
274	C₄	—	74	2	no	—
275	C₄	—	74	4	no	—
276	C₄	—	74	2	no	—
277	C₄	C₂×Q₈×D₃₇	1	4	yes	—
278	C₂	—	74	8	no	—
279	C₂	#14558239	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₃₇	1	1	yes	characteristic
281	C₂	C₂×#41×D₃₇	1	1	yes	characteristic
282	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×#125×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂²×D₁₄₈	C₂²
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	C₂×#554
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #14558239	C₂	direct (N × H)
2. C₂×#11924535	C₂	semidirect
3. C₂×#11924527	C₂	semidirect
4. C₂×#11924516	C₂	semidirect
5. C₂×#11924514	C₂	semidirect
6. C₂×#11924514	C₂	semidirect
7. C₂×C₄×Dic₇₄	C₂	semidirect
11. C₂×C₄⋊C₄×D₃₇	C₂	semidirect
12. C₂×#125×C₃₇	C₂	semidirect
19. #11924535	C₂²	semidirect
20. #11924527	C₂²	semidirect
21. #11924516	C₂²	semidirect
22. #11924514	C₂²	semidirect
23. #11924514	C₂²	semidirect
24. C₄×Dic₇₄	C₂²	semidirect
35. C₄⋊C₄×D₃₇	C₂²	semidirect
41. #125×C₃₇	C₂²	semidirect
58. C₂×D₁₄₈	Q₈	semidirect
112. D₁₄₈	C₂×Q₈	semidirect
137. C₂×C₃₇	#554	semidirect
145. C₂×#125	D₃₇	semidirect
154. C₃₇	C₂×#554	semidirect
163. #125	C₂×D₃₇	semidirect
279. C₂	#14558239	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558239	2
C₂×#11924535	2
C₂×#11924527	2
C₂×#11924516	2
C₂×#11924514	2
C₂×C₄×Dic₇₄	2
C₂×C₄×D₁₄₈	2
C₂×#11924498	2
C₂×C₄⋊C₄×D₃₇	2
C₂×#125×C₃₇	2
C₂×#554	37

Subgroups — C2×#14558239 #18008126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558239 #18008126