Subgroups — #18006339

Order 4736.

482 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	C₂×#11924558	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558192	C₂	1	2	yes	maximal, complemented
4	#14557917	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557774	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557769	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557756	C₂	1	2	yes	maximal, complemented
8	#14557697	C₂	1	2	yes	maximal, complemented
9	#14557674	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	2	yes	—
13	C₂²×Dic₇₄	C₂²	1	2	yes	—
14	#11924558	—	2	4	no	—
15	C₂×Dic₁₄₈	—	2	4	no	—
16	C₂×Dic₁₄₈	C₂²	1	2	yes	—
17	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
19	C₂×#10596930	C₂²	1	2	yes	—
20	C₂×#10596930	C₂²	1	2	yes	—
21	C₂×D₂₉₆	C₂²	1	1	yes	characteristic
22	#11924524	C₂²	1	2	yes	—
23	#11924522	—	2	2	no	—
24	#11924515	C₂²	1	2	yes	—
25	#11924514	—	2	2	no	—
26	#11924510	C₂²	1	2	yes	—
27	#11924508	—	2	2	no	—
28	C₄×Dic₇₄	C₂²	1	2	yes	—
29	#11924496	—	2	2	no	—
30	#11924470	C₂²	1	1	yes	characteristic
31	#11924461	C₂²	1	2	yes	—
32	#11924461	C₂²	1	2	yes	—
33	#11924458	C₂²	1	2	yes	—
34	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₇	—	2	2	no	—
45	C₂×C₂²⋊D₃₇	—	2	2	no	—
46	C₂²×Dic₃₇	—	2	2	no	—
47	#10596962	—	2	4	no	—
48	#10596962	—	4	2	no	—
49	C₂×D₁₄₈	C₂³	1	2	yes	—
50	C₂×C₄×D₃₇	—	2	2	no	—
51	C₂×Dic₇₄	—	2	4	no	—
52	C₂×Dic₇₄	—	2	2	no	—
53	C₂×Dic₇₄	—	4	2	no	—
54	C₂×Dic₇₄	C₂³	1	2	yes	—
55	C₂×Dic₇₄	C₂³	1	2	yes	—
56	C₂×C₈×C₃₇	—	2	2	no	—
57	C₂×C₈×C₃₇	—	2	2	no	—
58	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
65	C₄²×C₃₇	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₃₇	—	2	2	no	—
67	#10596938	—	2	2	no	—
68	#10596938	—	2	2	no	—
69	#10596938	—	2	2	no	—
70	C₄⋊Dic₃₇	C₂³	1	2	yes	—
71	#10596936	—	2	2	no	—
72	#10596936	—	2	2	no	—
73	C₄×Dic₃₇	—	2	2	no	—
74	Dic₁₄₈	—	2	4	no	—
75	Dic₁₄₈	—	2	2	no	—
76	Dic₁₄₈	D₄	1	4	yes	complemented
77	D₂₉₆	—	2	2	no	—
78	#10596930	—	2	4	no	—
79	#10596930	—	4	2	no	—
80	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₃₇	—	2	2	no	—
83	D₄×C₃₇	—	2	2	no	—
84	C₂×C₄×C₃₇	—	2	1	no	—
85	C₂×C₄×C₃₇	—	2	1	no	—
86	C₂×C₄×C₃₇	—	2	1	no	—
87	C₂×C₄×C₃₇	—	2	1	no	—
88	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₃₇	—	4	4	no	—
94	C₂²⋊D₃₇	—	4	4	no	—
95	C₂×Dic₃₇	—	2	2	no	—
96	C₂×Dic₃₇	—	2	2	no	—
97	C₂×Dic₃₇	—	2	2	no	—
98	C₂×Dic₃₇	—	4	4	no	—
99	D₁₄₈	—	2	2	no	—
100	D₁₄₈	—	2	2	no	—
101	C₄×D₃₇	—	4	4	no	—
102	Dic₇₄	—	2	2	no	—
103	Dic₇₄	—	2	2	no	—
104	Dic₇₄	—	2	2	no	—
105	Dic₇₄	—	4	4	no	—
106	Dic₇₄	C₂×D₄	1	4	yes	—
107	C₈×C₃₇	—	2	1	no	—
108	C₈×C₃₇	—	2	1	no	—
109	C₈×C₃₇	—	2	1	no	—
110	C₈×C₃₇	C₂×D₄	1	2	yes	—
111	C₂²×C₃₇	—	2	1	no	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	—	2	1	no	—
115	C₂²×C₃₇	—	2	1	no	—
116	C₂²×C₃₇	#135	1	2	yes	complemented
117	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₃₇	—	4	2	no	—
119	C₂×D₃₇	—	4	2	no	—
120	C₂×D₃₇	—	4	2	no	—
121	C₄×C₃₇	—	2	1	no	—
122	C₄×C₃₇	—	2	1	no	—
123	C₄×C₃₇	—	2	1	no	—
124	C₄×C₃₇	—	2	1	no	—
125	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₃₇	—	2	4	no	—
128	Dic₃₇	—	4	2	no	—
129	Dic₃₇	—	4	2	no	—
130	#3285	—	37	1	no	maximal, Sylow
131	C₂×C₃₇	—	2	2	no	—
132	C₂×C₃₇	—	4	1	no	—
133	C₂×C₃₇	D₄²	1	1	yes	characteristic
134	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
135	C₂×C₃₇	#578	1	1	yes	characteristic
136	D₃₇	—	8	2	no	—
137	C₂×#135	—	37	1	no	—
138	C₂²×Q₁₆	—	37	1	no	—
139	#548	—	37	2	no	—
140	#495	—	37	1	no	—
141	#467	—	37	1	no	—
142	#466	—	37	1	no	—
143	#464	—	37	2	no	—
144	#452	—	37	2	no	—
145	#448	—	37	2	no	—
146	C₄×Q₁₆	—	37	1	no	—
147	C₈×D₄	D₃₇	1	1	yes	characteristic, complemented
148	C₃₇	#3285	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	37	2	no	—
150	C₂²×Q₈	—	37	2	no	—
151	#135	—	74	4	no	—
152	C₂×Q₁₆	—	37	1	no	—
153	C₂×Q₁₆	—	37	2	no	—
154	C₂×Q₁₆	—	37	1	no	—
155	C₂×Q₁₆	—	74	4	no	—
156	C₂×SD₁₆	—	37	2	no	—
157	C₂×SD₁₆	—	37	2	no	—
158	C₂×D₈	—	37	1	no	—
159	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
161	#124	—	37	2	no	—
162	#124	—	74	2	no	—
163	C₂²⋊Q₈	—	37	2	no	—
164	C₂²⋊Q₈	—	74	2	no	—
165	#121	—	37	2	no	—
166	#121	—	74	2	no	—
167	C₄×Q₈	—	37	2	no	—
168	C₄×D₄	—	74	2	no	—
169	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
170	#107	—	37	1	no	—
171	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
172	#103	—	37	2	no	—
173	#103	—	37	2	no	—
174	#102	—	37	2	no	—
175	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
177	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
178	#41	—	74	4	no	—
179	#41	—	148	2	no	—
180	C₂×Q₈	—	37	2	no	—
181	C₂×Q₈	—	37	2	no	—
182	C₂×Q₈	—	74	4	no	—
183	C₂×Q₈	—	74	2	no	—
184	C₂×Q₈	—	148	2	no	—
185	C₂×D₄	—	37	2	no	—
186	C₂×D₄	—	74	2	no	—
187	C₂×D₄	—	74	2	no	—
188	C₂×D₄	D₁₄₈	1	1	yes	characteristic
189	C₂²×C₄	—	74	2	no	—
190	C₂²×C₄	—	74	2	no	—
191	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
193	Q₁₆	—	37	4	no	—
194	Q₁₆	—	74	4	no	—
195	Q₁₆	—	74	2	no	—
196	SD₁₆	—	74	4	no	—
197	SD₁₆	—	148	2	no	—
198	D₈	—	74	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
205	C₄⋊C₄	—	37	2	no	—
206	C₄⋊C₄	—	74	2	no	—
207	C₄⋊C₄	—	74	2	no	—
208	C₄⋊C₄	D₁₄₈	1	1	yes	characteristic
209	C₂²⋊C₄	—	74	2	no	—
210	C₂²⋊C₄	—	74	2	no	—
211	C₂²⋊C₄	—	74	2	no	—
212	C₂²⋊C₄	—	74	2	no	—
213	C₂²⋊C₄	D₁₄₈	1	1	yes	characteristic
214	C₂²⋊C₄	D₁₄₈	1	1	yes	characteristic
215	C₄²	—	74	2	no	—
216	C₄²	C₂²×D₃₇	1	1	yes	characteristic
217	C₂³	—	74	2	no	—
218	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
219	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
220	Q₈	—	37	4	no	—
221	Q₈	—	74	2	no	—
222	Q₈	—	74	2	no	—
223	Q₈	—	74	2	no	—
224	Q₈	—	148	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	74	2	no	—
227	D₄	—	74	2	no	—
228	D₄	—	148	4	no	—
229	D₄	—	148	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	74	2	no	—
235	C₂×C₄	—	74	2	no	—
236	C₂×C₄	—	74	2	no	—
237	C₂×C₄	—	148	4	no	—
238	C₂×C₄	—	148	4	no	—
239	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
240	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
241	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₃₇	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	148	2	no	—
254	C₂²	—	148	2	no	—
255	C₂²	—	148	2	no	—
256	C₂²	#11924558	1	2	yes	complemented
257	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	74	4	no	—
263	C₄	—	148	2	no	—
264	C₄	—	148	2	no	—
265	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
266	C₄	#11924582	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	296	2	no	—
270	C₂	D₄×D₁₄₈	1	1	yes	characteristic
271	C₂	C₂×#11924558	1	1	yes	characteristic
272	C₂	#14558350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₈×D₄×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	C₈×D₄	D₃₇
hypocenter	C₃₇	#3285
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11924558	C₂	semidirect
2. C₂²×Dic₁₄₈	C₂	semidirect
3. #14558192	C₂	semidirect
4. #14557917	C₂	semidirect
5. #14557774	C₂	semidirect
6. #14557769	C₂	semidirect
7. #14557756	C₂	semidirect
8. #14557697	C₂	semidirect
9. #14557674	C₂	semidirect
10. C₄×Dic₁₄₈	C₂	semidirect
11. C₈×D₄×C₃₇	C₂	semidirect
76. Dic₁₄₈	D₄	semidirect
116. C₂²×C₃₇	#135	semidirect
147. C₈×D₄	D₃₇	semidirect
148. C₃₇	#3285	semidirect
256. C₂²	#11924558	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924558	2
C₂²×Dic₁₄₈	2
#14558192	2
#14557917	2
#14557774	2
#14557769	2
#14557756	2
#14557697	2
#14557674	2
C₄×Dic₁₄₈	2
C₈×D₄×C₃₇	2
#3285	37