Subgroups — #18004380

Order 4736.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558086	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557842	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557838	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557835	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557831	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557826	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557821	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557816	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557811	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557571	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557569	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#11924465	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557529	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#416	C₂	1	1	yes	maximal, characteristic
16	C₂²×#64436	C₂²	1	1	yes	characteristic
17	#11924535	C₂²	1	1	yes	characteristic
18	#11924535	C₂²	1	1	yes	characteristic
19	#11924527	C₂²	1	1	yes	characteristic
20	#11924527	C₂²	1	1	yes	characteristic
21	#11924513	—	2	1	no	—
22	#11924513	—	2	1	no	—
23	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₇₄	—	2	1	no	—
26	C₄×Dic₇₄	—	2	1	no	—
27	D₄×Dic₃₇	—	2	1	no	—
28	D₄×Dic₃₇	—	2	1	no	—
29	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
31	#11924495	—	2	1	no	—
32	#11924495	C₂²	1	1	yes	characteristic
33	#11924495	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₃₇	—	2	1	no	—
35	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
36	#11924470	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic
38	#11924469	C₂²	1	1	yes	characteristic
39	#11924469	C₂²	1	1	yes	characteristic
40	#11924468	C₂²	1	1	yes	characteristic
41	#11924468	C₂²	1	1	yes	characteristic
42	#11924467	C₂²	1	1	yes	characteristic
43	#11924467	C₂²	1	1	yes	characteristic
44	#11924465	C₂²	1	1	yes	characteristic
45	#11924465	C₂²	1	1	yes	characteristic
46	#11924465	C₂²	1	1	yes	characteristic
47	#11924465	C₂²	1	1	yes	characteristic
48	#11924460	C₂²	1	1	yes	characteristic
49	#11924460	C₂²	1	1	yes	characteristic
50	#11924457	C₂²	1	1	yes	characteristic
51	#11924457	C₂²	1	1	yes	characteristic
52	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
54	#117×C₃₇	C₂²	1	1	yes	characteristic
55	#103×C₃₇	C₂²	1	1	yes	characteristic
56	#103×C₃₇	C₂²	1	1	yes	characteristic
57	#102×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	#41×C₃₇	—	2	2	no	—
60	#41×C₃₇	Q₈	1	4	yes	complemented
61	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₃₇	—	2	1	no	—
63	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	—	2	1	no	—
65	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
66	C₂²×Dic₃₇	—	2	1	no	—
67	C₂²×Dic₃₇	—	2	1	no	—
68	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
69	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
70	C₂×Dic₇₄	—	2	1	no	—
71	C₂×Dic₇₄	—	2	1	no	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₇	—	2	1	no	—
78	C₄²×C₃₇	—	2	1	no	—
79	C₂²⋊Dic₃₇	—	2	1	no	—
80	C₂²⋊Dic₃₇	—	2	1	no	—
81	C₂²⋊Dic₃₇	—	2	1	no	—
82	C₂²⋊Dic₃₇	—	2	1	no	—
83	C₄⋊Dic₃₇	—	2	1	no	—
84	C₄⋊Dic₃₇	—	2	1	no	—
85	C₄⋊Dic₃₇	—	2	1	no	—
86	C₄⋊Dic₃₇	—	2	1	no	—
87	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
91	#10596936	—	2	1	no	—
92	#10596936	—	2	1	no	—
93	#10596936	—	2	1	no	—
94	#10596936	—	2	1	no	—
95	C₄×Dic₃₇	—	2	1	no	—
96	C₄×Dic₃₇	—	2	1	no	—
97	C₄×Dic₃₇	—	2	1	no	—
98	C₄×Dic₃₇	—	2	1	no	—
99	C₄×Dic₃₇	D₄	1	1	yes	characteristic
100	C₄×Dic₃₇	D₄	1	1	yes	characteristic
101	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
103	C₂×#64436	—	2	2	no	—
104	C₂×#64436	C₂³	1	1	yes	characteristic
105	C₂×#64436	C₂³	1	1	yes	characteristic
106	C₂³×C₃₇	—	2	1	no	—
107	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₃₇	—	2	1	no	—
109	Q₈×C₃₇	C₂×Q₈	1	2	yes	—
110	D₄×C₃₇	—	2	4	no	—
111	D₄×C₃₇	—	2	1	no	—
112	D₄×C₃₇	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₃₇	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
119	C₂×C₄×C₃₇	#41	1	2	yes	—
120	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₃₇	—	2	2	no	—
122	C₂×Dic₃₇	—	2	1	no	—
123	C₂×Dic₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	4	1	no	—
127	C₂×Dic₃₇	—	4	2	no	—
128	C₂×Dic₃₇	—	4	2	no	—
129	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
133	Dic₇₄	—	4	2	no	—
134	Dic₇₄	—	4	2	no	—
135	C₈×C₃₇	—	2	1	no	—
136	C₈×C₃₇	—	2	1	no	—
137	#64436	—	2	1	no	—
138	#64436	—	2	1	no	—
139	C₂²×C₃₇	—	2	1	no	—
140	C₂²×C₃₇	—	2	1	no	—
141	C₂²×C₃₇	—	2	2	no	—
142	C₂²×C₃₇	—	4	1	no	—
143	C₂²×C₃₇	—	4	1	no	—
144	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
145	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₃₇	—	2	2	no	—
147	C₄×C₃₇	—	4	1	no	—
148	C₄×C₃₇	—	4	1	no	—
149	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
150	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
151	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
152	Dic₃₇	—	2	1	no	—
153	Dic₃₇	—	2	1	no	—
154	Dic₃₇	—	2	1	no	—
155	Dic₃₇	—	4	1	no	—
156	Dic₃₇	—	4	1	no	—
157	Dic₃₇	—	4	1	no	—
158	Dic₃₇	—	4	1	no	—
159	Dic₃₇	#135	1	2	yes	—
160	#3076	—	37	1	no	maximal, Sylow
161	C₂×C₃₇	—	2	2	no	—
162	C₂×C₃₇	—	4	2	no	—
163	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
164	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
165	C₂×C₃₇	#575	1	1	yes	characteristic
166	#533	—	37	1	no	—
167	C₄×#41	—	37	1	no	—
168	#479	—	37	1	no	—
169	#479	—	37	1	no	—
170	#478	—	37	1	no	—
171	#478	—	37	1	no	—
172	#477	—	37	1	no	—
173	#476	—	37	1	no	—
174	#475	—	37	1	no	—
175	#474	—	37	1	no	—
176	#427	—	37	1	no	—
177	#427	—	37	1	no	—
178	C₂×C₄⋊C₈	—	37	1	no	—
179	#416	—	37	1	no	—
180	#416	D₃₇	1	1	yes	characteristic
181	C₃₇	#3076	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
183	C₂²×C₈	—	37	1	no	—
184	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
185	C₄⋊Q₈	—	37	1	no	—
186	C₄⋊Q₈	—	37	1	no	—
187	#125	—	37	1	no	—
188	#125	—	37	1	no	—
189	C₂²⋊Q₈	—	74	1	no	—
190	C₂²⋊Q₈	—	74	1	no	—
191	C₄×Q₈	—	37	1	no	—
192	C₄×Q₈	—	37	1	no	—
193	C₄×Q₈	—	74	1	no	—
194	C₄×Q₈	—	74	1	no	—
195	C₄×D₄	—	37	1	no	—
196	C₄×D₄	—	37	1	no	—
197	C₄×D₄	—	74	1	no	—
198	C₄×D₄	—	74	1	no	—
199	#117	—	37	1	no	—
200	#117	—	37	1	no	—
201	#117	—	74	1	no	—
202	#117	C₂×D₃₇	1	1	yes	characteristic
203	C₂×C₄²	—	37	1	no	—
204	C₂×C₄²	—	74	1	no	—
205	#107	—	37	1	no	—
206	#107	—	37	1	no	—
207	#107	—	37	1	no	—
208	#107	—	37	1	no	—
209	#106	—	37	1	no	—
210	#106	—	37	1	no	—
211	#106	—	37	1	no	—
212	#106	—	37	1	no	—
213	C₄⋊C₈	—	37	1	no	—
214	C₄⋊C₈	—	37	1	no	—
215	C₄⋊C₈	—	37	1	no	—
216	C₄⋊C₈	—	37	1	no	—
217	#103	—	37	1	no	—
218	#103	—	37	1	no	—
219	#103	C₂×D₃₇	1	1	yes	characteristic
220	#103	C₂×D₃₇	1	1	yes	characteristic
221	#102	—	37	1	no	—
222	#102	—	37	1	no	—
223	#102	C₂×D₃₇	1	1	yes	characteristic
224	#102	C₂×D₃₇	1	1	yes	characteristic
225	#41	—	2	2	no	—
226	#41	Dic₇₄	1	4	yes	complemented
227	C₂×Q₈	—	74	1	no	—
228	C₂×Q₈	—	74	1	no	—
229	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	37	1	no	—
234	C₂²×C₄	—	37	1	no	—
235	C₂²×C₄	—	74	1	no	—
236	C₂²×C₄	—	74	1	no	—
237	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	37	1	no	—
240	C₂×C₈	—	37	1	no	—
241	C₂×C₈	—	74	2	no	—
242	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
244	C₄⋊C₄	—	37	1	no	—
245	C₄⋊C₄	—	37	1	no	—
246	C₄⋊C₄	—	37	1	no	—
247	C₄⋊C₄	—	37	1	no	—
248	C₄⋊C₄	—	74	1	no	—
249	C₄⋊C₄	—	74	1	no	—
250	C₄⋊C₄	—	74	1	no	—
251	C₄⋊C₄	—	74	1	no	—
252	C₄⋊C₄	—	74	1	no	—
253	C₄⋊C₄	—	74	1	no	—
254	C₄⋊C₄	—	74	1	no	—
255	C₄⋊C₄	—	74	1	no	—
256	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	74	1	no	—
260	C₂²⋊C₄	—	74	1	no	—
261	C₂²⋊C₄	—	74	1	no	—
262	C₂²⋊C₄	—	74	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	37	1	no	—
265	C₄²	—	37	1	no	—
266	C₄²	—	37	1	no	—
267	C₄²	—	37	1	no	—
268	C₄²	—	74	1	no	—
269	C₄²	—	74	1	no	—
270	C₄²	—	74	1	no	—
271	C₄²	—	74	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	D₄×D₃₇	1	1	yes	characteristic
274	Q₈	—	2	1	no	—
275	Q₈	—	148	2	no	—
276	Q₈	—	148	2	no	—
277	Q₈	C₂×Dic₇₄	1	2	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	1	no	—
280	D₄	C₂×Dic₇₄	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	37	1	no	—
286	C₂×C₄	—	37	1	no	—
287	C₂×C₄	—	37	1	no	—
288	C₂×C₄	—	37	1	no	—
289	C₂×C₄	—	74	2	no	—
290	C₂×C₄	—	74	1	no	—
291	C₂×C₄	—	74	1	no	—
292	C₂×C₄	—	74	1	no	—
293	C₂×C₄	—	74	1	no	—
294	C₂×C₄	—	148	2	no	—
295	C₂×C₄	—	148	1	no	—
296	C₂×C₄	—	148	2	no	—
297	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
298	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
299	C₂×C₄	#10596964	1	2	yes	—
300	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	74	1	no	—
304	C₈	—	74	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#11924513	1	2	yes	—
311	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	37	2	no	—
316	C₄	—	74	1	no	—
317	C₄	—	74	1	no	—
318	C₄	—	74	1	no	—
319	C₄	—	148	1	no	—
320	C₄	—	148	1	no	—
321	C₄	—	148	1	no	—
322	C₄	—	148	1	no	—
323	C₄	#11924513	1	2	yes	—
324	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
325	C₄	C₂×#10596964	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	C₂×#11924513	1	1	yes	characteristic
329	C₂	#135×D₃₇	1	1	yes	characteristic
330	C₂	#14558330	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#416	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#41×C₃₇	C₂²
hypercenter	#416	D₃₇
hypocenter	C₃₇	#3076
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #14558086	C₂	semidirect
3. #14557842	C₂	semidirect
4. #14557838	C₂	semidirect
5. #14557835	C₂	semidirect
6. #14557831	C₂	semidirect
7. #14557826	C₂	semidirect
8. #14557821	C₂	semidirect
9. #14557816	C₂	semidirect
10. #14557811	C₂	semidirect
11. #14557571	C₂	semidirect
12. #14557569	C₂	semidirect
13. C₂×#11924465	C₂	semidirect
60. #41×C₃₇	Q₈	semidirect
181. C₃₇	#3076	semidirect
226. #41	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#14558086	2
#41×Dic₃₇	2
#14557842	2
#14557838	2
#14557835	2
#14557831	2
#14557826	2
#14557821	2
#14557816	2
#14557811	2
#14557571	2
#14557569	2
C₂×#11924465	2
#14557529	2
C₃₇×#416	2
#3076	37