Subgroups — #18005184

Order 4736.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558078	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558069	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557941	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557919	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557896	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557894	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557786	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557784	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557781	C₂	1	1	yes	maximal, characteristic, complemented
10	SD₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557618	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557615	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557613	C₂	1	1	yes	maximal, characteristic, complemented
14	M₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596964	—	2	1	no	—
17	C₂×#10596941	C₂²	1	1	yes	characteristic
18	C₂×D₂₉₆	C₂²	1	1	yes	characteristic
19	C₂×#10596939	C₂²	1	1	yes	characteristic
20	C₂×#10596934	C₂²	1	1	yes	characteristic
21	#11924535	C₂²	1	1	yes	characteristic
22	#11924532	C₂²	1	1	yes	characteristic
23	#11924531	—	2	1	no	—
24	#11924527	C₂²	1	1	yes	characteristic
25	#11924525	C₂²	1	1	yes	characteristic
26	#11924522	—	2	1	no	—
27	#11924520	—	2	1	no	—
28	#11924510	C₂²	1	1	yes	characteristic
29	#11924510	C₂²	1	1	yes	characteristic
30	#11924508	—	2	1	no	—
31	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
32	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
33	#11924498	C₂²	1	1	yes	characteristic
34	#11924498	C₂²	1	1	yes	characteristic
35	#11924496	—	2	1	no	—
36	#11924496	—	2	1	no	—
37	#11924494	—	2	1	no	—
38	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
39	#11924469	C₂²	1	1	yes	characteristic
40	#11924468	C₂²	1	1	yes	characteristic
41	#11924467	C₂²	1	1	yes	characteristic
42	#11924460	C₂²	1	1	yes	characteristic
43	#11924458	C₂²	1	1	yes	characteristic
44	#11924458	C₂²	1	1	yes	characteristic
45	#11924457	C₂²	1	1	yes	characteristic
46	#11924456	C₂²	1	1	yes	characteristic
47	#11924456	C₂²	1	1	yes	characteristic
48	#11924446	C₂²	1	1	yes	characteristic
49	#11924446	C₂²	1	1	yes	characteristic
50	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
55	#121×C₃₇	C₂²	1	1	yes	characteristic
56	#107×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₃₇	—	2	1	no	—
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₃₇	—	2	1	no	—
64	C₂×C₂²⋊D₃₇	—	2	1	no	—
65	C₂×C₂²⋊D₃₇	—	2	1	no	—
66	C₂×C₂²⋊D₃₇	—	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	#10596964	—	4	4	no	—
71	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
72	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₃₇	—	2	1	no	—
74	C₂×C₄×D₃₇	—	2	1	no	—
75	C₂×Dic₇₄	—	2	1	no	—
76	SD₁₆×C₃₇	—	2	2	no	—
77	M₁₆×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₃₇	—	2	1	no	—
81	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₇	—	2	1	no	—
84	C₂²⋊C₄×C₃₇	—	2	1	no	—
85	C₂²⋊Dic₃₇	—	2	1	no	—
86	#10596941	—	2	2	no	—
87	#10596939	—	2	2	no	—
88	#10596938	—	2	1	no	—
89	#10596938	—	2	1	no	—
90	#10596938	—	2	1	no	—
91	#10596938	—	2	1	no	—
92	#10596938	—	2	1	no	—
93	#10596938	—	2	1	no	—
94	C₄⋊Dic₃₇	—	2	1	no	—
95	C₄⋊Dic₃₇	—	2	1	no	—
96	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
98	#10596936	—	2	1	no	—
99	#10596936	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	D₄	1	1	yes	characteristic
102	C₄×Dic₃₇	D₄	1	1	yes	characteristic
103	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
104	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
105	#10596934	—	2	2	no	—
106	C₂×#64436	C₂³	1	1	yes	characteristic
107	C₂×#64436	C₂³	1	1	yes	characteristic
108	D₂₉₆	—	2	2	no	—
109	C₂³×C₃₇	—	2	1	no	—
110	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₃₇	—	2	1	no	—
112	C₂²×D₃₇	—	2	1	no	—
113	Q₈×C₃₇	—	2	1	no	—
114	Q₈×C₃₇	—	2	1	no	—
115	D₄×C₃₇	—	2	1	no	—
116	D₄×C₃₇	—	2	1	no	—
117	D₄×C₃₇	—	4	2	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	—	2	1	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	2	1	no	—
123	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₃₇	—	4	2	no	—
126	C₂²⋊D₃₇	—	4	2	no	—
127	C₂²⋊D₃₇	—	4	2	no	—
128	C₂²⋊D₃₇	—	4	2	no	—
129	C₂×Dic₃₇	—	2	2	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	4	1	no	—
133	C₂×Dic₃₇	—	4	2	no	—
134	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
138	D₁₄₈	—	2	1	no	—
139	D₁₄₈	—	2	1	no	—
140	D₁₄₈	—	2	1	no	—
141	D₁₄₈	—	2	1	no	—
142	C₄×D₃₇	—	4	2	no	—
143	C₄×D₃₇	—	4	2	no	—
144	Dic₇₄	—	4	2	no	—
145	C₈×C₃₇	—	2	1	no	—
146	C₈×C₃₇	#41	1	2	yes	complemented
147	#64436	—	2	1	no	—
148	#64436	#41	1	2	yes	complemented
149	C₂²×C₃₇	—	2	1	no	—
150	C₂²×C₃₇	—	2	1	no	—
151	C₂²×C₃₇	—	2	1	no	—
152	C₂²×C₃₇	—	4	1	no	—
153	C₂²×C₃₇	—	4	1	no	—
154	C₂²×C₃₇	—	4	1	no	—
155	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₃₇	—	4	1	no	—
157	C₂×D₃₇	—	4	1	no	—
158	C₂×D₃₇	—	4	1	no	—
159	C₂×D₃₇	—	4	1	no	—
160	C₂×D₃₇	—	4	1	no	—
161	C₂×D₃₇	—	4	1	no	—
162	C₄×C₃₇	—	2	1	no	—
163	C₄×C₃₇	—	4	1	no	—
164	C₄×C₃₇	—	4	1	no	—
165	C₄×C₃₇	—	4	1	no	—
166	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
167	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
168	Dic₃₇	—	2	1	no	—
169	Dic₃₇	—	2	1	no	—
170	Dic₃₇	—	2	1	no	—
171	Dic₃₇	—	4	1	no	—
172	Dic₃₇	—	4	1	no	—
173	Dic₃₇	AGL(1,ℤ/8)	1	2	yes	—
174	#3171	—	37	1	no	maximal, Sylow
175	C₂×C₃₇	—	4	1	no	—
176	C₂×C₃₇	—	8	1	no	—
177	C₂×C₃₇	#532	1	1	yes	characteristic
178	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×C₃₇	#575	1	1	yes	characteristic
180	D₃₇	—	8	1	no	—
181	D₃₇	—	8	1	no	—
182	#532	—	37	1	no	—
183	#529	—	37	1	no	—
184	#501	—	37	1	no	—
185	#496	—	37	1	no	—
186	#490	—	37	1	no	—
187	#489	—	37	1	no	—
188	#469	—	37	1	no	—
189	#469	—	37	1	no	—
190	#468	—	37	1	no	—
191	#468	D₃₇	1	1	yes	characteristic, complemented
192	C₄×SD₁₆	—	37	1	no	—
193	C₄×SD₁₆	—	37	1	no	—
194	C₄×D₈	—	37	1	no	—
195	C₄×D₈	—	37	1	no	—
196	C₄×M₁₆	—	37	1	no	—
197	C₃₇	#3171	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	74	1	no	—
199	C₂×SD₁₆	—	37	1	no	—
200	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
201	C₂×D₈	—	37	1	no	—
202	C₂×D₈	—	37	1	no	—
203	C₂×M₁₆	—	37	1	no	—
204	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
205	C₄⋊Q₈	—	37	1	no	—
206	#127	—	37	1	no	—
207	#126	—	74	1	no	—
208	#125	—	37	1	no	—
209	#124	—	37	1	no	—
210	#124	—	74	1	no	—
211	#123	—	74	1	no	—
212	C₂²⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
213	#121	—	37	1	no	—
214	#121	—	37	1	no	—
215	#121	—	74	1	no	—
216	#121	C₂×D₃₇	1	1	yes	characteristic
217	C₄×Q₈	—	37	1	no	—
218	C₄×D₄	—	37	1	no	—
219	C₄×D₄	—	37	1	no	—
220	C₄×D₄	—	37	1	no	—
221	C₄×D₄	—	74	1	no	—
222	C₄×D₄	—	74	1	no	—
223	#117	—	74	1	no	—
224	C₂×C₄²	—	37	1	no	—
225	#107	—	37	1	no	—
226	#107	C₂×D₃₇	1	1	yes	characteristic
227	#106	—	37	1	no	—
228	#106	—	37	1	no	—
229	#103	—	37	1	no	—
230	#103	C₂×D₃₇	1	1	yes	characteristic
231	#102	—	37	1	no	—
232	#102	—	37	1	no	—
233	#102	—	37	1	no	—
234	#102	—	37	1	no	—
235	#102	—	37	1	no	—
236	#102	C₂×D₃₇	1	1	yes	characteristic
237	#97	—	37	1	no	—
238	#97	—	37	1	no	—
239	C₄×C₈	—	37	1	no	—
240	C₄×C₈	—	37	1	no	—
241	#41	—	148	4	no	—
242	C₂×Q₈	—	74	1	no	—
243	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	37	1	no	—
246	C₂×D₄	—	37	1	no	—
247	C₂×D₄	—	74	1	no	—
248	C₂×D₄	—	74	1	no	—
249	C₂×D₄	—	74	1	no	—
250	C₂×D₄	—	74	1	no	—
251	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
252	C₂²×C₄	—	37	1	no	—
253	C₂²×C₄	—	37	1	no	—
254	C₂²×C₄	—	74	1	no	—
255	C₂²×C₄	—	74	1	no	—
256	C₂²×C₄	—	74	1	no	—
257	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	74	2	no	—
260	D₈	—	74	2	no	—
261	D₈	—	74	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	74	2	no	—
264	C₂×C₈	—	37	1	no	—
265	C₂×C₈	—	37	1	no	—
266	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	37	1	no	—
270	C₄⋊C₄	—	37	1	no	—
271	C₄⋊C₄	—	74	1	no	—
272	C₄⋊C₄	—	74	1	no	—
273	C₄⋊C₄	—	74	1	no	—
274	C₄⋊C₄	—	74	1	no	—
275	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	74	1	no	—
280	C₂²⋊C₄	—	74	1	no	—
281	C₂²⋊C₄	—	74	1	no	—
282	C₂²⋊C₄	—	74	1	no	—
283	C₂²⋊C₄	—	74	1	no	—
284	C₂²⋊C₄	—	74	1	no	—
285	C₂²⋊C₄	—	74	1	no	—
286	C₄²	—	37	1	no	—
287	C₄²	—	37	1	no	—
288	C₄²	—	37	1	no	—
289	C₄²	—	37	1	no	—
290	C₄²	—	74	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	74	1	no	—
293	C₂³	—	74	1	no	—
294	C₂³	D₄×D₃₇	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	148	2	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	4	2	no	—
301	D₄	—	74	1	no	—
302	D₄	—	74	1	no	—
303	D₄	—	74	1	no	—
304	D₄	—	74	1	no	—
305	D₄	—	148	2	no	—
306	D₄	—	148	2	no	—
307	D₄	—	148	2	no	—
308	D₄	—	148	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	37	1	no	—
315	C₂×C₄	—	37	1	no	—
316	C₂×C₄	—	37	1	no	—
317	C₂×C₄	—	37	1	no	—
318	C₂×C₄	—	74	2	no	—
319	C₂×C₄	—	74	1	no	—
320	C₂×C₄	—	74	1	no	—
321	C₂×C₄	—	148	2	no	—
322	C₂×C₄	—	148	2	no	—
323	C₂×C₄	—	148	2	no	—
324	C₂×C₄	—	148	1	no	—
325	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	37	2	no	—
329	C₈	—	74	1	no	—
330	C₈	#10596964	1	2	yes	complemented
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	148	1	no	—
338	C₂²	—	148	1	no	—
339	C₂²	—	148	1	no	—
340	C₂²	—	148	1	no	—
341	C₂²	—	148	1	no	—
342	C₂²	—	148	1	no	—
343	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	37	2	no	—
349	C₄	—	74	1	no	—
350	C₄	—	74	1	no	—
351	C₄	—	74	1	no	—
352	C₄	—	148	1	no	—
353	C₄	—	148	1	no	—
354	C₄	#41×D₃₇	1	1	yes	characteristic
355	C₄	C₂×#10596964	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	8	1	no	—
358	C₂	—	296	1	no	—
359	C₂	—	296	1	no	—
360	C₂	#14558078	1	1	yes	characteristic
361	C₂	AGL(1,ℤ/8)×D₃₇	1	1	yes	characteristic
362	C₂	#14558332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#468	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#468	D₃₇
hypocenter	C₃₇	#3171
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #14558078	C₂	semidirect
2. #14558069	C₂	semidirect
3. #14557941	C₂	semidirect
4. #14557919	C₂	semidirect
5. #14557896	C₂	semidirect
6. #14557894	C₂	semidirect
7. #14557786	C₂	semidirect
8. #14557784	C₂	semidirect
9. #14557781	C₂	semidirect
10. SD₁₆×Dic₃₇	C₂	semidirect
11. #14557618	C₂	semidirect
12. #14557615	C₂	semidirect
13. #14557613	C₂	semidirect
14. M₁₆×Dic₃₇	C₂	semidirect
15. C₃₇×#468	C₂	semidirect
146. C₈×C₃₇	#41	semidirect
148. #64436	#41	semidirect
191. #468	D₃₇	semidirect
197. C₃₇	#3171	semidirect
330. C₈	#10596964	semidirect

Maximal subgroups

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

subgroup	index
#14558078	2
#14558069	2
#14557941	2
#14557919	2
#14557896	2
#14557894	2
#14557786	2
#14557784	2
#14557781	2
SD₁₆×Dic₃₇	2
#14557618	2
#14557615	2
#14557613	2
M₁₆×Dic₃₇	2
C₃₇×#468	2
#3171	37