Subgroups — #18006433

Order 4736.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558223	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558179	C₂	1	1	yes	maximal, characteristic
3	#14557888	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557864	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557858	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557834	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557774	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557762	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557756	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557697	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557686	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10596930	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557530	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#11924458	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596962	C₂²	1	1	yes	characteristic
17	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic, complemented
19	C₂×#10596930	C₂²	1	1	yes	characteristic
20	C₂×#10596930	C₂²	1	1	yes	characteristic
21	#11924529	—	2	1	no	—
22	#11924528	C₂²	1	1	yes	characteristic, complemented
23	#11924526	—	2	1	no	—
24	#11924524	C₂²	1	1	yes	characteristic
25	#11924521	—	2	1	no	—
26	#11924517	—	2	1	no	—
27	#11924515	C₂²	1	1	yes	characteristic
28	#11924515	C₂²	1	1	yes	characteristic
29	#11924515	C₂²	1	1	yes	characteristic, complemented
30	#11924510	C₂²	1	1	yes	characteristic
31	#11924509	—	2	1	no	—
32	#11924504	—	2	1	no	—
33	C₄×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
34	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
35	#11924495	C₂²	1	1	yes	characteristic, complemented
36	#11924492	—	2	1	no	—
37	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×D₃₇	—	2	1	no	—
39	#11924470	C₂²	1	1	yes	characteristic, complemented
40	#11924468	C₂²	1	1	yes	characteristic, complemented
41	#11924468	C₂²	1	1	yes	characteristic, complemented
42	#11924461	C₂²	1	1	yes	characteristic, complemented
43	#11924461	C₂²	1	1	yes	characteristic, complemented
44	#11924461	C₂²	1	1	yes	characteristic, complemented
45	#11924461	C₂²	1	1	yes	characteristic, complemented
46	#11924458	C₂²	1	1	yes	characteristic
47	#11924458	C₂²	1	1	yes	characteristic
48	#11924458	C₂²	1	1	yes	characteristic
49	#11924458	C₂²	1	1	yes	characteristic
50	#11924458	C₂²	1	1	yes	characteristic
51	#11924458	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
56	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₃₇	—	2	1	no	—
60	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
61	C₂²×C₄×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	C₂²×Dic₃₇	—	2	1	no	—
66	#10596962	—	2	2	no	—
67	#10596962	—	4	1	no	—
68	C₂×D₁₄₈	—	2	2	no	—
69	C₂×D₁₄₈	—	2	1	no	—
70	C₂×D₁₄₈	—	4	1	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	C₂³	1	1	yes	characteristic
76	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₃₇	—	4	1	no	—
78	C₂×C₈×C₃₇	—	4	1	no	—
79	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
86	C₄²×C₃₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₃₇	—	2	1	no	—
88	C₂²⋊Dic₃₇	—	2	1	no	—
89	C₂²⋊Dic₃₇	—	2	1	no	—
90	C₂²⋊Dic₃₇	—	2	1	no	—
91	#10596938	—	2	1	no	—
92	#10596938	—	2	1	no	—
93	#10596938	—	2	1	no	—
94	#10596938	—	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	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
100	#10596936	—	2	1	no	—
101	#10596936	—	2	1	no	—
102	#10596936	—	2	1	no	—
103	#10596936	—	2	1	no	—
104	#10596936	—	2	1	no	—
105	#10596936	—	2	1	no	—
106	C₄×Dic₃₇	—	2	1	no	—
107	C₄×Dic₃₇	—	2	1	no	—
108	Dic₁₄₈	—	4	1	no	—
109	#10596930	—	2	2	no	—
110	#10596930	—	4	1	no	—
111	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₃₇	—	2	1	no	—
114	C₂²×D₃₇	—	2	1	no	—
115	C₂²×D₃₇	—	4	2	no	—
116	C₂²×D₃₇	—	4	2	no	—
117	C₂²×D₃₇	—	8	1	no	—
118	D₄×C₃₇	—	2	2	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	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₃₇	—	4	2	no	—
129	C₂²⋊D₃₇	—	8	1	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	2	1	no	—
133	C₂×Dic₃₇	—	2	1	no	—
134	C₂×Dic₃₇	—	2	1	no	—
135	C₂×Dic₃₇	—	2	1	no	—
136	C₂×Dic₃₇	—	4	2	no	—
137	D₁₄₈	—	2	1	no	—
138	D₁₄₈	—	2	1	no	—
139	D₁₄₈	—	2	1	no	—
140	D₁₄₈	—	4	2	no	—
141	D₁₄₈	#41	1	2	yes	—
142	C₄×D₃₇	—	4	2	no	—
143	C₄×D₃₇	—	8	1	no	—
144	Dic₇₄	—	2	1	no	—
145	Dic₇₄	—	2	1	no	—
146	Dic₇₄	—	2	1	no	—
147	Dic₇₄	—	2	1	no	—
148	C₈×C₃₇	—	2	1	no	—
149	C₈×C₃₇	—	2	1	no	—
150	C₈×C₃₇	—	2	1	no	—
151	C₈×C₃₇	—	2	1	no	—
152	C₂²×C₃₇	—	2	1	no	—
153	C₂²×C₃₇	—	2	1	no	—
154	C₂²×C₃₇	—	2	1	no	—
155	C₂²×C₃₇	—	2	1	no	—
156	C₂²×C₃₇	—	2	1	no	—
157	C₂²×C₃₇	#135	1	2	yes	—
158	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₃₇	—	2	2	no	—
160	C₂×D₃₇	—	4	1	no	—
161	C₂×D₃₇	—	4	1	no	—
162	C₂×D₃₇	—	4	1	no	—
163	C₂×D₃₇	—	4	1	no	—
164	C₂×D₃₇	—	4	1	no	—
165	C₂×D₃₇	—	8	4	no	—
166	C₄×C₃₇	—	2	1	no	—
167	C₄×C₃₇	—	2	1	no	—
168	C₄×C₃₇	—	2	1	no	—
169	C₄×C₃₇	—	2	1	no	—
170	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
171	C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₃₇	—	4	1	no	—
173	Dic₃₇	—	4	1	no	—
174	Dic₃₇	—	4	1	no	—
175	Dic₃₇	—	4	1	no	—
176	Dic₃₇	—	4	1	no	—
177	Dic₃₇	—	4	1	no	—
178	#3296	—	37	1	no	maximal, Sylow
179	C₂×C₃₇	—	2	2	no	—
180	C₂×C₃₇	—	4	1	no	—
181	C₂×C₃₇	#546	1	1	yes	characteristic
182	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
183	C₂×C₃₇	#577	1	1	yes	characteristic
184	D₃₇	—	4	2	no	—
185	D₃₇	—	8	1	no	—
186	#552	—	37	1	no	—
187	#546	—	37	1	no	—
188	#488	—	37	1	no	—
189	#483	—	37	1	no	—
190	#482	—	37	1	no	—
191	#478	—	37	1	no	—
192	#467	—	37	1	no	—
193	#465	—	37	1	no	—
194	#464	—	37	1	no	—
195	#452	—	37	1	no	—
196	#450	—	37	1	no	—
197	C₄×SD₁₆	—	37	1	no	—
198	C₈×D₄	D₃₇	1	1	yes	characteristic, complemented
199	#416	—	37	1	no	—
200	C₂×#102	—	37	1	no	—
201	C₃₇	#3296	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	37	1	no	—
203	C₂²×D₄	—	37	1	no	—
204	C₂×Q₁₆	—	37	1	no	—
205	C₂×SD₁₆	—	37	1	no	—
206	C₂×SD₁₆	—	37	1	no	—
207	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
208	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
209	#126	—	74	1	no	—
210	#125	—	37	1	no	—
211	#125	—	74	1	no	—
212	#124	—	37	1	no	—
213	#123	—	74	1	no	—
214	#123	—	74	1	no	—
215	C₂²⋊Q₈	—	37	1	no	—
216	C₂²⋊Q₈	—	37	1	no	—
217	C₂²⋊Q₈	—	37	1	no	—
218	#121	—	37	1	no	—
219	#121	—	74	1	no	—
220	#120	—	74	1	no	—
221	C₄×Q₈	—	37	1	no	—
222	C₄×D₄	—	37	1	no	—
223	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
224	#117	—	37	1	no	—
225	#117	—	74	1	no	—
226	C₂×C₄⋊C₄	—	37	1	no	—
227	C₂×C₂²⋊C₄	—	74	1	no	—
228	#107	—	37	1	no	—
229	#106	—	37	1	no	—
230	#106	—	37	1	no	—
231	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
232	#103	—	37	1	no	—
233	#103	—	37	1	no	—
234	#103	—	37	1	no	—
235	#103	—	37	1	no	—
236	#102	—	37	1	no	—
237	#102	—	37	1	no	—
238	#102	—	37	1	no	—
239	#102	—	37	1	no	—
240	#102	—	37	1	no	—
241	#102	—	37	1	no	—
242	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
243	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
244	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
245	C₂⁴	—	74	1	no	—
246	#41	—	74	2	no	—
247	#41	—	148	1	no	—
248	C₂×Q₈	—	37	1	no	—
249	C₂×Q₈	—	37	1	no	—
250	C₂×D₄	—	37	1	no	—
251	C₂×D₄	—	37	1	no	—
252	C₂×D₄	—	74	2	no	—
253	C₂×D₄	—	74	1	no	—
254	C₂×D₄	—	74	1	no	—
255	C₂×D₄	—	74	1	no	—
256	C₂×D₄	—	148	1	no	—
257	C₂×D₄	D₁₄₈	1	1	yes	characteristic
258	C₂²×C₄	—	74	1	no	—
259	C₂²×C₄	—	74	1	no	—
260	C₂²×C₄	—	74	1	no	—
261	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
263	Q₁₆	—	148	1	no	—
264	SD₁₆	—	74	2	no	—
265	SD₁₆	—	148	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
269	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
272	C₄⋊C₄	—	37	1	no	—
273	C₄⋊C₄	—	37	1	no	—
274	C₄⋊C₄	—	37	1	no	—
275	C₄⋊C₄	—	37	1	no	—
276	C₄⋊C₄	—	74	1	no	—
277	C₄⋊C₄	—	74	1	no	—
278	C₄⋊C₄	—	74	1	no	—
279	C₄⋊C₄	—	74	1	no	—
280	C₄⋊C₄	—	74	1	no	—
281	C₄⋊C₄	—	74	1	no	—
282	C₄⋊C₄	—	74	1	no	—
283	C₄⋊C₄	D₁₄₈	1	1	yes	characteristic
284	C₂²⋊C₄	—	74	1	no	—
285	C₂²⋊C₄	—	74	1	no	—
286	C₂²⋊C₄	—	74	1	no	—
287	C₂²⋊C₄	—	74	1	no	—
288	C₂²⋊C₄	—	74	1	no	—
289	C₂²⋊C₄	—	74	1	no	—
290	C₂²⋊C₄	—	74	1	no	—
291	C₂²⋊C₄	—	74	1	no	—
292	C₂²⋊C₄	D₁₄₈	1	1	yes	characteristic
293	C₂²⋊C₄	D₁₄₈	1	1	yes	characteristic
294	C₄²	—	74	1	no	—
295	C₄²	—	74	1	no	—
296	C₄²	C₂²×D₃₇	1	1	yes	characteristic
297	C₂³	—	296	1	no	—
298	C₂³	—	74	1	no	—
299	C₂³	—	74	1	no	—
300	C₂³	—	148	2	no	—
301	C₂³	—	148	2	no	—
302	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
303	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
304	Q₈	—	74	1	no	—
305	Q₈	—	74	1	no	—
306	Q₈	—	74	1	no	—
307	Q₈	—	74	1	no	—
308	D₄	—	2	2	no	—
309	D₄	—	37	2	no	—
310	D₄	—	296	1	no	—
311	D₄	—	74	1	no	—
312	D₄	—	74	1	no	—
313	D₄	—	74	1	no	—
314	D₄	—	148	2	no	—
315	D₄	—	148	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	296	1	no	—
321	C₂×C₄	—	74	1	no	—
322	C₂×C₄	—	74	1	no	—
323	C₂×C₄	—	74	1	no	—
324	C₂×C₄	—	74	1	no	—
325	C₂×C₄	—	74	1	no	—
326	C₂×C₄	—	74	1	no	—
327	C₂×C₄	—	148	2	no	—
328	C₂×C₄	—	148	2	no	—
329	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
330	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
331	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
332	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	296	4	no	—
344	C₂²	—	74	2	no	—
345	C₂²	—	148	1	no	—
346	C₂²	—	148	1	no	—
347	C₂²	—	148	1	no	—
348	C₂²	—	148	1	no	—
349	C₂²	—	148	1	no	—
350	C₂²	#11924558	1	2	yes	—
351	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	148	1	no	—
357	C₄	—	148	1	no	—
358	C₄	—	148	1	no	—
359	C₄	—	148	1	no	—
360	C₄	—	148	1	no	—
361	C₄	—	148	1	no	—
362	C₄	#41×D₃₇	1	1	yes	characteristic
363	C₄	#11924581	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	—	296	1	no	—
367	C₂	—	148	2	no	—
368	C₂	#14558178	1	1	yes	characteristic
369	C₂	C₂×#11924558	1	1	yes	characteristic
370	C₂	#14558345	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
prime radical	#14558179	C₂
hypercenter	C₈×D₄	D₃₇
hypocenter	C₃₇	#3296
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. #14558223	C₂	semidirect
3. #14557888	C₂	semidirect
4. #14557864	C₂	semidirect
5. #14557858	C₂	semidirect
6. #14557834	C₂	semidirect
7. #14557774	C₂	semidirect
8. #14557762	C₂	semidirect
9. #14557756	C₂	semidirect
10. #14557697	C₂	semidirect
11. #14557686	C₂	semidirect
12. C₄×#10596930	C₂	semidirect
13. #14557530	C₂	semidirect
14. C₂×#11924458	C₂	semidirect
15. C₈×D₄×C₃₇	C₂	semidirect
18. C₂×Dic₁₄₈	C₂²	semidirect
22. #11924528	C₂²	semidirect
29. #11924515	C₂²	semidirect
33. C₄×Dic₇₄	C₂²	semidirect
35. #11924495	C₂²	semidirect
39. #11924470	C₂²	semidirect
40. #11924468	C₂²	semidirect
41. #11924468	C₂²	semidirect
42. #11924461	C₂²	semidirect
43. #11924461	C₂²	semidirect
44. #11924461	C₂²	semidirect
45. #11924461	C₂²	semidirect
53. C₂²×C₈×C₃₇	C₂²	semidirect
55. C₄⋊C₈×C₃₇	C₂²	semidirect
57. C₂²⋊C₈×C₃₇	C₂²	semidirect
58. C₄×C₈×C₃₇	C₂²	semidirect
198. C₈×D₄	D₃₇	semidirect
201. C₃₇	#3296	semidirect
208. C₂²×C₈	C₂×D₃₇	semidirect
231. C₄⋊C₈	C₂×D₃₇	semidirect
243. C₂²⋊C₈	C₂×D₃₇	semidirect
244. C₄×C₈	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558223	2
#14558179	2
#14557888	2
#14557864	2
#14557858	2
#14557834	2
#14557774	2
#14557762	2
#14557756	2
#14557697	2
#14557686	2
C₄×#10596930	2
#14557530	2
C₂×#11924458	2
C₈×D₄×C₃₇	2
#3296	37