Subgroups — D₃₇⋊#435 #18011304

Order 4736.

564 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	C₂×D₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558565	C₂	1	2	yes	maximal, complemented
4	#14558564	C₂	1	2	yes	maximal, complemented
5	#14558563	C₂	1	2	yes	maximal, complemented
6	#14558562	C₂	1	2	yes	maximal, complemented
7	D₃₇⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃₇⋊(C₂²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	D₃₇⋊(C₂²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#14558492	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₄×D₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596953	C₂²	1	2	yes	complemented
13	D₃₇⋊M₁₆	—	2	4	no	—
14	C₂×#10596957	C₂²	1	2	yes	complemented
15	C₂×D₃₇⋊C₈	—	2	4	no	—
16	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×D₃₇⋊C₈	C₂²	1	2	yes	complemented
21	C₂²×#64438	C₂²	1	2	yes	complemented
22	#11924603	C₂²	1	2	yes	complemented
23	C₄⋊#64438	C₂²	1	2	yes	complemented
24	#11924593	C₂²	1	2	yes	complemented
25	#11924593	C₂²	1	2	yes	complemented
26	C₂²⋊#64438	C₂²	1	2	yes	complemented
27	C₂²⋊#64438	C₂²	1	2	yes	complemented
28	#11924591	C₂²	1	2	yes	complemented
29	#11924591	C₂²	1	2	yes	complemented
30	C₂×D₄×D₃₇	C₄	1	1	yes	characteristic
31	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
33	C₄×D₁₄₈	C₄	1	1	yes	characteristic
34	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
35	#11924498	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
38	#11924496	C₂²	1	1	yes	characteristic
39	#11924496	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₃₇	C₄	1	1	yes	characteristic
41	C₂²⋊C₄×D₃₇	C₄	1	1	yes	characteristic
42	C₂²⋊C₄×D₃₇	C₄	1	1	yes	characteristic
43	C₄²×D₃₇	C₂²	1	1	yes	characteristic
44	C₄×D₄×C₃₇	C₄	1	1	yes	characteristic
45	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
54	D₄×D₃₇	—	2	4	no	—
55	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₃₇	—	2	2	no	—
57	C₂×C₄×D₃₇	—	2	2	no	—
58	C₂×C₄×D₃₇	—	2	2	no	—
59	C₂×C₄×D₃₇	—	2	2	no	—
60	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
69	#10596957	—	2	4	no	—
70	C₂×#64438	—	2	4	no	—
71	C₂×#64438	C₂³	1	2	yes	—
72	C₂×#64438	C₂³	1	2	yes	—
73	C₂×#64438	C₂³	1	2	yes	—
74	C₂×#64438	C₂³	1	2	yes	—
75	#10596953	—	2	4	no	—
76	D₃₇⋊C₈	—	2	2	no	—
77	D₃₇⋊C₈	—	2	2	no	—
78	D₃₇⋊C₈	—	2	2	no	—
79	D₃₇⋊C₈	—	2	4	no	—
80	D₃₇⋊C₈	D₄	1	4	yes	complemented
81	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
87	#10596938	C₂×C₄	1	1	yes	characteristic
88	#10596938	C₂×C₄	1	1	yes	characteristic
89	#10596938	C₂×C₄	1	1	yes	characteristic
90	#10596938	C₂×C₄	1	1	yes	characteristic
91	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
92	#10596936	C₂×C₄	1	1	yes	characteristic
93	#10596936	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
96	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
97	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₃₇	—	2	2	no	—
100	C₂²×D₃₇	—	2	2	no	—
101	C₂²×D₃₇	—	2	2	no	—
102	C₂²×D₃₇	—	2	2	no	—
103	C₂²×D₃₇	—	2	2	no	—
104	C₂²×D₃₇	M₁₆	1	4	yes	complemented
105	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
108	D₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	2	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²⋊D₃₇	—	2	2	no	—
117	C₂²⋊D₃₇	—	2	2	no	—
118	C₂×Dic₃₇	—	2	1	no	—
119	C₂×Dic₃₇	—	2	1	no	—
120	C₂×Dic₃₇	—	2	1	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
127	D₁₄₈	—	2	2	no	—
128	C₄×D₃₇	—	2	2	no	—
129	C₄×D₃₇	—	2	2	no	—
130	C₄×D₃₇	—	2	2	no	—
131	C₄×D₃₇	—	2	2	no	—
132	C₄×D₃₇	—	2	2	no	—
133	C₄×D₃₇	—	2	2	no	—
134	C₄×D₃₇	—	2	1	no	—
135	C₄×D₃₇	—	2	1	no	—
136	C₄×D₃₇	—	2	1	no	—
137	C₄×D₃₇	—	2	1	no	—
138	C₄×D₃₇	C₂×D₄	1	2	yes	—
139	C₄×D₃₇	#41	1	2	yes	—
140	#64438	—	2	2	no	—
141	#64438	—	2	2	no	—
142	#64438	—	2	2	no	—
143	#64438	C₂×D₄	1	4	yes	complemented
144	C₂²×C₃₇	—	2	1	no	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	—	2	1	no	—
147	C₂²×C₃₇	—	2	1	no	—
148	C₂²×C₃₇	—	2	1	no	—
149	C₂²×C₃₇	C₂×M₁₆	1	2	yes	complemented
150	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₃₇	—	2	1	no	—
152	C₂×D₃₇	—	2	1	no	—
153	C₂×D₃₇	—	2	8	no	—
154	C₂×D₃₇	—	2	1	no	—
155	C₂×D₃₇	—	2	1	no	—
156	C₂×D₃₇	—	2	1	no	—
157	C₂×D₃₇	—	4	4	no	—
158	C₂×D₃₇	C₄×D₄	1	2	yes	—
159	C₂×D₃₇	C₂×M₁₆	1	2	yes	—
160	C₂×D₃₇	C₂×M₁₆	1	2	yes	complemented
161	C₂×D₃₇	#131	1	2	yes	—
162	C₄×C₃₇	—	2	1	no	—
163	C₄×C₃₇	—	2	1	no	—
164	C₄×C₃₇	—	2	1	no	—
165	C₄×C₃₇	—	2	1	no	—
166	C₄×C₃₇	C₄×D₄	1	2	yes	—
167	Dic₃₇	—	2	1	no	—
168	Dic₃₇	—	2	1	no	—
169	Dic₃₇	—	2	1	no	—
170	Dic₃₇	—	2	1	no	—
171	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
172	Dic₃₇	C₂×#41	1	1	yes	characteristic
173	C₂×#435	—	37	1	no	maximal, Sylow
174	C₂×C₃₇	—	2	2	no	—
175	C₂×C₃₇	—	4	1	no	—
176	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
178	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
179	D₃₇	—	2	2	no	—
180	D₃₇	—	4	1	no	—
181	D₃₇	#435	1	4	yes	complemented
182	C₂²×M₁₆	—	37	1	no	—
183	C₂³×C₈	—	37	1	no	—
184	C₂×C₄×D₄	—	37	1	no	—
185	#435	—	37	2	no	—
186	#435	—	37	2	no	—
187	#435	—	37	2	no	—
188	#435	—	37	2	no	—
189	C₂×C₄⋊C₈	—	37	1	no	—
190	C₂×C₂²⋊C₈	—	37	1	no	—
191	C₂×C₂²⋊C₈	—	37	1	no	—
192	C₂×#97	—	37	1	no	—
193	C₃₇	C₂×#435	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	—	37	1	no	—
195	C₂³×C₄	—	37	1	no	—
196	C₂³×C₄	—	37	1	no	—
197	C₂×M₁₆	—	37	2	no	—
198	C₂×M₁₆	—	37	2	no	—
199	C₂×M₁₆	—	74	4	no	—
200	C₂²×C₈	—	37	1	no	—
201	C₂²×C₈	—	37	2	no	—
202	C₂²×C₈	—	37	1	no	—
203	C₂²×C₈	—	37	2	no	—
204	C₂²×C₈	—	37	1	no	—
205	C₂²×C₈	—	37	1	no	—
206	C₂²×C₈	—	74	4	no	—
207	C₄×D₄	—	37	1	no	—
208	C₄×D₄	—	37	1	no	—
209	C₄×D₄	—	37	1	no	—
210	C₄×D₄	—	37	1	no	—
211	C₄×D₄	—	37	1	no	—
212	C₄×D₄	—	37	1	no	—
213	C₄×D₄	—	37	1	no	—
214	C₄×D₄	#3866	1	1	yes	characteristic
215	C₂×C₄⋊C₄	—	37	1	no	—
216	C₂×C₂²⋊C₄	—	37	1	no	—
217	C₂×C₂²⋊C₄	—	37	1	no	—
218	C₂×C₄²	—	37	1	no	—
219	C₄⋊C₈	—	37	2	no	—
220	C₄⋊C₈	—	37	2	no	—
221	C₂²⋊C₈	—	37	2	no	—
222	C₂²⋊C₈	—	37	2	no	—
223	C₂²⋊C₈	—	37	2	no	—
224	C₂²⋊C₈	—	37	2	no	—
225	#97	—	37	2	no	—
226	#97	—	37	2	no	—
227	C₂⁴	—	37	1	no	—
228	C₂⁴	—	37	1	no	—
229	C₂×D₄	—	37	1	no	—
230	C₂×D₄	—	37	1	no	—
231	C₂×D₄	—	37	1	no	—
232	C₂×D₄	—	74	4	no	—
233	C₂×D₄	C₂×#3866	1	1	yes	characteristic
234	C₂²×C₄	—	37	1	no	—
235	C₂²×C₄	—	37	1	no	—
236	C₂²×C₄	—	37	1	no	—
237	C₂²×C₄	—	37	1	no	—
238	C₂²×C₄	—	37	1	no	—
239	C₂²×C₄	—	37	1	no	—
240	C₂²×C₄	—	37	1	no	—
241	C₂²×C₄	—	37	1	no	—
242	C₂²×C₄	—	37	1	no	—
243	C₂²×C₄	—	37	1	no	—
244	C₂²×C₄	—	37	1	no	—
245	C₂²×C₄	—	74	2	no	—
246	C₂²×C₄	—	74	2	no	—
247	C₂²×C₄	—	74	2	no	—
248	C₂²×C₄	—	74	2	no	—
249	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
250	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
251	M₁₆	—	74	4	no	—
252	M₁₆	—	74	4	no	—
253	C₂×C₈	—	37	4	no	—
254	C₂×C₈	—	37	2	no	—
255	C₂×C₈	—	37	2	no	—
256	C₂×C₈	—	37	2	no	—
257	C₂×C₈	—	37	2	no	—
258	C₂×C₈	—	74	4	no	—
259	C₂×C₈	—	74	4	no	—
260	C₂×C₈	—	74	2	no	—
261	C₂×C₈	—	74	2	no	—
262	C₂×C₈	—	74	2	no	—
263	C₄⋊C₄	—	37	1	no	—
264	C₄⋊C₄	—	37	1	no	—
265	C₄⋊C₄	—	37	1	no	—
266	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
267	C₂²⋊C₄	—	37	1	no	—
268	C₂²⋊C₄	—	37	1	no	—
269	C₂²⋊C₄	—	37	1	no	—
270	C₂²⋊C₄	—	37	1	no	—
271	C₂²⋊C₄	—	37	1	no	—
272	C₂²⋊C₄	—	37	1	no	—
273	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
275	C₄²	—	37	1	no	—
276	C₄²	—	37	1	no	—
277	C₄²	—	37	1	no	—
278	C₄²	C₂×#3866	1	1	yes	characteristic
279	C₂³	—	37	1	no	—
280	C₂³	—	37	4	no	—
281	C₂³	—	37	1	no	—
282	C₂³	—	37	1	no	—
283	C₂³	—	74	2	no	—
284	C₂³	—	74	2	no	—
285	C₂³	—	74	2	no	—
286	C₂³	—	74	2	no	—
287	C₂³	—	74	2	no	—
288	C₂³	C₂²×#3866	1	1	yes	characteristic
289	C₂³	C₂²×#3866	1	1	yes	characteristic
290	D₄	—	2	2	no	—
291	D₄	—	74	2	no	—
292	D₄	—	74	2	no	—
293	D₄	—	74	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	37	1	no	—
297	C₂×C₄	—	37	2	no	—
298	C₂×C₄	—	37	2	no	—
299	C₂×C₄	—	37	1	no	—
300	C₂×C₄	—	37	1	no	—
301	C₂×C₄	—	37	1	no	—
302	C₂×C₄	—	37	1	no	—
303	C₂×C₄	—	74	1	no	—
304	C₂×C₄	—	74	1	no	—
305	C₂×C₄	—	74	1	no	—
306	C₂×C₄	—	74	1	no	—
307	C₂×C₄	—	74	2	no	—
308	C₂×C₄	—	74	2	no	—
309	C₂×C₄	—	74	2	no	—
310	C₂×C₄	—	74	2	no	—
311	C₂×C₄	—	74	1	no	—
312	C₂×C₄	—	74	1	no	—
313	C₂×C₄	—	74	1	no	—
314	C₂×C₄	—	74	1	no	—
315	C₂×C₄	—	74	2	no	—
316	C₂×C₄	—	74	2	no	—
317	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
318	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
319	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
320	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
321	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
322	C₈	—	37	4	no	—
323	C₈	—	74	2	no	—
324	C₈	—	74	2	no	—
325	C₈	—	74	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	37	2	no	—
332	C₂²	—	37	2	no	—
333	C₂²	—	37	2	no	—
334	C₂²	—	37	2	no	—
335	C₂²	—	74	8	no	—
336	C₂²	—	74	1	no	—
337	C₂²	—	74	1	no	—
338	C₂²	—	74	1	no	—
339	C₂²	—	74	1	no	—
340	C₂²	—	74	1	no	—
341	C₂²	—	148	4	no	—
342	C₂²	D₃₇⋊M₁₆	1	2	yes	complemented
343	C₂²	C₂³×#3866	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	37	1	no	—
349	C₄	—	37	1	no	—
350	C₄	—	74	1	no	—
351	C₄	—	74	1	no	—
352	C₄	—	74	1	no	—
353	C₄	—	74	1	no	—
354	C₄	D₄×#3866	1	2	yes	—
355	C₂	—	2	2	no	—
356	C₂	—	4	1	no	—
357	C₂	—	37	4	no	—
358	C₂	—	74	2	no	—
359	C₂	—	148	1	no	—
360	C₂	C₂×D₄×#3866	1	1	yes	characteristic
361	C₂	C₂×D₃₇⋊M₁₆	1	1	yes	characteristic
362	C₂	D₃₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#3866
Fitting	C₄×D₄×C₃₇	C₄
prime radical	C₂×D₄×D₃₇	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×D₄	#3866
hypocenter	C₃₇	C₂×#435
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₇⋊M₁₆	C₂	semidirect
2. C₂²×D₃₇⋊C₈	C₂	semidirect
3. #14558565	C₂	semidirect
4. #14558564	C₂	semidirect
5. #14558563	C₂	semidirect
6. #14558562	C₂	semidirect
7. D₃₇⋊(C₄⋊C₈)	C₂	semidirect
8. D₃₇⋊(C₂²⋊C₈)	C₂	semidirect
9. D₃₇⋊(C₂²⋊C₈)	C₂	semidirect
10. #14558492	C₂	semidirect
12. C₂×#10596953	C₂²	semidirect
14. C₂×#10596957	C₂²	semidirect
20. C₂×D₃₇⋊C₈	C₂²	semidirect
21. C₂²×#64438	C₂²	semidirect
22. #11924603	C₂²	semidirect
23. C₄⋊#64438	C₂²	semidirect
24. #11924593	C₂²	semidirect
25. #11924593	C₂²	semidirect
26. C₂²⋊#64438	C₂²	semidirect
27. C₂²⋊#64438	C₂²	semidirect
28. #11924591	C₂²	semidirect
29. #11924591	C₂²	semidirect
80. D₃₇⋊C₈	D₄	semidirect
104. C₂²×D₃₇	M₁₆	semidirect
143. #64438	C₂×D₄	semidirect
149. C₂²×C₃₇	C₂×M₁₆	semidirect
160. C₂×D₃₇	C₂×M₁₆	semidirect
181. D₃₇	#435	semidirect
193. C₃₇	C₂×#435	semidirect
342. C₂²	D₃₇⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₇⋊M₁₆	2
C₂²×D₃₇⋊C₈	2
#14558565	2
#14558564	2
#14558563	2
#14558562	2
D₃₇⋊(C₄⋊C₈)	2
D₃₇⋊(C₂²⋊C₈)	2
#14558492	2
C₄×D₄×D₃₇	2
C₂×#435	37

Subgroups — D37⋊#435 #18011304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₇⋊#435 #18011304