Subgroups — C₃₇×#33633 #22654886

Order 9472.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#2951	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2229	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2228	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2227	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2224	C₂	1	2	yes	maximal, complemented
7	C₃₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₇×#2139	C₂	1	1	yes	maximal, characteristic
11	C₃₇×#2138	C₂	1	1	yes	maximal, characteristic
12	C₂×#135×C₃₇	C₂²	1	1	yes	characteristic
13	C₃₇×#544	—	2	1	no	—
14	C₃₇×#520	—	2	1	no	—
15	C₂×M₃₂×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	C₃₇×#501	—	2	2	no	—
18	C₃₇×#500	C₂²	1	2	yes	—
19	C₃₇×#499	C₂²	1	2	yes	—
20	C₃₇×#440	—	2	2	no	—
21	C₄×Q₁₆×C₃₇	C₂²	1	2	yes	—
22	C₄×D₈×C₃₇	C₂²	1	2	yes	—
23	C₃₇×#428	—	2	1	no	—
24	C₃₇×#427	C₂²	1	1	yes	characteristic
25	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
26	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
27	C₃₇×#417	—	2	1	no	—
28	C₃₇×#405	C₂²	1	1	yes	characteristic
29	C₃₇×#367	C₂²	1	2	yes	—
30	C₃₇×#366	C₂²	1	2	yes	—
31	C₃₇×#366	C₂²	1	2	yes	—
32	C₃₇×#366	C₂²	1	2	yes	—
33	C₃₇×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₃₇×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₃₇×#358	C₂²	1	1	yes	characteristic
36	C₃₇×#358	C₂²	1	1	yes	characteristic
37	C₃₇×#358	C₂²	1	2	yes	—
38	C₃₇×#357	C₂²	1	1	yes	characteristic
39	C₃₇×#357	C₂²	1	1	yes	characteristic
40	C₃₇×#357	C₂²	1	2	yes	—
41	C₂×#41×C₃₇	—	2	1	no	—
42	#135×C₃₇	—	2	2	no	—
43	#135×C₃₇	Q₈	1	4	yes	complemented
44	C₂×Q₁₆×C₃₇	C₂³	1	1	yes	characteristic
45	C₂×SD₁₆×C₃₇	—	2	1	no	—
46	C₂×D₈×C₃₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₃₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₃₇	—	2	2	no	—
50	C₄⋊Q₈×C₃₇	—	4	1	no	—
51	#125×C₃₇	—	2	2	no	—
52	#125×C₃₇	—	4	1	no	—
53	C₂²⋊Q₈×C₃₇	—	4	2	no	—
54	C₄×Q₈×C₃₇	—	2	2	no	—
55	C₄×D₄×C₃₇	—	2	2	no	—
56	C₄×D₄×C₃₇	—	4	2	no	—
57	#117×C₃₇	—	2	1	no	—
58	#117×C₃₇	—	4	1	no	—
59	#117×C₃₇	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₇	—	4	1	no	—
63	M₃₂×C₃₇	—	2	2	no	—
64	C₂×C₁₆×C₃₇	—	2	2	no	—
65	C₂×C₁₆×C₃₇	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₃₇	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₃₇	C₂³	1	2	yes	—
68	#107×C₃₇	—	2	1	no	—
69	#107×C₃₇	—	2	1	no	—
70	#107×C₃₇	—	2	2	no	—
71	#107×C₃₇	C₂³	1	2	yes	—
72	#107×C₃₇	C₂³	1	1	yes	characteristic
73	#107×C₃₇	C₂³	1	1	yes	characteristic
74	#107×C₃₇	C₂³	1	2	yes	—
75	#106×C₃₇	—	2	1	no	—
76	#106×C₃₇	—	2	2	no	—
77	#103×C₃₇	—	2	2	no	—
78	#102×C₃₇	—	2	2	no	—
79	#97×C₃₇	D₄	1	2	yes	—
80	C₄×C₈×C₃₇	C₂³	1	2	yes	—
81	#41×C₃₇	—	2	4	no	—
82	#41×C₃₇	—	4	2	no	—
83	C₂×Q₈×C₃₇	—	2	1	no	—
84	C₂×Q₈×C₃₇	—	4	2	no	—
85	C₂×D₄×C₃₇	—	2	1	no	—
86	C₂×D₄×C₃₇	—	4	1	no	—
87	C₂²×C₄×C₃₇	—	4	1	no	—
88	C₂²×C₄×C₃₇	—	4	1	no	—
89	C₂²×C₄×C₃₇	—	4	1	no	—
90	C₂²×C₄×C₃₇	—	4	2	no	—
91	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₃₇	—	2	1	no	—
93	Q₁₆×C₃₇	C₂×Q₈	1	2	yes	—
94	SD₁₆×C₃₇	—	2	4	no	—
95	D₈×C₃₇	—	2	1	no	—
96	D₈×C₃₇	C₂×Q₈	1	2	yes	—
97	M₁₆×C₃₇	—	2	2	no	—
98	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₃₇	#41	1	2	yes	—
102	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₃₇	—	2	2	no	—
104	C₄⋊C₄×C₃₇	—	2	1	no	—
105	C₄⋊C₄×C₃₇	—	2	2	no	—
106	C₄⋊C₄×C₃₇	—	2	1	no	—
107	C₄⋊C₄×C₃₇	—	4	1	no	—
108	C₄⋊C₄×C₃₇	—	4	1	no	—
109	C₄⋊C₄×C₃₇	—	4	2	no	—
110	C₄⋊C₄×C₃₇	—	4	2	no	—
111	C₄⋊C₄×C₃₇	—	4	2	no	—
112	C₄⋊C₄×C₃₇	—	4	2	no	—
113	C₄⋊C₄×C₃₇	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₃₇	D₈	1	1	yes	characteristic
115	C₂²⋊C₄×C₃₇	—	4	1	no	—
116	C₂²⋊C₄×C₃₇	—	4	2	no	—
117	C₂²⋊C₄×C₃₇	D₈	1	2	yes	—
118	C₄²×C₃₇	—	4	1	no	—
119	C₄²×C₃₇	—	4	2	no	—
120	C₄²×C₃₇	C₂×D₄	1	2	yes	—
121	C₁₆×C₃₇	—	2	1	no	—
122	C₁₆×C₃₇	—	2	1	no	—
123	C₁₆×C₃₇	—	2	2	no	—
124	C₂³×C₃₇	—	4	1	no	—
125	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₃₇	—	2	2	no	—
127	Q₈×C₃₇	—	4	1	no	—
128	Q₈×C₃₇	—	16	2	no	—
129	D₄×C₃₇	—	2	2	no	—
130	D₄×C₃₇	—	4	4	no	—
131	D₄×C₃₇	—	4	1	no	—
132	C₂×C₄×C₃₇	—	4	1	no	—
133	C₂×C₄×C₃₇	—	4	4	no	—
134	C₂×C₄×C₃₇	—	4	1	no	—
135	C₂×C₄×C₃₇	—	4	2	no	—
136	C₂×C₄×C₃₇	—	4	1	no	—
137	C₂×C₄×C₃₇	—	4	2	no	—
138	C₂×C₄×C₃₇	—	16	2	no	—
139	C₂×C₄×C₃₇	—	16	1	no	—
140	C₂×C₄×C₃₇	—	16	1	no	—
141	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₇	#137	1	2	yes	—
146	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₃₇	—	2	2	no	—
148	C₈×C₃₇	C₂²⋊Q₈	1	2	yes	—
149	C₈×C₃₇	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₃₇	C₂×#41	1	1	yes	characteristic
151	#33633	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₃₇	—	2	1	no	—
153	C₂²×C₃₇	—	2	1	no	—
154	C₂²×C₃₇	—	4	2	no	—
155	C₂²×C₃₇	—	8	1	no	—
156	C₂²×C₃₇	—	8	1	no	—
157	C₂²×C₃₇	#474	1	2	yes	—
158	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₃₇	—	2	2	no	—
160	C₄×C₃₇	—	2	2	no	—
161	C₄×C₃₇	—	4	2	no	—
162	C₄×C₃₇	—	8	2	no	—
163	C₄×C₃₇	—	8	2	no	—
164	C₄×C₃₇	—	8	1	no	—
165	C₄×C₃₇	—	8	1	no	—
166	C₄×C₃₇	#474	1	2	yes	—
167	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₄×C₃₇	C₂×#137	1	1	yes	characteristic
169	#3164	C₂×C₃₇	1	1	yes	characteristic, complemented
170	#2951	C₂×C₃₇	1	1	yes	characteristic
171	#2229	C₂×C₃₇	1	2	yes	complemented
172	#2228	C₂×C₃₇	1	2	yes	complemented
173	#2227	C₂×C₃₇	1	2	yes	complemented
174	#2224	C₂×C₃₇	1	2	yes	complemented
175	#2159	C₂×C₃₇	1	1	yes	characteristic, complemented
176	C₂×#366	C₂×C₃₇	1	1	yes	characteristic, complemented
177	#2151	C₂×C₃₇	1	1	yes	characteristic, complemented
178	#2139	C₂×C₃₇	1	1	yes	characteristic
179	#2138	C₂×C₃₇	1	1	yes	characteristic
180	C₂×C₃₇	—	2	2	no	—
181	C₂×C₃₇	—	8	2	no	—
182	C₂×C₃₇	C₂×#474	1	1	yes	characteristic
183	C₂×C₃₇	#3415	1	1	yes	characteristic
184	C₂×C₃₇	#3417	1	1	yes	characteristic
185	C₂×#135	C₂²×C₃₇	1	1	yes	characteristic
186	#544	—	2	1	no	—
187	#520	—	2	1	no	—
188	C₂×M₃₂	C₂²×C₃₇	1	1	yes	characteristic
189	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
190	#501	—	2	2	no	—
191	#500	C₂²×C₃₇	1	2	yes	—
192	#499	C₂²×C₃₇	1	2	yes	—
193	#440	—	2	2	no	—
194	C₄×Q₁₆	C₂²×C₃₇	1	2	yes	—
195	C₄×D₈	C₂²×C₃₇	1	2	yes	—
196	#428	—	2	1	no	—
197	#427	C₂²×C₃₇	1	1	yes	characteristic
198	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
199	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
200	#417	—	2	1	no	—
201	#405	C₂²×C₃₇	1	1	yes	characteristic
202	#367	C₂²×C₃₇	1	2	yes	—
203	#366	C₂²×C₃₇	1	2	yes	—
204	#366	C₂²×C₃₇	1	2	yes	—
205	#366	C₂²×C₃₇	1	2	yes	—
206	C₄⋊C₁₆	C₂²×C₃₇	1	2	yes	—
207	C₄⋊C₁₆	C₂²×C₃₇	1	2	yes	—
208	#358	C₂²×C₃₇	1	1	yes	characteristic
209	#358	C₂²×C₃₇	1	1	yes	characteristic
210	#358	C₂²×C₃₇	1	2	yes	—
211	#357	C₂²×C₃₇	1	1	yes	characteristic
212	#357	C₂²×C₃₇	1	1	yes	characteristic
213	#357	C₂²×C₃₇	1	2	yes	—
214	C₃₇	#33633	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×#41	—	2	1	no	—
216	#135	—	2	2	no	—
217	#135	Q₈×C₃₇	1	4	yes	complemented
218	C₂×Q₁₆	C₂³×C₃₇	1	1	yes	characteristic
219	C₂×SD₁₆	—	2	1	no	—
220	C₂×D₈	C₂³×C₃₇	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₃₇	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	4	1	no	—
225	#125	—	2	2	no	—
226	#125	—	4	1	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	4	2	no	—
231	#117	—	2	1	no	—
232	#117	—	4	1	no	—
233	#117	D₄×C₃₇	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	4	1	no	—
237	M₃₂	—	2	2	no	—
238	C₂×C₁₆	—	2	2	no	—
239	C₂×C₁₆	C₂³×C₃₇	1	1	yes	characteristic
240	C₂×C₁₆	C₂³×C₃₇	1	1	yes	characteristic
241	C₂×C₁₆	C₂³×C₃₇	1	2	yes	—
242	#107	—	2	1	no	—
243	#107	—	2	1	no	—
244	#107	—	2	2	no	—
245	#107	C₂³×C₃₇	1	2	yes	—
246	#107	C₂³×C₃₇	1	1	yes	characteristic
247	#107	C₂³×C₃₇	1	1	yes	characteristic
248	#107	C₂³×C₃₇	1	2	yes	—
249	#106	—	2	1	no	—
250	#106	—	2	2	no	—
251	#103	—	2	2	no	—
252	#102	—	2	2	no	—
253	#97	D₄×C₃₇	1	2	yes	—
254	C₄×C₈	C₂³×C₃₇	1	2	yes	—
255	#41	—	2	4	no	—
256	#41	—	4	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
266	Q₁₆	—	2	1	no	—
267	Q₁₆	C₂×Q₈×C₃₇	1	2	yes	—
268	SD₁₆	—	2	4	no	—
269	D₈	—	2	1	no	—
270	D₈	C₂×Q₈×C₃₇	1	2	yes	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
274	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
275	C₂×C₈	#41×C₃₇	1	2	yes	—
276	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	D₈×C₃₇	1	1	yes	characteristic
288	C₄⋊C₄	D₈×C₃₇	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	D₈×C₃₇	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×D₄×C₃₇	1	2	yes	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₂³	—	4	1	no	—
299	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	16	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	16	2	no	—
313	C₂×C₄	—	16	1	no	—
314	C₂×C₄	—	16	1	no	—
315	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
319	C₂×C₄	#137×C₃₇	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₂²⋊Q₈×C₃₇	1	2	yes	—
323	C₈	C₂²×Q₈×C₃₇	1	1	yes	characteristic
324	C₈	C₂×#41×C₃₇	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	C₃₇×#474	1	2	yes	—
331	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	C₃₇×#474	1	2	yes	—
340	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
341	C₄	C₂×#137×C₃₇	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×C₃₇×#474	1	1	yes	characteristic
345	C₂	C₃₇×#3415	1	1	yes	characteristic
346	C₂	C₃₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
prime radical	C₂×#135×C₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ #117×C₃₇ ◁ C₃₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3164	C₂	semidirect
3. C₃₇×#2229	C₂	semidirect
4. C₃₇×#2228	C₂	semidirect
5. C₃₇×#2227	C₂	semidirect
6. C₃₇×#2224	C₂	semidirect
7. C₃₇×#2159	C₂	semidirect
8. C₂×C₃₇×#366	C₂	semidirect
9. C₃₇×#2151	C₂	semidirect
43. #135×C₃₇	Q₈	semidirect
151. #33633	C₃₇	direct (N × H)
169. #3164	C₂×C₃₇	semidirect
171. #2229	C₂×C₃₇	semidirect
172. #2228	C₂×C₃₇	semidirect
173. #2227	C₂×C₃₇	semidirect
174. #2224	C₂×C₃₇	semidirect
175. #2159	C₂×C₃₇	semidirect
176. C₂×#366	C₂×C₃₇	semidirect
177. #2151	C₂×C₃₇	semidirect
214. C₃₇	#33633	direct (N × H)
217. #135	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3164	2
C₃₇×#2951	2
C₃₇×#2229	2
C₃₇×#2228	2
C₃₇×#2227	2
C₃₇×#2224	2
C₃₇×#2159	2
C₂×C₃₇×#366	2
C₃₇×#2151	2
C₃₇×#2139	2
C₃₇×#2138	2
#33633	37

Subgroups — C37×#33633 #22654886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33633 #22654886