Subgroups — C₃₇×#29880 #22651133

Order 9472.

688 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#433	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2628	C₂	1	2	yes	maximal
4	C₃₇×#2087	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2086	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2076	C₂	1	2	yes	maximal
7	C₃₇×#2075	C₂	1	2	yes	maximal
8	C₃₇×#1993	C₂	1	2	yes	maximal, complemented
9	C₃₇×#1835	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×#117×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#500	—	2	4	no	—
13	C₃₇×#433	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#400	—	2	2	no	—
23	C₃₇×#399	—	2	2	no	—
24	C₃₇×#395	—	2	2	no	—
25	C₃₇×#393	—	2	2	no	—
26	C₃₇×#389	C₂²	1	4	yes	—
27	C₃₇×#382	C₂²	1	2	yes	—
28	C₃₇×#380	C₂²	1	4	yes	—
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₃₇×#340	C₂²	1	4	yes	—
31	C₃₇×#340	C₂²	1	2	yes	—
32	C₃₇×#340	C₂²	1	2	yes	—
33	C₂²×Q₈×C₃₇	—	2	2	no	—
34	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₃₇	—	2	4	no	—
41	#117×C₃₇	—	2	2	no	—
42	#117×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	#107×C₃₇	—	2	4	no	—
55	#107×C₃₇	—	2	4	no	—
56	C₄⋊C₈×C₃₇	—	2	2	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	C₂²⋊C₈×C₃₇	—	2	2	no	—
59	C₂²⋊C₈×C₃₇	—	2	2	no	—
60	#97×C₃₇	—	2	2	no	—
61	C₄×C₈×C₃₇	D₄	1	4	yes	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	2	no	—
65	#95×C₃₇	—	2	4	no	—
66	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₃₇	—	4	4	no	—
68	C₂×Q₈×C₃₇	—	8	4	no	—
69	C₂²×C₄×C₃₇	—	2	1	no	—
70	C₂²×C₄×C₃₇	—	2	1	no	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	—	2	2	no	—
73	C₂²×C₄×C₃₇	—	2	4	no	—
74	C₂²×C₄×C₃₇	—	2	4	no	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	—	2	2	no	—
83	C₂×C₈×C₃₇	—	2	1	no	—
84	C₂×C₈×C₃₇	—	2	1	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₃₇	—	2	2	no	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	4	no	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₄⋊C₄×C₃₇	—	8	4	no	—
97	C₂²⋊C₄×C₃₇	—	2	2	no	—
98	C₂²⋊C₄×C₃₇	—	2	2	no	—
99	C₂²⋊C₄×C₃₇	—	8	4	no	—
100	C₄²×C₃₇	—	2	1	no	—
101	C₄²×C₃₇	—	2	1	no	—
102	C₄²×C₃₇	C₂×D₄	1	2	yes	—
103	C₄²×C₃₇	#41	1	2	yes	—
104	C₂³×C₃₇	—	2	1	no	—
105	C₂³×C₃₇	—	2	2	no	—
106	C₂³×C₃₇	—	2	2	no	—
107	C₂³×C₃₇	—	2	2	no	—
108	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₇	—	8	8	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	4	1	no	—
119	C₂×C₄×C₃₇	—	4	1	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	4	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	—	4	4	no	—
125	C₂×C₄×C₃₇	—	4	4	no	—
126	C₂×C₄×C₃₇	—	4	4	no	—
127	C₂×C₄×C₃₇	#121	1	2	yes	—
128	C₂×C₄×C₃₇	#121	1	2	yes	—
129	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
132	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₃₇	—	2	2	no	—
134	C₈×C₃₇	—	2	2	no	—
135	C₈×C₃₇	—	4	1	no	—
136	C₈×C₃₇	—	4	1	no	—
137	#29880	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₃₇	—	4	1	no	—
139	C₂²×C₃₇	—	4	4	no	—
140	C₂²×C₃₇	—	4	1	no	—
141	C₂²×C₃₇	—	4	1	no	—
142	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
143	C₂²×C₃₇	#536	1	2	yes	—
144	C₂²×C₃₇	#576	1	2	yes	—
145	C₂²×C₃₇	#578	1	2	yes	—
146	C₄×C₃₇	—	2	2	no	—
147	C₄×C₃₇	—	2	2	no	—
148	C₄×C₃₇	—	4	1	no	—
149	C₄×C₃₇	—	4	1	no	—
150	C₄×C₃₇	—	4	1	no	—
151	C₄×C₃₇	—	8	4	no	—
152	C₄×C₃₇	—	8	4	no	—
153	C₄×C₃₇	#536	1	1	yes	characteristic
154	C₄×C₃₇	#536	1	1	yes	characteristic
155	C₄×C₃₇	#541	1	2	yes	—
156	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
157	C₂×#433	C₂×C₃₇	1	1	yes	characteristic
158	#2628	C₂×C₃₇	1	2	yes	—
159	#2087	C₂×C₃₇	1	2	yes	complemented
160	#2086	C₂×C₃₇	1	2	yes	complemented
161	#2076	C₂×C₃₇	1	2	yes	—
162	#2075	C₂×C₃₇	1	2	yes	—
163	#1993	C₂×C₃₇	1	2	yes	complemented
164	#1835	C₂×C₃₇	1	1	yes	characteristic, complemented
165	C₂×C₃₇	—	8	1	no	—
166	C₂×C₃₇	#2621	1	1	yes	characteristic
167	C₂×C₃₇	#3058	1	1	yes	characteristic
168	C₂×C₃₇	#3059	1	1	yes	characteristic
169	C₂×C₃₇	#3243	1	4	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
171	C₂×#117	C₂²×C₃₇	1	1	yes	characteristic
172	#500	—	2	4	no	—
173	#433	—	2	4	no	—
174	C₂×#107	C₂²×C₃₇	1	2	yes	—
175	C₂×#107	C₂²×C₃₇	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×C₂²⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
180	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
182	#400	—	2	2	no	—
183	#399	—	2	2	no	—
184	#395	—	2	2	no	—
185	#393	—	2	2	no	—
186	#389	C₂²×C₃₇	1	4	yes	—
187	#382	C₂²×C₃₇	1	2	yes	—
188	#380	C₂²×C₃₇	1	4	yes	—
189	#340	C₂²×C₃₇	1	2	yes	—
190	#340	C₂²×C₃₇	1	4	yes	—
191	#340	C₂²×C₃₇	1	4	yes	—
192	#340	C₂²×C₃₇	1	2	yes	—
193	C₃₇	#29880	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×Q₈	—	2	2	no	—
195	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
209	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	D₄×C₃₇	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	D₄×C₃₇	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
215	#107	—	2	4	no	—
216	#107	—	2	4	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	#97	—	2	2	no	—
222	C₄×C₈	D₄×C₃₇	1	4	yes	—
223	#95	—	2	4	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	8	4	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
241	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	8	4	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	C₂×D₄×C₃₇	1	2	yes	—
264	C₄²	#41×C₃₇	1	2	yes	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
270	Q₈	—	8	8	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	#121×C₃₇	1	2	yes	—
289	C₂×C₄	#121×C₃₇	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
293	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	4	1	no	—
297	C₈	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
303	C₂²	C₃₇×#536	1	2	yes	—
304	C₂²	C₃₇×#576	1	2	yes	—
305	C₂²	C₃₇×#578	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	8	4	no	—
312	C₄	—	8	4	no	—
313	C₄	C₃₇×#536	1	1	yes	characteristic
314	C₄	C₃₇×#536	1	1	yes	characteristic
315	C₄	C₃₇×#541	1	2	yes	—
316	C₂	—	8	1	no	—
317	C₂	C₃₇×#2621	1	1	yes	characteristic
318	C₂	C₃₇×#3058	1	1	yes	characteristic
319	C₂	C₃₇×#3059	1	1	yes	characteristic
320	C₂	C₃₇×#3243	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂⁴×C₃₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×#117×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#500	C₂	semidirect
4. C₃₇×#2087	C₂	semidirect
5. C₃₇×#2086	C₂	semidirect
8. C₃₇×#1993	C₂	semidirect
9. C₃₇×#1835	C₂	semidirect
137. #29880	C₃₇	direct (N × H)
156. C₂×#500	C₂×C₃₇	semidirect
159. #2087	C₂×C₃₇	semidirect
160. #2086	C₂×C₃₇	semidirect
163. #1993	C₂×C₃₇	semidirect
164. #1835	C₂×C₃₇	semidirect
193. C₃₇	#29880	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#500	2
C₂×C₃₇×#433	2
C₃₇×#2628	2
C₃₇×#2087	2
C₃₇×#2086	2
C₃₇×#2076	2
C₃₇×#2075	2
C₃₇×#1993	2
C₃₇×#1835	2
#29880	37

Subgroups — C37×#29880 #22651133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29880 #22651133