Subgroups — C₃₇×#28778 #22650031

Order 9472.

702 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×#2441	C₂	1	2	yes	maximal
2	C₃₇×#2083	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2059	C₂	1	2	yes	maximal, complemented
4	C₃₇×#1992	C₂	1	2	yes	maximal, complemented
5	C₃₇×#1944	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1854	C₂	1	2	yes	maximal
7	C₃₇×#1808	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1752	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#1735	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	2	yes	—
12	C₂×#117×C₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₃₇	C₂²	1	2	yes	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#106×C₃₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₃₇	C₂²	1	2	yes	—
18	C₃₇×#399	—	2	2	no	—
19	C₃₇×#396	—	2	2	no	—
20	C₃₇×#389	—	2	2	no	—
21	C₃₇×#389	C₂²	1	2	yes	—
22	C₃₇×#389	C₂²	1	2	yes	—
23	C₃₇×#387	—	2	2	no	—
24	C₃₇×#384	C₂²	1	2	yes	—
25	C₃₇×#380	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
27	C₃₇×#340	C₂²	1	2	yes	—
28	C₃₇×#340	C₂²	1	2	yes	—
29	C₃₇×#340	C₂²	1	1	yes	characteristic
30	C₃₇×#340	C₂²	1	2	yes	—
31	C₃₇×#340	C₂²	1	1	yes	characteristic
32	C₃₇×#340	C₂²	1	1	yes	characteristic
33	C₃₇×#340	C₂²	1	1	yes	characteristic
34	C₃₇×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	—	4	2	no	—
37	C₂²×C₈×C₃₇	Q₈	1	4	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
41	#125×C₃₇	—	2	4	no	—
42	#125×C₃₇	—	4	2	no	—
43	#117×C₃₇	—	2	4	no	—
44	#117×C₃₇	—	4	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
58	C₂×C₄²×C₃₇	—	2	2	no	—
59	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
60	#107×C₃₇	—	4	2	no	—
61	#107×C₃₇	—	4	2	no	—
62	#106×C₃₇	—	2	4	no	—
63	C₂²⋊C₈×C₃₇	Q₈	1	4	yes	—
64	#97×C₃₇	—	4	2	no	—
65	#95×C₃₇	—	2	2	no	—
66	#95×C₃₇	—	2	2	no	—
67	#95×C₃₇	—	2	2	no	—
68	#95×C₃₇	D₄	1	1	yes	characteristic
69	#95×C₃₇	D₄	1	1	yes	characteristic
70	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	—	2	2	no	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	2	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₃₇	—	2	2	no	—
84	C₂×C₈×C₃₇	—	2	2	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	4	4	no	—
88	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	2	no	—
98	C₄⋊C₄×C₃₇	—	4	4	no	—
99	C₄⋊C₄×C₃₇	—	4	4	no	—
100	C₄⋊C₄×C₃₇	—	4	4	no	—
101	C₄⋊C₄×C₃₇	—	4	4	no	—
102	C₄⋊C₄×C₃₇	—	8	2	no	—
103	C₄⋊C₄×C₃₇	—	8	2	no	—
104	C₄⋊C₄×C₃₇	#41	1	4	yes	—
105	C₂²⋊C₄×C₃₇	—	4	2	no	—
106	C₂²⋊C₄×C₃₇	—	4	4	no	—
107	C₂²⋊C₄×C₃₇	—	8	2	no	—
108	C₄²×C₃₇	—	2	2	no	—
109	C₄²×C₃₇	—	2	2	no	—
110	C₄²×C₃₇	—	4	4	no	—
111	C₂³×C₃₇	—	2	2	no	—
112	C₂³×C₃₇	—	2	2	no	—
113	C₂³×C₃₇	—	2	1	no	—
114	C₂³×C₃₇	—	2	1	no	—
115	C₂³×C₃₇	C₄⋊Q₈	1	2	yes	—
116	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	2	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	4	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	4	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	2	2	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	2	no	—
131	C₂×C₄×C₃₇	—	4	2	no	—
132	C₂×C₄×C₃₇	—	4	2	no	—
133	C₂×C₄×C₃₇	—	4	2	no	—
134	C₂×C₄×C₃₇	—	4	2	no	—
135	C₂×C₄×C₃₇	—	4	2	no	—
136	C₂×C₄×C₃₇	—	4	2	no	—
137	C₂×C₄×C₃₇	—	4	2	no	—
138	C₂×C₄×C₃₇	—	4	2	no	—
139	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
144	C₈×C₃₇	—	2	2	no	—
145	C₈×C₃₇	—	2	2	no	—
146	C₈×C₃₇	—	2	4	no	—
147	#28778	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₃₇	—	2	2	no	—
149	C₂²×C₃₇	—	2	2	no	—
150	C₂²×C₃₇	—	2	2	no	—
151	C₂²×C₃₇	—	4	2	no	—
152	C₂²×C₃₇	—	4	2	no	—
153	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂²×C₃₇	#546	1	2	yes	—
155	C₂²×C₃₇	#575	1	2	yes	—
156	C₂²×C₃₇	#576	1	1	yes	characteristic
157	C₂²×C₃₇	#578	1	1	yes	characteristic
158	C₄×C₃₇	—	2	2	no	—
159	C₄×C₃₇	—	4	2	no	—
160	C₄×C₃₇	—	4	2	no	—
161	C₄×C₃₇	—	4	4	no	—
162	C₄×C₃₇	—	8	2	no	—
163	C₄×C₃₇	—	8	2	no	—
164	C₄×C₃₇	—	8	2	no	—
165	C₄×C₃₇	#533	1	1	yes	characteristic
166	C₄×C₃₇	#543	1	1	yes	characteristic
167	C₄×C₃₇	#554	1	2	yes	—
168	#2441	C₂×C₃₇	1	2	yes	—
169	#2083	C₂×C₃₇	1	2	yes	complemented
170	#2059	C₂×C₃₇	1	2	yes	complemented
171	#1992	C₂×C₃₇	1	2	yes	complemented
172	#1944	C₂×C₃₇	1	2	yes	complemented
173	#1854	C₂×C₃₇	1	2	yes	—
174	#1808	C₂×C₃₇	1	1	yes	characteristic
175	#1752	C₂×C₃₇	1	1	yes	characteristic
176	#1735	C₂×C₃₇	1	1	yes	characteristic
177	C₂×C₃₇	—	4	2	no	—
178	C₂×C₃₇	#2549	1	1	yes	characteristic
179	C₂×C₃₇	#3162	1	1	yes	characteristic
180	C₂×C₃₇	#3164	1	1	yes	characteristic
181	C₂×C₃₇	#3309	1	2	yes	—
182	C₂×C₃₇	#3315	1	2	yes	—
183	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×#125	C₂²×C₃₇	1	2	yes	—
185	C₂×#117	C₂²×C₃₇	1	2	yes	—
186	C₂×#107	C₂²×C₃₇	1	2	yes	—
187	C₂×#107	C₂²×C₃₇	1	2	yes	—
188	C₂×#106	C₂²×C₃₇	1	2	yes	—
189	C₂×C₂²⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
190	C₂×#97	C₂²×C₃₇	1	2	yes	—
191	#399	—	2	2	no	—
192	#396	—	2	2	no	—
193	#389	—	2	2	no	—
194	#389	C₂²×C₃₇	1	2	yes	—
195	#389	C₂²×C₃₇	1	2	yes	—
196	#387	—	2	2	no	—
197	#384	C₂²×C₃₇	1	2	yes	—
198	#380	C₂²×C₃₇	1	2	yes	—
199	C₄×C₂²⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
200	#340	C₂²×C₃₇	1	1	yes	characteristic
201	#340	C₂²×C₃₇	1	2	yes	—
202	#340	C₂²×C₃₇	1	1	yes	characteristic
203	#340	C₂²×C₃₇	1	1	yes	characteristic
204	#340	C₂²×C₃₇	1	1	yes	characteristic
205	#340	C₂²×C₃₇	1	2	yes	—
206	#340	C₂²×C₃₇	1	2	yes	—
207	#336	C₂²×C₃₇	1	2	yes	—
208	C₃₇	#28778	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
210	C₂²×C₈	—	4	2	no	—
211	C₂²×C₈	Q₈×C₃₇	1	4	yes	—
212	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
215	#125	—	2	4	no	—
216	#125	—	4	2	no	—
217	#117	—	2	4	no	—
218	#117	—	4	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
234	#107	—	4	2	no	—
235	#107	—	4	2	no	—
236	#106	—	2	4	no	—
237	C₂²⋊C₈	Q₈×C₃₇	1	4	yes	—
238	#97	—	4	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	D₄×C₃₇	1	1	yes	characteristic
243	#95	D₄×C₃₇	1	1	yes	characteristic
244	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
255	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	4	no	—
262	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
263	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
264	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	8	2	no	—
277	C₄⋊C₄	—	8	2	no	—
278	C₄⋊C₄	#41×C₃₇	1	4	yes	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₂²⋊C₄	—	8	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	4	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₄⋊Q₈×C₃₇	1	2	yes	—
290	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
314	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
317	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	4	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
327	C₂²	C₃₇×#546	1	2	yes	—
328	C₂²	C₃₇×#575	1	2	yes	—
329	C₂²	C₃₇×#576	1	1	yes	characteristic
330	C₂²	C₃₇×#578	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	4	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	4	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₃₇×#533	1	1	yes	characteristic
339	C₄	C₃₇×#543	1	1	yes	characteristic
340	C₄	C₃₇×#554	1	2	yes	—
341	C₂	—	4	2	no	—
342	C₂	C₃₇×#2549	1	1	yes	characteristic
343	C₂	C₃₇×#3162	1	1	yes	characteristic
344	C₂	C₃₇×#3164	1	1	yes	characteristic
345	C₂	C₃₇×#3309	1	2	yes	—
346	C₂	C₃₇×#3315	1	2	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 ◁ C₄×C₂²⋊C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#2083	C₂	semidirect
3. C₃₇×#2059	C₂	semidirect
4. C₃₇×#1992	C₂	semidirect
5. C₃₇×#1944	C₂	semidirect
147. #28778	C₃₇	direct (N × H)
169. #2083	C₂×C₃₇	semidirect
170. #2059	C₂×C₃₇	semidirect
171. #1992	C₂×C₃₇	semidirect
172. #1944	C₂×C₃₇	semidirect
208. C₃₇	#28778	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2441	2
C₃₇×#2083	2
C₃₇×#2059	2
C₃₇×#1992	2
C₃₇×#1944	2
C₃₇×#1854	2
C₃₇×#1808	2
C₃₇×#1752	2
C₃₇×#1735	2
#28778	37

Subgroups — C37×#28778 #22650031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28778 #22650031