Subgroups — C₃₇×#28779 #22650032

Order 9472.

718 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2439	C₂	1	2	yes	maximal
2	C₃₇×#2085	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2058	C₂	1	2	yes	maximal, complemented
4	C₃₇×#1993	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×#1807	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×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	C₂×#117×C₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₃₇	C₂²	1	2	yes	—
14	C₂×#106×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×#396	—	2	2	no	—
19	C₃₇×#393	—	2	2	no	—
20	C₃₇×#391	—	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×#382	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	C₂²×Q₈×C₃₇	—	2	2	no	—
36	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	—	4	2	no	—
38	C₂²×C₈×C₃₇	Q₈	1	4	yes	complemented
39	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
42	C₄⋊Q₈×C₃₇	—	2	4	no	—
43	C₄⋊Q₈×C₃₇	—	4	2	no	—
44	#117×C₃₇	—	2	4	no	—
45	#117×C₃₇	—	4	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	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₃₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₇	—	2	2	no	—
58	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
59	#107×C₃₇	—	4	2	no	—
60	#106×C₃₇	—	2	4	no	—
61	#106×C₃₇	—	4	2	no	—
62	C₂²⋊C₈×C₃₇	Q₈	1	4	yes	complemented
63	#97×C₃₇	—	4	2	no	—
64	#95×C₃₇	—	2	2	no	—
65	#95×C₃₇	—	2	2	no	—
66	#95×C₃₇	—	2	2	no	—
67	#95×C₃₇	—	2	2	no	—
68	#95×C₃₇	—	2	2	no	—
69	#95×C₃₇	D₄	1	2	yes	—
70	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₃₇	—	4	4	no	—
72	C₂×Q₈×C₃₇	—	4	4	no	—
73	C₂×Q₈×C₃₇	—	8	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	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	2	2	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	—	4	4	no	—
91	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
92	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
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	—	2	2	no	—
99	C₄⋊C₄×C₃₇	—	2	2	no	—
100	C₄⋊C₄×C₃₇	—	2	2	no	—
101	C₄⋊C₄×C₃₇	—	4	4	no	—
102	C₄⋊C₄×C₃₇	—	4	4	no	—
103	C₄⋊C₄×C₃₇	—	8	2	no	—
104	C₄⋊C₄×C₃₇	#41	1	4	yes	—
105	C₂²⋊C₄×C₃₇	—	4	1	no	—
106	C₂²⋊C₄×C₃₇	—	4	1	no	—
107	C₂²⋊C₄×C₃₇	—	4	4	no	—
108	C₂²⋊C₄×C₃₇	—	8	2	no	—
109	C₄²×C₃₇	—	2	2	no	—
110	C₄²×C₃₇	—	2	2	no	—
111	C₄²×C₃₇	—	4	4	no	—
112	C₂³×C₃₇	—	2	2	no	—
113	C₂³×C₃₇	—	2	1	no	—
114	C₂³×C₃₇	—	2	1	no	—
115	C₂³×C₃₇	—	2	2	no	—
116	C₂³×C₃₇	C₄⋊Q₈	1	2	yes	—
117	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₃₇	—	8	8	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	4	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	2	4	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	2	2	no	—
128	C₂×C₄×C₃₇	—	2	2	no	—
129	C₂×C₄×C₃₇	—	2	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	—	4	2	no	—
140	C₂×C₄×C₃₇	—	4	2	no	—
141	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
142	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₃₇	—	2	2	no	—
147	C₈×C₃₇	—	2	2	no	—
148	C₈×C₃₇	—	2	4	no	—
149	#28779	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₃₇	—	2	2	no	—
151	C₂²×C₃₇	—	2	2	no	—
152	C₂²×C₃₇	—	2	2	no	—
153	C₂²×C₃₇	—	4	2	no	—
154	C₂²×C₃₇	—	4	2	no	—
155	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₂²×C₃₇	#546	1	2	yes	—
157	C₂²×C₃₇	#575	1	1	yes	characteristic
158	C₂²×C₃₇	#575	1	1	yes	characteristic
159	C₂²×C₃₇	#577	1	2	yes	—
160	C₄×C₃₇	—	2	2	no	—
161	C₄×C₃₇	—	4	2	no	—
162	C₄×C₃₇	—	4	4	no	—
163	C₄×C₃₇	—	4	2	no	—
164	C₄×C₃₇	—	8	2	no	—
165	C₄×C₃₇	—	8	2	no	—
166	C₄×C₃₇	—	8	2	no	—
167	C₄×C₃₇	#533	1	1	yes	characteristic
168	C₄×C₃₇	#543	1	1	yes	characteristic
169	C₄×C₃₇	#554	1	2	yes	—
170	#2439	C₂×C₃₇	1	2	yes	—
171	#2085	C₂×C₃₇	1	2	yes	complemented
172	#2058	C₂×C₃₇	1	2	yes	complemented
173	#1993	C₂×C₃₇	1	2	yes	complemented
174	#1944	C₂×C₃₇	1	2	yes	complemented
175	#1854	C₂×C₃₇	1	2	yes	—
176	#1807	C₂×C₃₇	1	1	yes	characteristic
177	#1752	C₂×C₃₇	1	1	yes	characteristic
178	#1735	C₂×C₃₇	1	1	yes	characteristic
179	C₂×C₃₇	—	4	2	no	—
180	C₂×C₃₇	#2549	1	1	yes	characteristic
181	C₂×C₃₇	#3162	1	1	yes	characteristic
182	C₂×C₃₇	#3163	1	1	yes	characteristic
183	C₂×C₃₇	#3311	1	2	yes	—
184	C₂×C₃₇	#3321	1	2	yes	—
185	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
187	C₂×#117	C₂²×C₃₇	1	2	yes	—
188	C₂×#107	C₂²×C₃₇	1	2	yes	—
189	C₂×#106	C₂²×C₃₇	1	2	yes	—
190	C₂×#106	C₂²×C₃₇	1	2	yes	—
191	C₂×C₂²⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
192	C₂×#97	C₂²×C₃₇	1	2	yes	—
193	#396	—	2	2	no	—
194	#393	—	2	2	no	—
195	#391	—	2	2	no	—
196	#389	C₂²×C₃₇	1	2	yes	—
197	#389	C₂²×C₃₇	1	2	yes	—
198	#387	—	2	2	no	—
199	#382	C₂²×C₃₇	1	2	yes	—
200	#380	C₂²×C₃₇	1	2	yes	—
201	C₄×C₂²⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
202	#340	C₂²×C₃₇	1	1	yes	characteristic
203	#340	C₂²×C₃₇	1	2	yes	—
204	#340	C₂²×C₃₇	1	1	yes	characteristic
205	#340	C₂²×C₃₇	1	1	yes	characteristic
206	#340	C₂²×C₃₇	1	1	yes	characteristic
207	#340	C₂²×C₃₇	1	2	yes	—
208	#340	C₂²×C₃₇	1	2	yes	—
209	#336	C₂²×C₃₇	1	2	yes	—
210	C₃₇	#28779	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×Q₈	—	2	2	no	—
212	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
213	C₂²×C₈	—	4	2	no	—
214	C₂²×C₈	Q₈×C₃₇	1	4	yes	complemented
215	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
218	C₄⋊Q₈	—	2	4	no	—
219	C₄⋊Q₈	—	4	2	no	—
220	#117	—	2	4	no	—
221	#117	—	4	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	1	yes	characteristic
232	C₂×C₂²⋊C₄	D₄×C₃₇	1	1	yes	characteristic
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
235	#107	—	4	2	no	—
236	#106	—	2	4	no	—
237	#106	—	4	2	no	—
238	C₂²⋊C₈	Q₈×C₃₇	1	4	yes	complemented
239	#97	—	4	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	D₄×C₃₇	1	2	yes	—
246	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×Q₈	—	8	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
260	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
268	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
269	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	#41×C₃₇	1	4	yes	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	4	4	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	2	no	—
292	C₂³	C₄⋊Q₈×C₃₇	1	2	yes	—
293	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
294	Q₈	—	8	8	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	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	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
318	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
321	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	4	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
331	C₂²	C₃₇×#546	1	2	yes	—
332	C₂²	C₃₇×#575	1	1	yes	characteristic
333	C₂²	C₃₇×#575	1	1	yes	characteristic
334	C₂²	C₃₇×#577	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	4	no	—
338	C₄	—	4	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₃₇×#533	1	1	yes	characteristic
343	C₄	C₃₇×#543	1	1	yes	characteristic
344	C₄	C₃₇×#554	1	2	yes	—
345	C₂	—	4	2	no	—
346	C₂	C₃₇×#2549	1	1	yes	characteristic
347	C₂	C₃₇×#3162	1	1	yes	characteristic
348	C₂	C₃₇×#3163	1	1	yes	characteristic
349	C₂	C₃₇×#3311	1	2	yes	—
350	C₂	C₃₇×#3321	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×#2085	C₂	semidirect
3. C₃₇×#2058	C₂	semidirect
4. C₃₇×#1993	C₂	semidirect
5. C₃₇×#1944	C₂	semidirect
38. C₂²×C₈×C₃₇	Q₈	semidirect
62. C₂²⋊C₈×C₃₇	Q₈	semidirect
149. #28779	C₃₇	direct (N × H)
171. #2085	C₂×C₃₇	semidirect
172. #2058	C₂×C₃₇	semidirect
173. #1993	C₂×C₃₇	semidirect
174. #1944	C₂×C₃₇	semidirect
210. C₃₇	#28779	direct (N × H)
214. C₂²×C₈	Q₈×C₃₇	semidirect
238. C₂²⋊C₈	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×#2439	2
C₃₇×#2085	2
C₃₇×#2058	2
C₃₇×#1993	2
C₃₇×#1944	2
C₃₇×#1854	2
C₃₇×#1807	2
C₃₇×#1752	2
C₃₇×#1735	2
#28779	37

Subgroups — C37×#28779 #22650032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28779 #22650032