Subgroups — C₃₇×#29819 #22651072

Order 9472.

832 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#3039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#3038	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2046	C₂	1	4	yes	maximal, complemented
6	C₃₇×#2019	C₂	1	4	yes	maximal, complemented
7	C₃₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₃₇	C₂²	1	2	yes	—
11	C₂×AGL(1,ℤ/8)×C₃₇	C₂²	1	2	yes	—
12	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
13	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₇	C₂²	1	1	yes	characteristic
16	C₃₇×#492	—	2	2	no	—
17	C₃₇×#492	—	2	2	no	—
18	C₃₇×#462	—	2	2	no	—
19	C₃₇×#461	—	2	2	no	—
20	C₃₇×#460	—	2	2	no	—
21	C₃₇×#459	—	2	2	no	—
22	C₃₇×#433	—	2	2	no	—
23	C₃₇×#431	—	2	2	no	—
24	C₃₇×#424	C₂²	1	2	yes	—
25	C₃₇×#423	C₂²	1	1	yes	characteristic
26	C₃₇×#423	C₂²	1	1	yes	characteristic
27	C₃₇×#419	C₂²	1	2	yes	—
28	C₃₇×#418	C₂²	1	2	yes	—
29	C₂×#101×C₃₇	C₂²	1	4	yes	—
30	C₂×#100×C₃₇	C₂²	1	4	yes	—
31	C₃₇×#408	C₂²	1	2	yes	—
32	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
34	C₃₇×#341	C₂²	1	4	yes	—
35	C₂×#41×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×D₄×C₃₇	—	2	2	no	—
38	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
39	#137×C₃₇	—	4	4	no	—
40	AGL(1,ℤ/8)×C₃₇	—	4	4	no	—
41	C₂×Q₁₆×C₃₇	—	2	2	no	—
42	C₂×SD₁₆×C₃₇	—	2	2	no	—
43	C₂×SD₁₆×C₃₇	—	2	2	no	—
44	C₂×SD₁₆×C₃₇	—	2	8	no	—
45	C₂×SD₁₆×C₃₇	—	4	4	no	—
46	C₂×SD₁₆×C₃₇	—	4	4	no	—
47	C₂×D₈×C₃₇	—	2	2	no	—
48	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
49	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
50	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₃₇	—	2	2	no	—
53	C₄⋊Q₈×C₃₇	—	2	1	no	—
54	C₄⋊Q₈×C₃₇	—	2	1	no	—
55	#127×C₃₇	—	2	2	no	—
56	#127×C₃₇	—	2	1	no	—
57	#127×C₃₇	—	2	1	no	—
58	C₄×Q₈×C₃₇	D₄	1	2	yes	—
59	C₄×D₄×C₃₇	—	2	2	no	—
60	C₄×D₄×C₃₇	D₄	1	2	yes	—
61	#117×C₃₇	—	2	1	no	—
62	#117×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
64	C₂×C₄²×C₃₇	—	2	1	no	—
65	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₃₇	—	2	4	no	—
67	C₄⋊C₈×C₃₇	—	2	2	no	—
68	C₄⋊C₈×C₃₇	—	2	2	no	—
69	#103×C₃₇	—	2	4	no	—
70	#102×C₃₇	—	2	4	no	—
71	#101×C₃₇	—	4	4	no	—
72	#100×C₃₇	—	4	4	no	—
73	C₂²⋊C₈×C₃₇	—	2	4	no	—
74	#97×C₃₇	—	2	1	no	—
75	C₄×C₈×C₃₇	—	2	1	no	—
76	C₄×C₈×C₃₇	D₄	1	2	yes	—
77	C₄×C₈×C₃₇	D₄	1	2	yes	—
78	C₂⁴×C₃₇	—	2	2	no	—
79	#41×C₃₇	—	2	2	no	—
80	#41×C₃₇	—	2	2	no	—
81	C₂×Q₈×C₃₇	—	2	2	no	—
82	C₂×Q₈×C₃₇	—	2	2	no	—
83	C₂×Q₈×C₃₇	—	2	4	no	—
84	C₂×Q₈×C₃₇	—	2	4	no	—
85	C₂×Q₈×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂×D₄×C₃₇	—	2	1	no	—
87	C₂×D₄×C₃₇	—	2	2	no	—
88	C₂×D₄×C₃₇	—	2	2	no	—
89	C₂×D₄×C₃₇	—	2	4	no	—
90	C₂×D₄×C₃₇	—	2	4	no	—
91	C₂×D₄×C₃₇	—	4	2	no	—
92	C₂×D₄×C₃₇	—	4	2	no	—
93	C₂×D₄×C₃₇	—	8	4	no	—
94	C₂×D₄×C₃₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₇	—	2	2	no	—
96	C₂²×C₄×C₃₇	—	2	1	no	—
97	C₂²×C₄×C₃₇	—	2	2	no	—
98	C₂²×C₄×C₃₇	#41	1	2	yes	—
99	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
100	Q₁₆×C₃₇	—	4	4	no	—
101	SD₁₆×C₃₇	—	4	16	no	—
102	SD₁₆×C₃₇	—	4	4	no	—
103	SD₁₆×C₃₇	—	4	4	no	—
104	D₈×C₃₇	—	4	4	no	—
105	M₁₆×C₃₇	—	2	2	no	—
106	M₁₆×C₃₇	—	2	4	no	—
107	M₁₆×C₃₇	—	2	2	no	—
108	M₁₆×C₃₇	—	2	4	no	—
109	C₂×C₈×C₃₇	—	2	2	no	—
110	C₂×C₈×C₃₇	—	2	2	no	—
111	C₂×C₈×C₃₇	—	2	2	no	—
112	C₂×C₈×C₃₇	—	2	2	no	—
113	C₂×C₈×C₃₇	—	2	2	no	—
114	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₃₇	—	2	2	no	—
116	C₄⋊C₄×C₃₇	—	4	2	no	—
117	C₄⋊C₄×C₃₇	—	4	2	no	—
118	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄×C₃₇	—	2	1	no	—
120	C₂²⋊C₄×C₃₇	—	2	2	no	—
121	C₄²×C₃₇	—	2	2	no	—
122	C₄²×C₃₇	—	2	1	no	—
123	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₃₇	C₂×D₄	1	2	yes	—
126	C₂³×C₃₇	—	2	1	no	—
127	C₂³×C₃₇	—	4	2	no	—
128	C₂³×C₃₇	—	4	4	no	—
129	C₂³×C₃₇	—	8	4	no	—
130	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
131	Q₈×C₃₇	—	2	1	no	—
132	Q₈×C₃₇	—	2	1	no	—
133	Q₈×C₃₇	—	4	4	no	—
134	Q₈×C₃₇	—	4	2	no	—
135	Q₈×C₃₇	—	4	2	no	—
136	D₄×C₃₇	—	2	1	no	—
137	D₄×C₃₇	—	2	1	no	—
138	D₄×C₃₇	—	4	2	no	—
139	D₄×C₃₇	—	4	2	no	—
140	D₄×C₃₇	—	4	4	no	—
141	D₄×C₃₇	—	4	2	no	—
142	D₄×C₃₇	—	8	4	no	—
143	D₄×C₃₇	—	8	4	no	—
144	C₂×C₄×C₃₇	—	2	4	no	—
145	C₂×C₄×C₃₇	—	2	2	no	—
146	C₂×C₄×C₃₇	—	2	1	no	—
147	C₂×C₄×C₃₇	—	4	2	no	—
148	C₂×C₄×C₃₇	—	4	4	no	—
149	C₂×C₄×C₃₇	—	4	2	no	—
150	C₂×C₄×C₃₇	—	4	4	no	—
151	C₂×C₄×C₃₇	#121	1	2	yes	—
152	C₂×C₄×C₃₇	#121	1	2	yes	—
153	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
156	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
157	C₈×C₃₇	—	2	4	no	—
158	C₈×C₃₇	—	4	2	no	—
159	C₈×C₃₇	—	4	2	no	—
160	C₈×C₃₇	—	4	2	no	—
161	#29819	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂²×C₃₇	—	2	2	no	—
163	C₂²×C₃₇	—	4	1	no	—
164	C₂²×C₃₇	—	4	2	no	—
165	C₂²×C₃₇	—	8	2	no	—
166	C₂²×C₃₇	—	8	4	no	—
167	C₂²×C₃₇	—	16	4	no	—
168	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
169	C₂²×C₃₇	#540	1	2	yes	—
170	C₄×C₃₇	—	2	2	no	—
171	C₄×C₃₇	—	4	1	no	—
172	C₄×C₃₇	—	4	2	no	—
173	C₄×C₃₇	—	8	2	no	—
174	C₄×C₃₇	#471	1	4	yes	—
175	C₄×C₃₇	#535	1	2	yes	—
176	C₄×C₃₇	#535	1	1	yes	characteristic
177	C₄×C₃₇	#535	1	1	yes	characteristic
178	C₂×#492	C₂×C₃₇	1	1	yes	characteristic, complemented
179	#3039	C₂×C₃₇	1	1	yes	characteristic, complemented
180	#3038	C₂×C₃₇	1	1	yes	characteristic
181	#2922	C₂×C₃₇	1	1	yes	characteristic, complemented
182	#2046	C₂×C₃₇	1	4	yes	complemented
183	#2019	C₂×C₃₇	1	4	yes	complemented
184	#1966	C₂×C₃₇	1	1	yes	characteristic, complemented
185	#1965	C₂×C₃₇	1	1	yes	characteristic, complemented
186	#1837	C₂×C₃₇	1	1	yes	characteristic, complemented
187	C₂×C₃₇	—	2	2	no	—
188	C₂×C₃₇	—	8	1	no	—
189	C₂×C₃₇	—	16	2	no	—
190	C₂×C₃₇	#2619	1	1	yes	characteristic
191	C₂×C₃₇	C₂×#471	1	2	yes	—
192	C₂×#137	C₂²×C₃₇	1	2	yes	—
193	C₂×AGL(1,ℤ/8)	C₂²×C₃₇	1	2	yes	—
194	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
196	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
197	C₄×#41	C₂²×C₃₇	1	1	yes	characteristic
198	#492	—	2	2	no	—
199	#492	—	2	2	no	—
200	#462	—	2	2	no	—
201	#461	—	2	2	no	—
202	#460	—	2	2	no	—
203	#459	—	2	2	no	—
204	#433	—	2	2	no	—
205	#431	—	2	2	no	—
206	#424	C₂²×C₃₇	1	2	yes	—
207	#423	C₂²×C₃₇	1	1	yes	characteristic
208	#423	C₂²×C₃₇	1	1	yes	characteristic
209	#419	C₂²×C₃₇	1	2	yes	—
210	#418	C₂²×C₃₇	1	2	yes	—
211	C₂×#101	C₂²×C₃₇	1	4	yes	—
212	C₂×#100	C₂²×C₃₇	1	4	yes	—
213	#408	C₂²×C₃₇	1	2	yes	—
214	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
216	#341	C₂²×C₃₇	1	4	yes	—
217	C₃₇	#29819	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×#41	C₂³×C₃₇	1	1	yes	characteristic
219	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
220	C₂²×D₄	—	2	2	no	—
221	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
222	#137	—	4	4	no	—
223	AGL(1,ℤ/8)	—	4	4	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	8	no	—
228	C₂×SD₁₆	—	4	4	no	—
229	C₂×SD₁₆	—	4	4	no	—
230	C₂×D₈	—	2	2	no	—
231	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
232	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
233	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
234	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
235	C₄⋊Q₈	—	2	2	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	#127	—	2	2	no	—
239	#127	—	2	1	no	—
240	#127	—	2	1	no	—
241	C₄×Q₈	D₄×C₃₇	1	2	yes	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	D₄×C₃₇	1	2	yes	—
244	#117	—	2	1	no	—
245	#117	C₂³×C₃₇	1	1	yes	characteristic
246	C₂×C₄⋊C₄	—	2	2	no	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	4	no	—
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	2	2	no	—
252	#103	—	2	4	no	—
253	#102	—	2	4	no	—
254	#101	—	4	4	no	—
255	#100	—	4	4	no	—
256	C₂²⋊C₈	—	2	4	no	—
257	#97	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	D₄×C₃₇	1	2	yes	—
260	C₄×C₈	D₄×C₃₇	1	2	yes	—
261	C₂⁴	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	4	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	C₂×D₄×C₃₇	1	1	yes	characteristic
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	4	no	—
273	C₂×D₄	—	2	4	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	8	4	no	—
277	C₂×D₄	C₂×D₄×C₃₇	1	1	yes	characteristic
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	#41×C₃₇	1	2	yes	—
282	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
283	Q₁₆	—	4	4	no	—
284	SD₁₆	—	4	16	no	—
285	SD₁₆	—	4	4	no	—
286	SD₁₆	—	4	4	no	—
287	D₈	—	4	4	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	1	no	—
306	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₃₇	1	2	yes	—
309	C₂³	—	2	1	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	4	4	no	—
312	C₂³	—	8	4	no	—
313	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
314	Q₈	—	2	1	no	—
315	Q₈	—	2	1	no	—
316	Q₈	—	4	4	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	D₄	—	2	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	4	no	—
324	D₄	—	4	2	no	—
325	D₄	—	8	4	no	—
326	D₄	—	8	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	#121×C₃₇	1	2	yes	—
335	C₂×C₄	#121×C₃₇	1	2	yes	—
336	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
338	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
339	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
340	C₈	—	2	4	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	4	no	—
349	C₂²	—	16	4	no	—
350	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
351	C₂²	C₃₇×#540	1	2	yes	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₃₇×#471	1	4	yes	—
357	C₄	C₃₇×#535	1	2	yes	—
358	C₄	C₃₇×#535	1	1	yes	characteristic
359	C₄	C₃₇×#535	1	1	yes	characteristic
360	C₂	—	2	2	no	—
361	C₂	—	8	1	no	—
362	C₂	—	16	2	no	—
363	C₂	C₃₇×#2619	1	1	yes	characteristic
364	C₂	C₂×C₃₇×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#492	C₂	semidirect
2. C₃₇×#3039	C₂	semidirect
4. C₃₇×#2922	C₂	semidirect
5. C₃₇×#2046	C₂	semidirect
6. C₃₇×#2019	C₂	semidirect
7. C₃₇×#1966	C₂	semidirect
8. C₃₇×#1965	C₂	semidirect
9. C₃₇×#1837	C₂	semidirect
161. #29819	C₃₇	direct (N × H)
178. C₂×#492	C₂×C₃₇	semidirect
179. #3039	C₂×C₃₇	semidirect
181. #2922	C₂×C₃₇	semidirect
182. #2046	C₂×C₃₇	semidirect
183. #2019	C₂×C₃₇	semidirect
184. #1966	C₂×C₃₇	semidirect
185. #1965	C₂×C₃₇	semidirect
186. #1837	C₂×C₃₇	semidirect
217. C₃₇	#29819	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×#492	2
C₃₇×#3039	2
C₃₇×#3038	2
C₃₇×#2922	2
C₃₇×#2046	2
C₃₇×#2019	2
C₃₇×#1966	2
C₃₇×#1965	2
C₃₇×#1837	2
#29819	37

Subgroups — C37×#29819 #22651072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29819 #22651072