Subgroups — C₃₇×#33780 #22655033

Order 9472.

632 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2250	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2243	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2208	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2206	C₂	1	2	yes	maximal, complemented
7	C₃₇×#2186	C₂	1	4	yes	maximal, complemented
8	C₃₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₃₇	—	2	1	no	—
10	C₂×#135×C₃₇	C₂²	1	2	yes	—
11	C₃₇×#532	—	2	1	no	—
12	C₃₇×#532	—	2	1	no	—
13	C₂×SD₃₂×C₃₇	C₂²	1	4	yes	—
14	C₂×D₁₆×C₃₇	C₂²	1	4	yes	—
15	C₃₇×#500	C₂²	1	1	yes	characteristic
16	C₃₇×#495	—	2	1	no	—
17	C₃₇×#495	C₂²	1	2	yes	—
18	C₃₇×#494	C₂²	1	1	yes	characteristic
19	C₃₇×#493	C₂²	1	1	yes	characteristic
20	C₃₇×#493	C₂²	1	1	yes	characteristic
21	C₃₇×#492	—	2	1	no	—
22	C₃₇×#486	—	2	1	no	—
23	C₃₇×#466	—	2	1	no	—
24	C₃₇×#466	C₂²	1	2	yes	—
25	C₄×D₈×C₃₇	—	2	1	no	—
26	C₄×D₈×C₃₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
28	C₃₇×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₇×#358	C₂²	1	4	yes	—
30	C₃₇×#357	C₂²	1	4	yes	—
31	C₃₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₇×#346	C₂²	1	2	yes	—
33	C₂×#41×C₃₇	—	2	2	no	—
34	C₂×#41×C₃₇	—	4	1	no	—
35	#135×C₃₇	—	4	4	no	—
36	#135×C₃₇	—	4	4	no	—
37	C₂×Q₁₆×C₃₇	—	2	1	no	—
38	C₂×Q₁₆×C₃₇	C₂³	1	2	yes	—
39	C₂×SD₁₆×C₃₇	—	2	2	no	—
40	C₂×SD₁₆×C₃₇	—	2	2	no	—
41	C₂×D₈×C₃₇	—	2	1	no	—
42	C₂×D₈×C₃₇	—	2	1	no	—
43	C₂×D₈×C₃₇	C₂³	1	2	yes	—
44	C₂×D₈×C₃₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₃₇	—	2	1	no	—
48	C₄⋊Q₈×C₃₇	—	2	1	no	—
49	#127×C₃₇	—	2	1	no	—
50	#127×C₃₇	—	2	1	no	—
51	#124×C₃₇	—	2	2	no	—
52	#124×C₃₇	—	4	1	no	—
53	#121×C₃₇	—	2	2	no	—
54	#121×C₃₇	—	4	1	no	—
55	#121×C₃₇	—	4	1	no	—
56	#121×C₃₇	—	4	1	no	—
57	C₄×D₄×C₃₇	—	2	2	no	—
58	C₄×D₄×C₃₇	—	4	1	no	—
59	C₄×D₄×C₃₇	—	4	1	no	—
60	C₄×D₄×C₃₇	—	4	1	no	—
61	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
62	SD₃₂×C₃₇	—	4	4	no	—
63	D₁₆×C₃₇	—	4	4	no	—
64	C₂×C₁₆×C₃₇	C₂³	1	4	yes	—
65	#107×C₃₇	—	2	1	no	—
66	#107×C₃₇	C₂³	1	2	yes	—
67	#102×C₃₇	—	2	2	no	—
68	#102×C₃₇	—	2	2	no	—
69	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
73	#41×C₃₇	—	4	4	no	—
74	#41×C₃₇	—	4	4	no	—
75	#41×C₃₇	—	8	4	no	—
76	C₂×Q₈×C₃₇	—	2	2	no	—
77	C₂×Q₈×C₃₇	—	4	1	no	—
78	C₂×D₄×C₃₇	—	2	2	no	—
79	C₂×D₄×C₃₇	—	2	2	no	—
80	C₂×D₄×C₃₇	—	4	2	no	—
81	C₂×D₄×C₃₇	—	4	2	no	—
82	C₂×D₄×C₃₇	—	4	1	no	—
83	C₂×D₄×C₃₇	—	4	1	no	—
84	C₂×D₄×C₃₇	—	4	1	no	—
85	C₂×D₄×C₃₇	—	4	1	no	—
86	C₂²×C₄×C₃₇	—	4	2	no	—
87	C₂²×C₄×C₃₇	—	4	2	no	—
88	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₃₇	—	2	4	no	—
92	Q₁₆×C₃₇	—	4	2	no	—
93	SD₁₆×C₃₇	—	4	4	no	—
94	SD₁₆×C₃₇	—	4	4	no	—
95	D₈×C₃₇	—	2	2	no	—
96	D₈×C₃₇	—	2	4	no	—
97	D₈×C₃₇	—	2	2	no	—
98	D₈×C₃₇	—	4	2	no	—
99	D₈×C₃₇	—	4	2	no	—
100	C₂×C₈×C₃₇	—	2	2	no	—
101	C₂×C₈×C₃₇	—	2	2	no	—
102	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₃₇	—	2	2	no	—
107	C₄⋊C₄×C₃₇	—	4	1	no	—
108	C₄⋊C₄×C₃₇	—	4	1	no	—
109	C₄⋊C₄×C₃₇	—	4	1	no	—
110	C₂²⋊C₄×C₃₇	—	4	2	no	—
111	C₂²⋊C₄×C₃₇	—	4	2	no	—
112	C₂²⋊C₄×C₃₇	—	4	2	no	—
113	C₂²⋊C₄×C₃₇	—	4	2	no	—
114	C₄²×C₃₇	D₈	1	1	yes	characteristic
115	C₄²×C₃₇	D₈	1	1	yes	characteristic
116	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₃₇	—	2	4	no	—
119	C₂³×C₃₇	—	4	2	no	—
120	C₂³×C₃₇	—	4	2	no	—
121	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₃₇	—	4	2	no	—
123	Q₈×C₃₇	—	4	2	no	—
124	Q₈×C₃₇	—	8	2	no	—
125	D₄×C₃₇	—	4	2	no	—
126	D₄×C₃₇	—	4	2	no	—
127	D₄×C₃₇	—	4	2	no	—
128	D₄×C₃₇	—	4	2	no	—
129	D₄×C₃₇	—	8	2	no	—
130	D₄×C₃₇	—	8	2	no	—
131	D₄×C₃₇	—	8	2	no	—
132	D₄×C₃₇	—	8	4	no	—
133	D₄×C₃₇	—	8	4	no	—
134	D₄×C₃₇	—	8	2	no	—
135	C₂×C₄×C₃₇	—	2	1	no	—
136	C₂×C₄×C₃₇	—	2	1	no	—
137	C₂×C₄×C₃₇	—	2	2	no	—
138	C₂×C₄×C₃₇	—	4	1	no	—
139	C₂×C₄×C₃₇	—	4	2	no	—
140	C₂×C₄×C₃₇	—	4	2	no	—
141	C₂×C₄×C₃₇	—	8	4	no	—
142	C₂×C₄×C₃₇	—	8	4	no	—
143	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₃₇	—	2	1	no	—
150	C₈×C₃₇	—	2	1	no	—
151	C₈×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
152	C₈×C₃₇	H(5,𝔽₂)	1	2	yes	—
153	#33780	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₃₇	—	2	1	no	—
155	C₂²×C₃₇	—	2	1	no	—
156	C₂²×C₃₇	—	2	1	no	—
157	C₂²×C₃₇	—	8	2	no	—
158	C₂²×C₃₇	—	8	2	no	—
159	C₂²×C₃₇	—	8	2	no	—
160	C₂²×C₃₇	—	8	2	no	—
161	C₂²×C₃₇	—	8	2	no	—
162	C₂²×C₃₇	—	8	2	no	—
163	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₃₇	—	2	1	no	—
165	C₄×C₃₇	—	2	1	no	—
166	C₄×C₃₇	—	2	1	no	—
167	C₄×C₃₇	—	2	1	no	—
168	C₄×C₃₇	—	8	2	no	—
169	C₄×C₃₇	—	8	2	no	—
170	C₄×C₃₇	#509	1	2	yes	—
171	C₄×C₃₇	#535	1	1	yes	characteristic
172	C₄×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	#3169	C₂×C₃₇	1	1	yes	characteristic, complemented
174	#3147	C₂×C₃₇	1	1	yes	characteristic, complemented
175	#2250	C₂×C₃₇	1	2	yes	complemented
176	#2243	C₂×C₃₇	1	2	yes	complemented
177	#2208	C₂×C₃₇	1	2	yes	complemented
178	#2206	C₂×C₃₇	1	2	yes	complemented
179	#2186	C₂×C₃₇	1	4	yes	complemented
180	#2163	C₂×C₃₇	1	1	yes	characteristic, complemented
181	C₂×C₃₇	—	4	1	no	—
182	C₂×C₃₇	—	16	2	no	—
183	C₂×C₃₇	—	16	2	no	—
184	C₂×C₃₇	#3205	1	1	yes	characteristic
185	C₂×C₃₇	C₂×#509	1	1	yes	characteristic
186	C₂×C₃₇	#3414	1	1	yes	characteristic
187	C₂×#135	—	2	1	no	—
188	C₂×#135	C₂²×C₃₇	1	2	yes	—
189	#532	—	2	1	no	—
190	#532	—	2	1	no	—
191	C₂×SD₃₂	C₂²×C₃₇	1	4	yes	—
192	C₂×D₁₆	C₂²×C₃₇	1	4	yes	—
193	#500	C₂²×C₃₇	1	1	yes	characteristic
194	#495	—	2	1	no	—
195	#495	C₂²×C₃₇	1	2	yes	—
196	#494	C₂²×C₃₇	1	1	yes	characteristic
197	#493	C₂²×C₃₇	1	1	yes	characteristic
198	#493	C₂²×C₃₇	1	1	yes	characteristic
199	#492	—	2	1	no	—
200	#486	—	2	1	no	—
201	#466	—	2	1	no	—
202	#466	C₂²×C₃₇	1	2	yes	—
203	C₄×D₈	—	2	1	no	—
204	C₄×D₈	C₂²×C₃₇	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
206	C₄⋊C₁₆	C₂²×C₃₇	1	2	yes	—
207	#358	C₂²×C₃₇	1	4	yes	—
208	#357	C₂²×C₃₇	1	4	yes	—
209	C₂²⋊C₁₆	C₂²×C₃₇	1	2	yes	—
210	#346	C₂²×C₃₇	1	2	yes	—
211	C₃₇	#33780	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×#41	—	2	2	no	—
213	C₂×#41	—	4	1	no	—
214	#135	—	4	4	no	—
215	#135	—	4	4	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	C₂³×C₃₇	1	2	yes	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	C₂³×C₃₇	1	2	yes	—
223	C₂×D₈	C₂³×C₃₇	1	2	yes	—
224	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	#127	—	2	1	no	—
229	#127	—	2	1	no	—
230	#124	—	2	2	no	—
231	#124	—	4	1	no	—
232	#121	—	2	2	no	—
233	#121	—	4	1	no	—
234	#121	—	4	1	no	—
235	#121	—	4	1	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	4	1	no	—
238	C₄×D₄	—	4	1	no	—
239	C₄×D₄	—	4	1	no	—
240	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
241	SD₃₂	—	4	4	no	—
242	D₁₆	—	4	4	no	—
243	C₂×C₁₆	C₂³×C₃₇	1	4	yes	—
244	#107	—	2	1	no	—
245	#107	C₂³×C₃₇	1	2	yes	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
249	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
252	#41	—	4	4	no	—
253	#41	—	4	4	no	—
254	#41	—	8	4	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
268	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
270	Q₁₆	—	2	4	no	—
271	Q₁₆	—	4	2	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	4	4	no	—
274	D₈	—	2	2	no	—
275	D₈	—	2	4	no	—
276	D₈	—	2	2	no	—
277	D₈	—	4	2	no	—
278	D₈	—	4	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
284	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	D₈×C₃₇	1	1	yes	characteristic
294	C₄²	D₈×C₃₇	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
296	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
297	C₁₆	—	2	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
323	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
324	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
325	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
326	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
331	C₈	H(5,𝔽₂)×C₃₇	1	2	yes	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₃₇×#509	1	2	yes	—
349	C₄	C₃₇×#535	1	1	yes	characteristic
350	C₄	C₂×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	16	2	no	—
353	C₂	—	16	2	no	—
354	C₂	C₃₇×#3205	1	1	yes	characteristic
355	C₂	C₂×C₃₇×#509	1	1	yes	characteristic
356	C₂	C₃₇×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3169	C₂	semidirect
2. C₃₇×#3147	C₂	semidirect
3. C₃₇×#2250	C₂	semidirect
4. C₃₇×#2243	C₂	semidirect
5. C₃₇×#2208	C₂	semidirect
6. C₃₇×#2206	C₂	semidirect
7. C₃₇×#2186	C₂	semidirect
8. C₃₇×#2163	C₂	semidirect
153. #33780	C₃₇	direct (N × H)
173. #3169	C₂×C₃₇	semidirect
174. #3147	C₂×C₃₇	semidirect
175. #2250	C₂×C₃₇	semidirect
176. #2243	C₂×C₃₇	semidirect
177. #2208	C₂×C₃₇	semidirect
178. #2206	C₂×C₃₇	semidirect
179. #2186	C₂×C₃₇	semidirect
180. #2163	C₂×C₃₇	semidirect
211. C₃₇	#33780	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×#3169	2
C₃₇×#3147	2
C₃₇×#2250	2
C₃₇×#2243	2
C₃₇×#2208	2
C₃₇×#2206	2
C₃₇×#2186	2
C₃₇×#2163	2
#33780	37

Subgroups — C37×#33780 #22655033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33780 #22655033