Subgroups — C₃₇×#28828 #22650081

Order 9472.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2571	C₂	1	2	yes	maximal
2	C₃₇×#2085	C₂	1	4	yes	maximal
3	C₃₇×#1948	C₂	1	2	yes	maximal
4	C₃₇×#1942	C₂	1	2	yes	maximal
5	C₃₇×#1846	C₂	1	2	yes	maximal
6	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#107×C₃₇	C₂²	1	2	yes	—
10	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#107×C₃₇	C₂²	1	2	yes	—
13	C₂×#106×C₃₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₃₇×#400	—	2	4	no	—
18	C₃₇×#389	—	2	2	no	—
19	C₃₇×#389	C₂²	1	4	yes	—
20	C₃₇×#384	C₂²	1	2	yes	—
21	C₃₇×#382	—	2	2	no	—
22	C₃₇×#382	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#340	C₂²	1	4	yes	—
26	C₃₇×#340	C₂²	1	2	yes	—
27	C₃₇×#336	C₂²	1	2	yes	—
28	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₃₇	—	2	2	no	—
40	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	#107×C₃₇	—	2	2	no	—
43	#107×C₃₇	—	2	4	no	—
44	#107×C₃₇	—	2	2	no	—
45	#107×C₃₇	—	4	2	no	—
46	#106×C₃₇	—	4	4	no	—
47	C₄⋊C₈×C₃₇	Q₈	1	4	yes	—
48	C₄×C₈×C₃₇	—	2	4	no	—
49	C₄×C₈×C₃₇	Q₈	1	4	yes	—
50	#95×C₃₇	—	2	2	no	—
51	#95×C₃₇	—	2	4	no	—
52	#95×C₃₇	—	2	4	no	—
53	#95×C₃₇	—	2	4	no	—
54	#95×C₃₇	D₄	1	2	yes	—
55	C₂²×C₄×C₃₇	—	2	2	no	—
56	C₂²×C₄×C₃₇	—	2	2	no	—
57	C₂²×C₄×C₃₇	—	2	4	no	—
58	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₃₇	—	2	2	no	—
64	C₂×C₈×C₃₇	—	2	2	no	—
65	C₂×C₈×C₃₇	—	2	1	no	—
66	C₂×C₈×C₃₇	—	2	1	no	—
67	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₃₇	#41	1	2	yes	—
71	C₂×C₈×C₃₇	#41	1	2	yes	—
72	C₄⋊C₄×C₃₇	—	2	2	no	—
73	C₄⋊C₄×C₃₇	—	2	2	no	—
74	C₄⋊C₄×C₃₇	—	2	2	no	—
75	C₄⋊C₄×C₃₇	—	2	2	no	—
76	C₄⋊C₄×C₃₇	—	2	4	no	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	4	no	—
79	C₄⋊C₄×C₃₇	—	2	2	no	—
80	C₄⋊C₄×C₃₇	—	4	4	no	—
81	C₄⋊C₄×C₃₇	—	8	2	no	—
82	C₄⋊C₄×C₃₇	—	8	4	no	—
83	C₄²×C₃₇	—	2	2	no	—
84	C₄²×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	—	4	4	no	—
86	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
87	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
88	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₃₇	—	2	1	no	—
90	C₂×C₄×C₃₇	—	2	1	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	4	2	no	—
98	C₂×C₄×C₃₇	—	4	2	no	—
99	C₂×C₄×C₃₇	—	4	4	no	—
100	C₂×C₄×C₃₇	—	4	2	no	—
101	C₂×C₄×C₃₇	—	4	2	no	—
102	C₂×C₄×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₃₇	#135	1	2	yes	—
109	C₂×C₄×C₃₇	#135	1	2	yes	—
110	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₈×C₃₇	—	2	2	no	—
117	C₈×C₃₇	—	2	4	no	—
118	C₈×C₃₇	#125	1	4	yes	—
119	#28828	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₃₇	#547	1	1	yes	characteristic
122	C₂²×C₃₇	#550	1	1	yes	characteristic
123	C₂²×C₃₇	C₂×#135	1	1	yes	characteristic
124	C₂²×C₃₇	C₂×#135	1	1	yes	characteristic
125	C₂²×C₃₇	#576	1	1	yes	characteristic
126	C₂²×C₃₇	#578	1	1	yes	characteristic
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	—	4	2	no	—
130	C₄×C₃₇	—	8	2	no	—
131	C₄×C₃₇	—	8	2	no	—
132	C₄×C₃₇	—	8	4	no	—
133	C₄×C₃₇	#499	1	4	yes	—
134	C₄×C₃₇	C₂×#125	1	1	yes	characteristic
135	C₄×C₃₇	#544	1	1	yes	characteristic
136	C₄×C₃₇	#554	1	1	yes	characteristic
137	C₄×C₃₇	#557	1	1	yes	characteristic
138	#2571	C₂×C₃₇	1	2	yes	—
139	#2085	C₂×C₃₇	1	4	yes	—
140	#1948	C₂×C₃₇	1	2	yes	—
141	#1942	C₂×C₃₇	1	2	yes	—
142	#1846	C₂×C₃₇	1	2	yes	—
143	#1845	C₂×C₃₇	1	1	yes	characteristic
144	C₄×#107	C₂×C₃₇	1	1	yes	characteristic
145	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
146	C₂×C₃₇	#2552	1	1	yes	characteristic
147	C₂×C₃₇	C₂×#499	1	1	yes	characteristic
148	C₂×C₃₇	#3164	1	1	yes	characteristic
149	C₂×C₃₇	#3341	1	1	yes	characteristic
150	C₂×C₃₇	#3342	1	1	yes	characteristic
151	C₂×C₃₇	#3393	1	1	yes	characteristic
152	C₂×C₃₇	#3396	1	1	yes	characteristic
153	C₂×#107	C₂²×C₃₇	1	2	yes	—
154	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
155	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
156	C₂×#107	C₂²×C₃₇	1	2	yes	—
157	C₂×#106	C₂²×C₃₇	1	4	yes	—
158	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
161	#400	—	2	4	no	—
162	#389	—	2	2	no	—
163	#389	C₂²×C₃₇	1	4	yes	—
164	#384	C₂²×C₃₇	1	2	yes	—
165	#382	—	2	2	no	—
166	#382	C₂²×C₃₇	1	4	yes	—
167	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
169	#340	C₂²×C₃₇	1	2	yes	—
170	#340	C₂²×C₃₇	1	4	yes	—
171	#336	C₂²×C₃₇	1	2	yes	—
172	C₃₇	#28828	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
180	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
187	#107	—	2	2	no	—
188	#107	—	2	4	no	—
189	#107	—	2	2	no	—
190	#107	—	4	2	no	—
191	#106	—	4	4	no	—
192	C₄⋊C₈	Q₈×C₃₇	1	4	yes	—
193	C₄×C₈	—	2	4	no	—
194	C₄×C₈	Q₈×C₃₇	1	4	yes	—
195	#95	—	2	2	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	#95	—	2	4	no	—
199	#95	D₄×C₃₇	1	2	yes	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
213	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
214	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
215	C₂×C₈	#41×C₃₇	1	2	yes	—
216	C₂×C₈	#41×C₃₇	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	8	2	no	—
227	C₄⋊C₄	—	8	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
232	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
233	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
252	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
253	C₂×C₄	#135×C₃₇	1	2	yes	—
254	C₂×C₄	#135×C₃₇	1	2	yes	—
255	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
260	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	4	no	—
263	C₈	#125×C₃₇	1	4	yes	—
264	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
265	C₂²	C₃₇×#547	1	1	yes	characteristic
266	C₂²	C₃₇×#550	1	1	yes	characteristic
267	C₂²	C₂×#135×C₃₇	1	1	yes	characteristic
268	C₂²	C₂×#135×C₃₇	1	1	yes	characteristic
269	C₂²	C₃₇×#576	1	1	yes	characteristic
270	C₂²	C₃₇×#578	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	2	no	—
275	C₄	—	8	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₃₇×#499	1	4	yes	—
278	C₄	C₂×#125×C₃₇	1	1	yes	characteristic
279	C₄	C₃₇×#544	1	1	yes	characteristic
280	C₄	C₃₇×#554	1	1	yes	characteristic
281	C₄	C₃₇×#557	1	1	yes	characteristic
282	C₂	C₃₇×#2552	1	1	yes	characteristic
283	C₂	C₂×C₃₇×#499	1	1	yes	characteristic
284	C₂	C₃₇×#3164	1	1	yes	characteristic
285	C₂	C₃₇×#3341	1	1	yes	characteristic
286	C₂	C₃₇×#3342	1	1	yes	characteristic
287	C₂	C₃₇×#3393	1	1	yes	characteristic
288	C₂	C₃₇×#3396	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
119. #28828	C₃₇	direct (N × H)
172. C₃₇	#28828	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×#2571	2
C₃₇×#2085	2
C₃₇×#1948	2
C₃₇×#1942	2
C₃₇×#1846	2
C₃₇×#1845	2
C₄×#107×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
#28828	37

Subgroups — C37×#28828 #22650081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28828 #22650081