Subgroups — C₃₇×#11230 #22632483

Order 9472.

622 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#424	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1742	C₂	1	2	yes	maximal
4	C₂×C₃₇×#341	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₇×#338	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₃₇	—	2	4	no	—
12	C₂×#108×C₃₇	—	2	2	no	—
13	C₃₇×#424	—	2	2	no	—
14	C₃₇×#424	—	2	2	no	—
15	C₂×C₄⋊C₈×C₃₇	C₄	1	4	yes	—
16	C₃₇×#343	—	2	4	no	—
17	C₃₇×#341	—	2	2	no	—
18	C₃₇×#338	—	2	4	no	—
19	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₃₇	—	2	2	no	—
21	C₂×M₁₆×C₃₇	—	2	2	no	—
22	C₂×M₁₆×C₃₇	—	2	1	no	—
23	C₂×M₁₆×C₃₇	—	2	1	no	—
24	C₂×M₁₆×C₃₇	—	2	2	no	—
25	C₂×M₁₆×C₃₇	—	2	2	no	—
26	C₂×M₁₆×C₃₇	—	2	2	no	—
27	C₂×M₁₆×C₃₇	—	2	2	no	—
28	C₂×M₁₆×C₃₇	—	2	2	no	—
29	C₂×M₁₆×C₃₇	—	2	2	no	—
30	C₂×M₁₆×C₃₇	—	2	2	no	—
31	C₂×M₁₆×C₃₇	—	2	2	no	—
32	C₂×M₁₆×C₃₇	—	2	2	no	—
33	C₂×M₁₆×C₃₇	—	2	2	no	—
34	C₂×M₁₆×C₃₇	D₄	1	4	yes	—
35	C₂×M₁₆×C₃₇	Q₈	1	4	yes	—
36	C₂²×C₈×C₃₇	—	2	2	no	—
37	C₂²×C₈×C₃₇	—	2	2	no	—
38	C₂²×C₈×C₃₇	—	2	2	no	—
39	C₂²×C₈×C₃₇	—	2	2	no	—
40	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₃₇	D₄	1	2	yes	—
43	C₂²×C₈×C₃₇	D₄	1	2	yes	—
44	#117×C₃₇	—	2	1	no	—
45	#117×C₃₇	—	2	1	no	—
46	#117×C₃₇	—	2	1	no	—
47	#117×C₃₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
51	#108×C₃₇	—	2	8	no	—
52	C₄⋊C₈×C₃₇	—	2	8	no	—
53	C₂⁴×C₃₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₃₇	#41	1	2	yes	—
64	C₂²×C₄×C₃₇	#41	1	2	yes	—
65	M₁₆×C₃₇	—	2	4	no	—
66	M₁₆×C₃₇	—	2	4	no	—
67	M₁₆×C₃₇	—	2	4	no	—
68	M₁₆×C₃₇	—	2	4	no	—
69	M₁₆×C₃₇	—	4	1	no	—
70	M₁₆×C₃₇	—	4	2	no	—
71	M₁₆×C₃₇	—	4	2	no	—
72	M₁₆×C₃₇	—	4	2	no	—
73	M₁₆×C₃₇	—	4	2	no	—
74	M₁₆×C₃₇	—	4	1	no	—
75	M₁₆×C₃₇	—	4	2	no	—
76	C₂×C₈×C₃₇	—	2	4	no	—
77	C₂×C₈×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	—	2	4	no	—
79	C₂×C₈×C₃₇	—	2	2	no	—
80	C₂×C₈×C₃₇	—	2	2	no	—
81	C₂×C₈×C₃₇	—	2	4	no	—
82	C₂×C₈×C₃₇	—	2	4	no	—
83	C₂×C₈×C₃₇	—	2	4	no	—
84	C₂×C₈×C₃₇	—	4	2	no	—
85	C₂×C₈×C₃₇	—	4	2	no	—
86	C₂×C₈×C₃₇	—	4	2	no	—
87	C₂×C₈×C₃₇	—	4	2	no	—
88	C₂×C₈×C₃₇	C₂²⋊C₄	1	4	yes	—
89	C₄⋊C₄×C₃₇	—	2	2	no	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₂²⋊C₄×C₃₇	—	2	2	no	—
92	C₂²⋊C₄×C₃₇	—	2	2	no	—
93	C₄²×C₃₇	—	2	2	no	—
94	C₄²×C₃₇	—	2	2	no	—
95	C₂³×C₃₇	—	4	2	no	—
96	C₂³×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₃₇	#125	1	2	yes	—
100	C₂³×C₃₇	C₄⋊Q₈	1	2	yes	—
101	C₂×C₄×C₃₇	—	2	4	no	—
102	C₂×C₄×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₃₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₃₇	#120	1	1	yes	characteristic
108	C₂×C₄×C₃₇	#121	1	2	yes	—
109	C₂×C₄×C₃₇	#121	1	2	yes	—
110	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₃₇	#123	1	1	yes	characteristic
116	C₂×C₄×C₃₇	#123	1	2	yes	—
117	C₂×C₄×C₃₇	#124	1	1	yes	characteristic
118	C₂×C₄×C₃₇	#125	1	2	yes	—
119	C₂×C₄×C₃₇	#126	1	2	yes	—
120	C₂×C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
121	C₈×C₃₇	—	2	4	no	—
122	C₈×C₃₇	—	4	2	no	—
123	C₈×C₃₇	—	4	4	no	—
124	C₈×C₃₇	—	4	4	no	—
125	#11230	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₃₇	—	2	2	no	—
127	C₂²×C₃₇	—	2	2	no	—
128	C₂²×C₃₇	—	2	2	no	—
129	C₂²×C₃₇	—	4	2	no	—
130	C₂²×C₃₇	—	4	2	no	—
131	C₂²×C₃₇	#380	1	1	yes	characteristic
132	C₂²×C₃₇	#389	1	2	yes	—
133	C₂²×C₃₇	#395	1	1	yes	characteristic
134	C₂²×C₃₇	#398	1	2	yes	—
135	C₂²×C₃₇	#400	1	1	yes	characteristic
136	C₄×C₃₇	—	4	4	no	—
137	C₄×C₃₇	#386	1	1	yes	characteristic
138	C₄×C₃₇	#387	1	2	yes	—
139	C₄×C₃₇	#391	1	1	yes	characteristic
140	C₄×C₃₇	#394	1	1	yes	characteristic
141	C₄×C₃₇	#396	1	2	yes	—
142	C₄×C₃₇	#398	1	1	yes	characteristic
143	C₂²×#108	C₂×C₃₇	1	1	yes	characteristic
144	C₂×#424	C₂×C₃₇	1	1	yes	characteristic
145	#1742	C₂×C₃₇	1	2	yes	—
146	C₂×#341	C₂×C₃₇	1	1	yes	characteristic
147	C₂×#338	C₂×C₃₇	1	2	yes	—
148	C₂×C₃₇	—	2	2	no	—
149	C₂×C₃₇	—	2	1	no	—
150	C₂×C₃₇	—	2	2	no	—
151	C₂×C₃₇	—	2	1	no	—
152	C₂×C₃₇	#1441	1	1	yes	characteristic
153	C₂×C₃₇	#1813	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
155	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×#117	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×#108	—	2	4	no	—
160	C₂×#108	—	2	2	no	—
161	#424	—	2	2	no	—
162	#424	—	2	2	no	—
163	C₂×C₄⋊C₈	C₄×C₃₇	1	4	yes	—
164	#343	—	2	4	no	—
165	#341	—	2	2	no	—
166	#338	—	2	4	no	—
167	C₃₇	#11230	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	D₄×C₃₇	1	4	yes	—
184	C₂×M₁₆	Q₈×C₃₇	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
190	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
191	C₂²×C₈	D₄×C₃₇	1	2	yes	—
192	C₂²×C₈	D₄×C₃₇	1	2	yes	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
198	C₂×C₂²⋊C₄	Q₈×C₃₇	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
200	#108	—	2	8	no	—
201	C₄⋊C₈	—	2	8	no	—
202	C₂⁴	C₂×Q₈×C₃₇	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
204	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
210	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
211	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
212	C₂²×C₄	#41×C₃₇	1	2	yes	—
213	C₂²×C₄	#41×C₃₇	1	2	yes	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂²⋊C₄×C₃₇	1	4	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
246	C₂³	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
247	C₂³	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
248	C₂³	#125×C₃₇	1	2	yes	—
249	C₂³	C₄⋊Q₈×C₃₇	1	2	yes	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
253	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₃₇	1	2	yes	—
255	C₂×C₄	C₄×Q₈×C₃₇	1	2	yes	—
256	C₂×C₄	#120×C₃₇	1	1	yes	characteristic
257	C₂×C₄	#121×C₃₇	1	2	yes	—
258	C₂×C₄	#121×C₃₇	1	2	yes	—
259	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
260	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
262	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
263	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
264	C₂×C₄	#123×C₃₇	1	1	yes	characteristic
265	C₂×C₄	#123×C₃₇	1	2	yes	—
266	C₂×C₄	#124×C₃₇	1	1	yes	characteristic
267	C₂×C₄	#125×C₃₇	1	2	yes	—
268	C₂×C₄	#126×C₃₇	1	2	yes	—
269	C₂×C₄	C₄⋊Q₈×C₃₇	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	4	no	—
273	C₈	—	4	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₃₇×#380	1	1	yes	characteristic
280	C₂²	C₃₇×#389	1	2	yes	—
281	C₂²	C₃₇×#395	1	1	yes	characteristic
282	C₂²	C₃₇×#398	1	2	yes	—
283	C₂²	C₃₇×#400	1	1	yes	characteristic
284	C₄	—	4	4	no	—
285	C₄	C₃₇×#386	1	1	yes	characteristic
286	C₄	C₃₇×#387	1	2	yes	—
287	C₄	C₃₇×#391	1	1	yes	characteristic
288	C₄	C₃₇×#394	1	1	yes	characteristic
289	C₄	C₃₇×#396	1	2	yes	—
290	C₄	C₃₇×#398	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	2	no	—
294	C₂	—	2	1	no	—
295	C₂	C₃₇×#1441	1	1	yes	characteristic
296	C₂	C₃₇×#1813	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	#380
prime radical	C₂⁴×C₃₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂²×M₁₆×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
125. #11230	C₃₇	direct (N × H)
167. C₃₇	#11230	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#108×C₃₇	2
C₂×C₃₇×#424	2
C₃₇×#1742	2
C₂×C₃₇×#341	2
C₂×C₃₇×#338	2
#11230	37

Subgroups — C37×#11230 #22632483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#11230 #22632483