Subgroups — C₃₇×#29796 #22651049

Order 9472.

1178 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#460	C₂	1	2	yes	maximal, complemented
2	C₃₇×#2867	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2041	C₂	1	4	yes	maximal, complemented
5	C₃₇×#2014	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1982	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
9	C₂×#127×C₃₇	—	2	1	no	—
10	C₂×#127×C₃₇	—	2	2	no	—
11	C₂×#127×C₃₇	—	2	1	no	—
12	C₂×#127×C₃₇	C₂²	1	2	yes	—
13	C₂×#127×C₃₇	C₂²	1	4	yes	—
14	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#460	—	2	8	no	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#102×C₃₇	C₂²	1	4	yes	—
18	C₂×#102×C₃₇	C₂²	1	8	yes	—
19	C₃₇×#391	C₂²	1	2	yes	—
20	C₃₇×#389	C₂²	1	2	yes	—
21	C₃₇×#387	—	2	2	no	—
22	C₄×C₄⋊C₄×C₃₇	—	2	1	no	—
23	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#376	—	2	1	no	—
25	C₄³×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#336	C₂²	1	4	yes	—
27	C₂²×Q₈×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₃₇	—	2	2	no	—
29	C₂²×D₄×C₃₇	—	2	4	no	—
30	C₂²×D₄×C₃₇	—	2	4	no	—
31	C₂²×D₄×C₃₇	—	2	2	no	—
32	C₂²×D₄×C₃₇	C₂³	1	4	yes	—
33	C₂×SD₁₆×C₃₇	—	2	8	no	—
34	C₂×SD₁₆×C₃₇	—	4	8	no	—
35	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
36	#127×C₃₇	—	2	4	no	—
37	#127×C₃₇	—	2	4	no	—
38	#127×C₃₇	—	4	8	no	—
39	#127×C₃₇	—	4	8	no	—
40	#127×C₃₇	—	4	4	no	—
41	#127×C₃₇	—	4	4	no	—
42	C₄×Q₈×C₃₇	—	2	2	no	—
43	C₄×Q₈×C₃₇	D₄	1	4	yes	complemented
44	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₇	—	2	1	no	—
49	C₂×C₄²×C₃₇	—	2	1	no	—
50	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₃₇	—	2	4	no	—
56	#102×C₃₇	—	2	8	no	—
57	#102×C₃₇	—	4	8	no	—
58	#95×C₃₇	—	2	2	no	—
59	#95×C₃₇	—	2	2	no	—
60	#95×C₃₇	—	2	2	no	—
61	#95×C₃₇	—	2	1	no	—
62	C₂⁴×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	—	2	1	no	—
64	C₂×Q₈×C₃₇	—	2	4	no	—
65	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₃₇	—	2	4	no	—
67	C₂×D₄×C₃₇	—	2	4	no	—
68	C₂×D₄×C₃₇	—	2	8	no	—
69	C₂×D₄×C₃₇	—	2	8	no	—
70	C₂×D₄×C₃₇	—	4	8	no	—
71	C₂×D₄×C₃₇	—	4	8	no	—
72	C₂×D₄×C₃₇	—	4	8	no	—
73	C₂×D₄×C₃₇	—	4	8	no	—
74	C₂×D₄×C₃₇	—	4	8	no	—
75	C₂×D₄×C₃₇	—	4	4	no	—
76	C₂×D₄×C₃₇	—	4	4	no	—
77	C₂×D₄×C₃₇	—	4	8	no	—
78	C₂×D₄×C₃₇	—	4	4	no	—
79	C₂×D₄×C₃₇	—	4	8	no	—
80	C₂×D₄×C₃₇	—	4	4	no	—
81	C₂²×C₄×C₃₇	—	2	1	no	—
82	C₂²×C₄×C₃₇	—	2	1	no	—
83	C₂²×C₄×C₃₇	—	2	2	no	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
89	SD₁₆×C₃₇	—	4	16	no	—
90	C₂×C₈×C₃₇	—	2	4	no	—
91	C₂×C₈×C₃₇	—	2	8	no	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	1	no	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
97	C₄²×C₃₇	—	2	4	no	—
98	C₄²×C₃₇	—	2	4	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	4	no	—
101	C₄²×C₃₇	—	4	2	no	—
102	C₄²×C₃₇	SD₁₆	1	4	yes	complemented
103	C₄²×C₃₇	C₂×D₄	1	2	yes	—
104	C₄²×C₃₇	#41	1	2	yes	—
105	C₂³×C₃₇	—	4	4	no	—
106	C₂³×C₃₇	—	4	8	no	—
107	C₂³×C₃₇	—	4	4	no	—
108	C₂³×C₃₇	—	4	8	no	—
109	C₂³×C₃₇	—	4	4	no	—
110	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₃₇	—	2	4	no	—
112	Q₈×C₃₇	—	4	2	no	—
113	D₄×C₃₇	—	4	4	no	—
114	D₄×C₃₇	—	4	4	no	—
115	D₄×C₃₇	—	4	8	no	—
116	D₄×C₃₇	—	8	16	no	—
117	D₄×C₃₇	—	8	16	no	—
118	D₄×C₃₇	—	8	8	no	—
119	D₄×C₃₇	—	8	8	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	2	no	—
131	C₂×C₄×C₃₇	—	4	2	no	—
132	C₂×C₄×C₃₇	#121	1	2	yes	—
133	C₂×C₄×C₃₇	#121	1	2	yes	—
134	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
135	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₇	H(5,𝔽₂)	1	4	yes	—
141	C₈×C₃₇	—	4	4	no	—
142	#29796	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₃₇	—	8	4	no	—
144	C₂²×C₃₇	—	8	8	no	—
145	C₂²×C₃₇	—	8	8	no	—
146	C₂²×C₃₇	—	8	4	no	—
147	C₂²×C₃₇	—	8	4	no	—
148	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
149	C₂²×C₃₇	#535	1	2	yes	—
150	C₂²×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
151	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₃₇	—	2	4	no	—
154	C₄×C₃₇	—	2	4	no	—
155	C₄×C₃₇	—	2	2	no	—
156	C₄×C₃₇	—	4	2	no	—
157	C₄×C₃₇	—	4	2	no	—
158	C₄×C₃₇	—	8	2	no	—
159	C₄×C₃₇	#460	1	4	yes	—
160	C₄×C₃₇	#535	1	2	yes	—
161	C₄×C₃₇	#540	1	2	yes	—
162	C₂×#460	C₂×C₃₇	1	2	yes	complemented
163	#2867	C₂×C₃₇	1	1	yes	characteristic
164	#2295	C₂×C₃₇	1	1	yes	characteristic, complemented
165	#2041	C₂×C₃₇	1	4	yes	complemented
166	#2014	C₂×C₃₇	1	4	yes	complemented
167	#1982	C₂×C₃₇	1	2	yes	complemented
168	#1831	C₂×C₃₇	1	1	yes	characteristic, complemented
169	C₂×C₃₇	—	16	4	no	—
170	C₂×C₃₇	#2619	1	1	yes	characteristic
171	C₂×C₃₇	C₂×#460	1	1	yes	characteristic
172	C₂×C₃₇	#3038	1	1	yes	characteristic
173	C₂×C₃₇	#3208	1	2	yes	—
174	C₂×C₃₇	#3217	1	2	yes	—
175	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
176	C₂×#127	—	2	1	no	—
177	C₂×#127	—	2	2	no	—
178	C₂×#127	—	2	1	no	—
179	C₂×#127	C₂²×C₃₇	1	2	yes	—
180	C₂×#127	C₂²×C₃₇	1	4	yes	—
181	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
182	#460	—	2	8	no	—
183	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
184	C₂×#102	C₂²×C₃₇	1	4	yes	—
185	C₂×#102	C₂²×C₃₇	1	8	yes	—
186	#391	C₂²×C₃₇	1	2	yes	—
187	#389	C₂²×C₃₇	1	2	yes	—
188	#387	—	2	2	no	—
189	C₄×C₄⋊C₄	—	2	1	no	—
190	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
191	#376	—	2	1	no	—
192	C₄³	C₂²×C₃₇	1	1	yes	characteristic
193	#336	C₂²×C₃₇	1	4	yes	—
194	C₃₇	#29796	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×Q₈	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×D₄	—	2	2	no	—
197	C₂²×D₄	—	2	4	no	—
198	C₂²×D₄	—	2	4	no	—
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
201	C₂×SD₁₆	—	2	8	no	—
202	C₂×SD₁₆	—	4	8	no	—
203	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
204	#127	—	2	4	no	—
205	#127	—	2	4	no	—
206	#127	—	4	8	no	—
207	#127	—	4	8	no	—
208	#127	—	4	4	no	—
209	#127	—	4	4	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	D₄×C₃₇	1	4	yes	complemented
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₃₇	1	2	yes	—
220	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
222	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	#102	—	2	8	no	—
225	#102	—	4	8	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	1	no	—
230	C₂⁴	—	2	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	8	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	4	8	no	—
239	C₂×D₄	—	4	8	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	—	4	8	no	—
242	C₂×D₄	—	4	8	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	8	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	8	no	—
248	C₂×D₄	—	4	4	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
257	SD₁₆	—	4	16	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	8	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	4	2	no	—
270	C₄²	SD₁₆×C₃₇	1	4	yes	complemented
271	C₄²	C₂×D₄×C₃₇	1	2	yes	—
272	C₄²	#41×C₃₇	1	2	yes	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	8	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	8	no	—
277	C₂³	—	4	4	no	—
278	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	8	no	—
284	D₄	—	8	16	no	—
285	D₄	—	8	16	no	—
286	D₄	—	8	8	no	—
287	D₄	—	8	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	#121×C₃₇	1	2	yes	—
301	C₂×C₄	#121×C₃₇	1	2	yes	—
302	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
303	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
304	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
305	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
306	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
308	C₂×C₄	H(5,𝔽₂)×C₃₇	1	4	yes	—
309	C₈	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	8	no	—
312	C₂²	—	8	8	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
316	C₂²	C₃₇×#535	1	2	yes	—
317	C₂²	C₂²×SD₁₆×C₃₇	1	1	yes	characteristic
318	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
319	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₃₇×#460	1	4	yes	—
327	C₄	C₃₇×#535	1	2	yes	—
328	C₄	C₃₇×#540	1	2	yes	—
329	C₂	—	16	4	no	—
330	C₂	C₃₇×#2619	1	1	yes	characteristic
331	C₂	C₂×C₃₇×#460	1	1	yes	characteristic
332	C₂	C₃₇×#3038	1	1	yes	characteristic
333	C₂	C₃₇×#3208	1	2	yes	—
334	C₂	C₃₇×#3217	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#460	C₂	semidirect
3. C₃₇×#2295	C₂	semidirect
4. C₃₇×#2041	C₂	semidirect
5. C₃₇×#2014	C₂	semidirect
6. C₃₇×#1982	C₂	semidirect
7. C₃₇×#1831	C₂	semidirect
43. C₄×Q₈×C₃₇	D₄	semidirect
102. C₄²×C₃₇	SD₁₆	semidirect
142. #29796	C₃₇	direct (N × H)
162. C₂×#460	C₂×C₃₇	semidirect
164. #2295	C₂×C₃₇	semidirect
165. #2041	C₂×C₃₇	semidirect
166. #2014	C₂×C₃₇	semidirect
167. #1982	C₂×C₃₇	semidirect
168. #1831	C₂×C₃₇	semidirect
194. C₃₇	#29796	direct (N × H)
211. C₄×Q₈	D₄×C₃₇	semidirect
270. C₄²	SD₁₆×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#460	2
C₃₇×#2867	2
C₃₇×#2295	2
C₃₇×#2041	2
C₃₇×#2014	2
C₃₇×#1982	2
C₃₇×#1831	2
#29796	37

Subgroups — C37×#29796 #22651049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29796 #22651049