Subgroups — #17997674

Order 4736.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924513	C₂	1	1	yes	maximal, characteristic
2	#14557439	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557437	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557420	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557420	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557417	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557416	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557408	C₂	1	1	yes	maximal, characteristic
9	#14557406	C₂	1	1	yes	maximal, characteristic
10	C₄×#10596936	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²⋊C₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
14	C₂×#11924443	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂²⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
17	C₂³×Dic₃₇	C₂²	1	1	yes	characteristic
18	C₂³×Dic₃₇	C₂²	1	1	yes	characteristic
19	#11924513	C₄	1	8	yes	complemented
20	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
23	C₂×#10596936	C₂²	1	1	yes	characteristic
24	C₂×#10596936	C₂²	1	1	yes	characteristic
25	C₂×#10596936	C₂²	1	1	yes	characteristic
26	C₂×#10596936	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
33	#11924443	C₂²	1	1	yes	characteristic
34	#11924443	C₂²	1	1	yes	characteristic
35	#11924443	C₂²	1	1	yes	characteristic
36	#11924443	C₂²	1	1	yes	characteristic
37	#11924443	C₂²	1	1	yes	characteristic
38	#11924443	C₂²	1	1	yes	characteristic
39	#11924443	C₂²	1	1	yes	characteristic
40	#11924443	C₂²	1	1	yes	characteristic
41	#11924443	C₂²	1	1	yes	characteristic
42	#11924443	C₂²	1	1	yes	characteristic
43	#11924443	C₂²	1	1	yes	characteristic
44	#11924443	C₂²	1	1	yes	characteristic
45	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
49	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
50	#95×C₃₇	C₂²	1	1	yes	characteristic
51	#95×C₃₇	C₂²	1	1	yes	characteristic
52	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₇	—	2	2	no	—
54	C₂²×C₄×C₃₇	—	2	2	no	—
55	C₂²×C₄×C₃₇	—	4	1	no	—
56	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₇	—	2	2	no	—
63	C₂²×Dic₃₇	—	2	2	no	—
64	C₂²×Dic₃₇	—	4	1	no	—
65	C₂²×Dic₃₇	—	4	1	no	—
66	C₂²×Dic₃₇	—	4	1	no	—
67	C₂²×Dic₃₇	C₂×C₄	1	4	yes	—
68	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
69	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
70	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
71	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
72	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
73	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
74	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
75	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
76	C₂×Dic₇₄	—	4	1	no	—
77	C₂×Dic₇₄	—	4	1	no	—
78	C₂×Dic₇₄	C₂×C₄	1	4	yes	complemented
79	C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
80	C₂²⋊C₄×C₃₇	Q₈	1	4	yes	complemented
81	C₄²×C₃₇	—	2	2	no	—
82	C₄²×C₃₇	—	4	1	no	—
83	C₂²⋊Dic₃₇	—	2	2	no	—
84	C₂²⋊Dic₃₇	C₂×C₄	1	4	yes	—
85	C₄⋊Dic₃₇	—	2	2	no	—
86	C₄⋊Dic₃₇	—	4	1	no	—
87	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	complemented
88	#10596936	—	2	2	no	—
89	#10596936	—	4	1	no	—
90	#10596936	C₂×C₄	1	4	yes	complemented
91	#10596936	C₂×C₄	1	4	yes	complemented
92	C₄×Dic₃₇	—	2	2	no	—
93	C₄×Dic₃₇	—	4	1	no	—
94	C₄×Dic₃₇	—	4	1	no	—
95	C₄×Dic₃₇	—	4	1	no	—
96	C₂³×C₃₇	—	2	1	no	—
97	C₂³×C₃₇	—	2	1	no	—
98	C₂³×C₃₇	—	2	1	no	—
99	C₂³×C₃₇	—	2	1	no	—
100	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
101	C₂³×C₃₇	C₂×Q₈	1	2	yes	—
102	C₂³×C₃₇	#41	1	2	yes	—
103	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₃₇	—	2	1	no	—
105	C₂×C₄×C₃₇	—	2	1	no	—
106	C₂×C₄×C₃₇	—	2	1	no	—
107	C₂×C₄×C₃₇	—	2	1	no	—
108	C₂×C₄×C₃₇	—	2	1	no	—
109	C₂×C₄×C₃₇	—	2	1	no	—
110	C₂×C₄×C₃₇	—	2	1	no	—
111	C₂×C₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	4	4	no	—
118	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
121	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
122	C₂×C₄×C₃₇	#41	1	2	yes	—
123	C₂×Dic₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	2	1	no	—
127	C₂×Dic₃₇	—	2	1	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	—	2	1	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	2	1	no	—
133	C₂×Dic₃₇	—	2	1	no	—
134	C₂×Dic₃₇	—	2	1	no	—
135	C₂×Dic₃₇	—	2	1	no	—
136	C₂×Dic₃₇	—	2	1	no	—
137	C₂×Dic₃₇	—	2	1	no	—
138	C₂×Dic₃₇	—	2	1	no	—
139	C₂×Dic₃₇	—	2	1	no	—
140	C₂×Dic₃₇	—	2	1	no	—
141	C₂×Dic₃₇	—	2	1	no	—
142	C₂×Dic₃₇	—	4	4	no	—
143	C₂×Dic₃₇	—	4	4	no	—
144	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
145	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
146	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
147	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
148	C₂×Dic₃₇	#41	1	2	yes	—
149	Dic₇₄	—	4	4	no	—
150	C₂²×C₃₇	—	2	2	no	—
151	C₂²×C₃₇	—	2	2	no	—
152	C₂²×C₃₇	—	2	2	no	—
153	C₂²×C₃₇	—	2	2	no	—
154	C₂²×C₃₇	—	2	2	no	—
155	C₂²×C₃₇	—	2	2	no	—
156	C₂²×C₃₇	C₄×Q₈	1	4	yes	—
157	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
158	C₂²×C₃₇	C₂²×Q₈	1	1	yes	characteristic
159	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
160	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
161	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
162	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
164	C₄×C₃₇	—	2	2	no	—
165	C₄×C₃₇	—	2	2	no	—
166	C₄×C₃₇	—	2	2	no	—
167	C₄×C₃₇	—	2	2	no	—
168	C₄×C₃₇	—	2	2	no	—
169	C₄×C₃₇	—	4	1	no	—
170	Dic₃₇	—	2	2	no	—
171	Dic₃₇	—	2	2	no	—
172	Dic₃₇	—	2	2	no	—
173	Dic₃₇	—	2	2	no	—
174	Dic₃₇	—	2	2	no	—
175	Dic₃₇	—	4	1	no	—
176	Dic₃₇	—	4	1	no	—
177	Dic₃₇	—	4	1	no	—
178	#2368	—	37	1	no	maximal, Sylow
179	C₂×C₃₇	—	2	4	no	—
180	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₂×C₃₇	C₄×#41	1	1	yes	characteristic
182	C₂×C₃₇	#518	1	1	yes	characteristic
183	C₂×C₃₇	#551	1	1	yes	characteristic
184	C₂×C₃₇	#551	1	1	yes	characteristic
185	C₂×C₃₇	#554	1	1	yes	characteristic
186	C₂×C₃₇	#554	1	1	yes	characteristic
187	C₂×C₂²⋊Q₈	—	37	1	no	—
188	#391	—	37	1	no	—
189	#391	—	37	1	no	—
190	#387	—	37	1	no	—
191	#387	—	37	1	no	—
192	#387	—	37	1	no	—
193	#387	—	37	1	no	—
194	#385	—	37	1	no	—
195	#385	—	37	1	no	—
196	C₄×C₄⋊C₄	—	37	1	no	—
197	C₄×C₄⋊C₄	—	37	1	no	—
198	C₄×C₂²⋊C₄	—	37	1	no	—
199	C₄×C₂²⋊C₄	—	37	1	no	—
200	C₄×C₂²⋊C₄	D₃₇	1	1	yes	characteristic
201	C₂×#95	—	37	1	no	—
202	C₃₇	#2368	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	37	1	no	—
204	C₂³×C₄	—	37	1	no	—
205	C₂³×C₄	—	37	1	no	—
206	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
207	C₂²⋊Q₈	—	37	8	no	—
208	C₂×C₄⋊C₄	—	37	1	no	—
209	C₂×C₄⋊C₄	—	37	1	no	—
210	C₂×C₄⋊C₄	—	37	1	no	—
211	C₂×C₄⋊C₄	—	37	1	no	—
212	C₂×C₄⋊C₄	—	37	1	no	—
213	C₂×C₄⋊C₄	—	37	1	no	—
214	C₂×C₄⋊C₄	—	37	1	no	—
215	C₂×C₂²⋊C₄	—	37	1	no	—
216	C₂×C₂²⋊C₄	—	37	1	no	—
217	C₂×C₂²⋊C₄	C₂×D₃₇	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	C₂×D₃₇	1	1	yes	characteristic
219	C₂×C₄²	—	37	1	no	—
220	C₂×C₄²	—	37	1	no	—
221	C₂×C₄²	—	37	1	no	—
222	C₂×C₄²	—	37	1	no	—
223	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
224	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
225	#95	—	37	1	no	—
226	#95	—	37	1	no	—
227	#95	—	37	1	no	—
228	#95	—	37	1	no	—
229	#95	—	37	1	no	—
230	#95	—	37	1	no	—
231	#95	—	37	1	no	—
232	#95	—	37	1	no	—
233	#95	—	37	1	no	—
234	#95	—	37	1	no	—
235	#95	—	37	1	no	—
236	#95	—	37	1	no	—
237	#95	C₂×D₃₇	1	1	yes	characteristic
238	#95	C₂×D₃₇	1	1	yes	characteristic
239	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
240	C₂×Q₈	—	37	4	no	—
241	C₂×Q₈	—	148	1	no	—
242	C₂×Q₈	—	148	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	37	1	no	—
247	C₂²×C₄	—	37	1	no	—
248	C₂²×C₄	—	37	1	no	—
249	C₂²×C₄	—	37	1	no	—
250	C₂²×C₄	—	37	1	no	—
251	C₂²×C₄	—	37	4	no	—
252	C₂²×C₄	—	37	1	no	—
253	C₂²×C₄	—	37	1	no	—
254	C₂²×C₄	—	37	1	no	—
255	C₂²×C₄	—	74	2	no	—
256	C₂²×C₄	—	74	2	no	—
257	C₂²×C₄	—	148	1	no	—
258	C₂²×C₄	—	148	1	no	—
259	C₂²×C₄	—	148	1	no	—
260	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
266	C₄⋊C₄	—	37	4	no	—
267	C₄⋊C₄	—	37	4	no	—
268	C₄⋊C₄	—	37	4	no	—
269	C₄⋊C₄	—	74	2	no	—
270	C₄⋊C₄	—	74	2	no	—
271	C₄⋊C₄	—	148	1	no	—
272	C₄⋊C₄	—	148	1	no	—
273	C₂²⋊C₄	—	37	4	no	—
274	C₂²⋊C₄	—	74	2	no	—
275	C₂²⋊C₄	Dic₇₄	1	4	yes	complemented
276	C₂²⋊C₄	C₄×D₃₇	1	4	yes	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	74	2	no	—
280	C₄²	—	148	1	no	—
281	C₄²	—	148	1	no	—
282	C₄²	—	148	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	C₂×Dic₇₄	1	2	yes	—
288	C₂³	C₂×C₄×D₃₇	1	2	yes	—
289	C₂³	#10596962	1	2	yes	—
290	C₂³	C₂³×D₃₇	1	1	yes	characteristic
291	Q₈	—	148	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	37	2	no	—
307	C₂×C₄	—	37	2	no	—
308	C₂×C₄	—	37	2	no	—
309	C₂×C₄	—	37	2	no	—
310	C₂×C₄	—	37	2	no	—
311	C₂×C₄	—	74	1	no	—
312	C₂×C₄	—	74	1	no	—
313	C₂×C₄	—	74	1	no	—
314	C₂×C₄	—	74	1	no	—
315	C₂×C₄	—	74	1	no	—
316	C₂×C₄	—	74	1	no	—
317	C₂×C₄	—	74	1	no	—
318	C₂×C₄	—	74	1	no	—
319	C₂×C₄	—	74	1	no	—
320	C₂×C₄	—	74	1	no	—
321	C₂×C₄	—	74	1	no	—
322	C₂×C₄	—	74	1	no	—
323	C₂×C₄	—	74	1	no	—
324	C₂×C₄	—	74	1	no	—
325	C₂×C₄	—	74	1	no	—
326	C₂×C₄	—	74	1	no	—
327	C₂×C₄	—	74	1	no	—
328	C₂×C₄	—	74	1	no	—
329	C₂×C₄	—	74	1	no	—
330	C₂×C₄	—	148	4	no	—
331	C₂×C₄	—	148	4	no	—
332	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
333	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
334	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
335	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
336	C₂×C₄	#10596964	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	C₄×Dic₇₄	1	4	yes	—
344	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
345	C₂²	C₂²×Dic₇₄	1	1	yes	characteristic
346	C₂²	#41×D₃₇	1	1	yes	characteristic
347	C₂²	C₂×#10596964	1	1	yes	characteristic
348	C₂²	C₂×#10596962	1	1	yes	characteristic
349	C₂²	#11924581	1	1	yes	characteristic
350	C₂²	#11924582	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	74	2	no	—
358	C₄	—	74	2	no	—
359	C₄	—	74	2	no	—
360	C₄	—	74	2	no	—
361	C₄	—	74	2	no	—
362	C₄	—	148	1	no	—
363	C₄	—	148	1	no	—
364	C₄	—	148	1	no	—
365	C₂	—	2	4	no	—
366	C₂	C₂×C₄×Dic₇₄	1	1	yes	characteristic
367	C₂	C₄×#10596964	1	1	yes	characteristic
368	C₂	#14558005	1	1	yes	characteristic
369	C₂	#14558207	1	1	yes	characteristic
370	C₂	#14558209	1	1	yes	characteristic
371	C₂	#14558240	1	1	yes	characteristic
372	C₂	#14558242	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₃₇	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂⁴×C₃₇	C₂³
hypercenter	C₄×C₂²⋊C₄	D₃₇
hypocenter	C₃₇	#2368
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #14557439	C₂	semidirect
3. #14557437	C₂	semidirect
4. #14557420	C₂	semidirect
5. #14557420	C₂	semidirect
6. #14557417	C₂	semidirect
7. #14557416	C₂	semidirect
10. C₄×#10596936	C₂	semidirect
11. C₄×C₄⋊Dic₃₇	C₂	semidirect
19. #11924513	C₄	semidirect
78. C₂×Dic₇₄	C₂×C₄	semidirect
80. C₂²⋊C₄×C₃₇	Q₈	semidirect
87. C₄⋊Dic₃₇	C₂×C₄	semidirect
90. #10596936	C₂×C₄	semidirect
91. #10596936	C₂×C₄	semidirect
202. C₃₇	#2368	semidirect
275. C₂²⋊C₄	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924513	2
#14557439	2
#14557437	2
#14557420	2
#14557417	2
#14557416	2
#14557408	2
#14557406	2
C₄×#10596936	2
C₄×C₄⋊Dic₃₇	2
C₂²⋊C₄×Dic₃₇	2
C₂×#11924443	2
C₄×C₂²⋊C₄×C₃₇	2
#2368	37