Subgroups — #18003412

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924568	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596966	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557638	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557635	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557631	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557625	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557620	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557618	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557572	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557540	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557533	C₂	1	1	yes	maximal, characteristic
14	#14557485	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596966	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
18	#11924568	C₄	1	8	yes	complemented
19	C₂×#10596942	C₂²	1	1	yes	characteristic
20	C₂×#10596930	C₂²	1	1	yes	characteristic
21	C₂×#10596941	C₂²	1	1	yes	characteristic
22	C₂×#10596929	C₂²	1	1	yes	characteristic
23	C₂×#10596929	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₇₄	—	2	1	no	—
27	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
28	#11924501	—	2	1	no	—
29	#11924501	C₂²	1	1	yes	characteristic
30	C₄×D₁₄₈	—	2	1	no	—
31	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
32	#11924498	—	2	1	no	—
33	#11924498	C₂²	1	1	yes	characteristic
34	#11924494	—	2	1	no	—
35	#11924494	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₃₇	—	2	1	no	—
37	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
38	C₄²×D₃₇	—	2	1	no	—
39	C₄²×D₃₇	—	2	1	no	—
40	C₄²×D₃₇	C₂²	1	1	yes	characteristic
41	#11924470	C₂²	1	1	yes	characteristic
42	#11924469	C₂²	1	1	yes	characteristic
43	#11924467	C₂²	1	1	yes	characteristic
44	#11924461	C₂²	1	1	yes	characteristic
45	#11924460	C₂²	1	1	yes	characteristic
46	#11924460	C₂²	1	1	yes	characteristic
47	#11924459	C₂²	1	1	yes	characteristic
48	#11924458	C₂²	1	1	yes	characteristic
49	#11924456	C₂²	1	1	yes	characteristic
50	#11924447	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
56	#106×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#103×C₃₇	C₂²	1	1	yes	characteristic
59	#97×C₃₇	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
62	#10596966	—	2	2	no	—
63	#10596966	C₂×C₄	1	4	yes	—
64	Q₈×D₃₇	—	2	2	no	—
65	Q₈×D₃₇	C₂×C₄	1	4	yes	complemented
66	C₂×D₁₄₈	—	2	1	no	—
67	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₃₇	—	2	1	no	—
69	C₂×C₄×D₃₇	—	2	1	no	—
70	C₂×C₄×D₃₇	—	2	1	no	—
71	C₂×C₄×D₃₇	—	2	1	no	—
72	C₂×C₄×D₃₇	—	2	1	no	—
73	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
74	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
75	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
76	C₂×Dic₇₄	—	2	1	no	—
77	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
78	Q₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₃₇	—	2	1	no	—
82	C₄⋊C₄×C₃₇	—	2	1	no	—
83	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
85	C₄²×C₃₇	—	2	1	no	—
86	C₄²×C₃₇	—	2	1	no	—
87	C₄²×C₃₇	C₂³	1	1	yes	characteristic
88	#10596942	C₂×C₄	1	4	yes	complemented
89	#10596941	C₂×C₄	1	4	yes	complemented
90	#10596938	—	2	1	no	—
91	#10596938	—	2	1	no	—
92	#10596938	—	2	1	no	—
93	C₄⋊Dic₃₇	—	2	1	no	—
94	C₄⋊Dic₃₇	—	2	1	no	—
95	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
97	#10596936	—	2	1	no	—
98	#10596936	—	2	1	no	—
99	#10596936	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	—	2	1	no	—
102	C₄×Dic₃₇	—	2	1	no	—
103	C₄×Dic₃₇	—	2	1	no	—
104	C₄×Dic₃₇	—	2	1	no	—
105	C₄×Dic₃₇	D₄	1	1	yes	characteristic
106	C₄×Dic₃₇	D₄	1	1	yes	characteristic
107	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
108	C₂×#64436	C₂³	1	1	yes	characteristic
109	C₂×#64436	C₂³	1	1	yes	characteristic
110	#10596930	C₂×C₄	1	4	yes	complemented
111	#10596929	—	2	2	no	—
112	#10596929	C₂×C₄	1	4	yes	complemented
113	C₂²×D₃₇	—	2	1	no	—
114	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₃₇	—	2	1	no	—
116	Q₈×C₃₇	—	2	1	no	—
117	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
118	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₃₇	—	2	1	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	2	1	no	—
123	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₃₇	—	2	1	no	—
127	C₂×Dic₃₇	—	2	1	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
132	D₁₄₈	—	2	4	no	—
133	D₁₄₈	—	2	1	no	—
134	D₁₄₈	C₂²×C₄	1	2	yes	—
135	C₄×D₃₇	—	2	4	no	—
136	C₄×D₃₇	—	2	4	no	—
137	C₄×D₃₇	—	2	2	no	—
138	C₄×D₃₇	—	4	1	no	—
139	C₄×D₃₇	—	4	2	no	—
140	C₄×D₃₇	—	4	2	no	—
141	C₄×D₃₇	—	4	2	no	—
142	C₄×D₃₇	C₂²×C₄	1	2	yes	—
143	C₄×D₃₇	#41	1	2	yes	—
144	Dic₇₄	—	2	1	no	—
145	Dic₇₄	—	2	4	no	—
146	Dic₇₄	C₂²×C₄	1	2	yes	—
147	C₈×C₃₇	—	2	1	no	—
148	C₈×C₃₇	C₂²×C₄	1	2	yes	—
149	#64436	—	2	1	no	—
150	#64436	C₂²×C₄	1	2	yes	—
151	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₃₇	—	2	1	no	—
153	C₂×D₃₇	—	2	1	no	—
154	C₂×D₃₇	—	2	2	no	—
155	C₂×D₃₇	—	4	1	no	—
156	C₂×D₃₇	—	4	1	no	—
157	C₂×D₃₇	C₄×D₄	1	2	yes	—
158	C₄×C₃₇	—	2	2	no	—
159	C₄×C₃₇	—	2	2	no	—
160	C₄×C₃₇	—	2	1	no	—
161	C₄×C₃₇	—	2	1	no	—
162	C₄×C₃₇	—	4	1	no	—
163	C₄×C₃₇	—	4	1	no	—
164	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
165	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
166	Dic₃₇	—	2	2	no	—
167	Dic₃₇	—	2	1	no	—
168	Dic₃₇	—	4	1	no	—
169	Dic₃₇	—	4	1	no	—
170	Dic₃₇	C₄×D₄	1	2	yes	—
171	Dic₃₇	#137	1	2	yes	complemented
172	C₄×#137	—	37	1	no	maximal, Sylow
173	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
175	C₂×C₃₇	#575	1	1	yes	characteristic
176	D₃₇	—	2	2	no	—
177	D₃₇	—	4	2	no	—
178	C₂×#137	—	37	1	no	—
179	C₄×#41	—	37	1	no	—
180	C₂×C₄×Q₈	—	37	1	no	—
181	#441	—	37	1	no	—
182	#441	D₃₇	1	1	yes	characteristic, complemented
183	#440	—	37	1	no	—
184	#440	—	37	1	no	—
185	C₄×Q₁₆	—	37	1	no	—
186	C₄×Q₁₆	—	37	1	no	—
187	C₄×SD₁₆	—	37	1	no	—
188	C₄×SD₁₆	—	37	1	no	—
189	#428	—	37	1	no	—
190	#419	—	37	1	no	—
191	#417	—	37	1	no	—
192	C₄×M₁₆	—	37	1	no	—
193	C₃₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	37	1	no	—
195	C₂²×Q₈	—	37	1	no	—
196	#137	—	37	8	no	—
197	C₂×Q₁₆	—	37	1	no	—
198	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
199	C₂×SD₁₆	—	37	1	no	—
200	C₂×SD₁₆	—	37	1	no	—
201	C₂×M₁₆	—	37	1	no	—
202	C₂×M₁₆	—	37	1	no	—
203	C₄×Q₈	—	37	1	no	—
204	C₄×Q₈	—	37	1	no	—
205	C₄×Q₈	—	37	1	no	—
206	C₄×Q₈	—	37	1	no	—
207	C₄×Q₈	—	74	1	no	—
208	C₄×Q₈	—	74	1	no	—
209	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
211	C₄×D₄	—	37	1	no	—
212	C₄×D₄	—	37	1	no	—
213	C₄×D₄	—	74	1	no	—
214	C₄×D₄	—	74	1	no	—
215	#117	—	37	1	no	—
216	#117	—	74	1	no	—
217	C₂×C₄⋊C₄	—	37	1	no	—
218	C₂×C₄⋊C₄	—	74	1	no	—
219	C₂×C₄²	—	37	1	no	—
220	C₂×C₄²	—	74	1	no	—
221	C₂×C₄²	—	74	1	no	—
222	#107	—	37	1	no	—
223	#107	—	37	1	no	—
224	#106	—	37	1	no	—
225	#106	C₂×D₃₇	1	1	yes	characteristic
226	#103	—	37	1	no	—
227	#103	—	37	1	no	—
228	#103	—	37	1	no	—
229	#103	—	37	1	no	—
230	#103	C₂×D₃₇	1	1	yes	characteristic
231	#103	C₂×D₃₇	1	1	yes	characteristic
232	#102	—	37	1	no	—
233	#102	—	37	1	no	—
234	#97	—	37	1	no	—
235	#97	C₂×D₃₇	1	1	yes	characteristic
236	C₄×C₈	—	37	1	no	—
237	C₄×C₈	—	37	1	no	—
238	#41	—	37	4	no	—
239	#41	—	74	2	no	—
240	C₂×Q₈	—	37	4	no	—
241	C₂×Q₈	—	37	1	no	—
242	C₂×Q₈	—	74	1	no	—
243	C₂×Q₈	—	74	2	no	—
244	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
246	C₂×D₄	—	37	1	no	—
247	C₂×D₄	—	74	1	no	—
248	C₂²×C₄	—	37	1	no	—
249	C₂²×C₄	—	37	1	no	—
250	C₂²×C₄	—	37	1	no	—
251	C₂²×C₄	—	74	1	no	—
252	C₂²×C₄	—	74	1	no	—
253	C₂²×C₄	—	74	1	no	—
254	C₂²×C₄	—	74	1	no	—
255	C₂²×C₄	—	74	1	no	—
256	Q₁₆	—	37	4	no	—
257	Q₁₆	C₄×D₃₇	1	4	yes	complemented
258	SD₁₆	—	37	4	no	—
259	SD₁₆	—	37	4	no	—
260	M₁₆	—	37	4	no	—
261	M₁₆	—	74	2	no	—
262	C₂×C₈	—	37	1	no	—
263	C₂×C₈	—	37	1	no	—
264	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	37	1	no	—
269	C₄⋊C₄	—	37	1	no	—
270	C₄⋊C₄	—	74	1	no	—
271	C₄⋊C₄	—	74	1	no	—
272	C₄⋊C₄	—	74	1	no	—
273	C₄⋊C₄	—	74	1	no	—
274	C₄⋊C₄	—	74	1	no	—
275	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	74	1	no	—
278	C₂²⋊C₄	—	74	1	no	—
279	C₂²⋊C₄	—	74	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	37	1	no	—
283	C₄²	—	37	1	no	—
284	C₄²	—	37	1	no	—
285	C₄²	—	74	1	no	—
286	C₄²	—	74	1	no	—
287	C₄²	—	74	1	no	—
288	C₄²	—	74	1	no	—
289	C₄²	—	74	1	no	—
290	C₄²	C₂²×D₃₇	1	1	yes	characteristic
291	C₂³	—	37	1	no	—
292	C₂³	—	74	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	37	2	no	—
296	Q₈	—	74	4	no	—
297	Q₈	—	74	1	no	—
298	Q₈	C₂×C₄×D₃₇	1	2	yes	—
299	Q₈	C₂×C₄×D₃₇	1	2	yes	—
300	D₄	—	37	2	no	—
301	D₄	—	74	4	no	—
302	D₄	—	74	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	37	1	no	—
308	C₂×C₄	—	37	1	no	—
309	C₂×C₄	—	37	2	no	—
310	C₂×C₄	—	37	2	no	—
311	C₂×C₄	—	37	1	no	—
312	C₂×C₄	—	74	2	no	—
313	C₂×C₄	—	74	4	no	—
314	C₂×C₄	—	74	4	no	—
315	C₂×C₄	—	74	1	no	—
316	C₂×C₄	—	74	1	no	—
317	C₂×C₄	—	74	1	no	—
318	C₂×C₄	—	148	1	no	—
319	C₂×C₄	—	148	2	no	—
320	C₂×C₄	—	148	2	no	—
321	C₂×C₄	—	148	2	no	—
322	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
323	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	37	2	no	—
327	C₈	—	74	1	no	—
328	C₈	C₂×C₄×D₃₇	1	2	yes	—
329	C₂²	—	37	2	no	—
330	C₂²	—	74	1	no	—
331	C₂²	—	74	1	no	—
332	C₂²	—	74	2	no	—
333	C₂²	—	148	1	no	—
334	C₂²	—	148	1	no	—
335	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	37	2	no	—
343	C₄	—	37	2	no	—
344	C₄	—	74	2	no	—
345	C₄	—	74	1	no	—
346	C₄	—	148	1	no	—
347	C₄	—	148	1	no	—
348	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
349	C₄	#41×D₃₇	1	1	yes	characteristic
350	C₂	—	74	2	no	—
351	C₂	—	148	2	no	—
352	C₂	C₄×D₄×D₃₇	1	1	yes	characteristic
353	C₂	#137×D₃₇	1	1	yes	characteristic
354	C₂	#14558332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#441	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#10596966	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#441	D₃₇
hypocenter	C₃₇	C₄×#137
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₃₇	C₂	semidirect
4. #14557638	C₂	semidirect
5. #14557635	C₂	semidirect
6. #14557631	C₂	semidirect
7. Q₁₆×Dic₃₇	C₂	semidirect
8. #14557625	C₂	semidirect
9. #14557620	C₂	semidirect
10. #14557618	C₂	semidirect
11. #14557572	C₂	semidirect
12. #14557540	C₂	semidirect
14. #14557485	C₂	semidirect
15. C₃₇×#441	C₂	semidirect
18. #11924568	C₄	semidirect
65. Q₈×D₃₇	C₂×C₄	semidirect
78. Q₁₆×C₃₇	C₂×C₄	semidirect
88. #10596942	C₂×C₄	semidirect
89. #10596941	C₂×C₄	semidirect
110. #10596930	C₂×C₄	semidirect
112. #10596929	C₂×C₄	semidirect
171. Dic₃₇	#137	semidirect
182. #441	D₃₇	semidirect
193. C₃₇	C₄×#137	semidirect
257. Q₁₆	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924568	2
C₄×#10596966	2
C₄×Q₈×D₃₇	2
#14557638	2
#14557635	2
#14557631	2
Q₁₆×Dic₃₇	2
#14557625	2
#14557620	2
#14557618	2
#14557572	2
#14557540	2
#14557533	2
#14557485	2
C₃₇×#441	2
C₄×#137	37