Subgroups — #18011370

Order 4736.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558612	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558611	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558580	C₂	1	2	yes	maximal, complemented
4	#14558578	C₂	1	2	yes	maximal, complemented
5	#14558576	C₂	1	2	yes	maximal, complemented
6	#14558575	C₂	1	2	yes	maximal, complemented
7	#14558548	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃₇⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11924600	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#3866	C₂	1	1	yes	maximal, characteristic, complemented
11	#14558329	C₂	1	1	yes	maximal, characteristic
12	#11924624	C₂²	1	2	yes	complemented
13	#11924623	—	2	2	no	—
14	#11924623	—	2	1	no	—
15	#11924623	—	2	1	no	—
16	#11924623	C₂²	1	2	yes	complemented
17	#11924623	C₂²	1	2	yes	complemented
18	#11924623	C₂²	1	2	yes	complemented
19	C₂×#10596953	—	2	1	no	—
20	C₂×#10596953	C₂²	1	1	yes	characteristic
21	D₃₇⋊M₁₆	—	2	1	no	—
22	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
23	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
24	C₂×#10596957	—	2	2	no	—
25	C₂×C₄×#3866	C₂²	1	1	yes	characteristic
26	#11924607	C₂²	1	2	yes	complemented
27	#11924606	C₂²	1	2	yes	complemented
28	#11924601	C₂²	1	1	yes	characteristic, complemented
29	#11924601	C₂²	1	1	yes	characteristic, complemented
30	#11924600	C₂²	1	2	yes	complemented
31	#11924600	C₂²	1	2	yes	complemented
32	#11924600	C₂²	1	1	yes	characteristic, complemented
33	#11924600	C₂²	1	1	yes	characteristic, complemented
34	#11924590	C₂²	1	2	yes	complemented
35	C₈×#3866	C₂²	1	2	yes	complemented
36	C₂×#10596964	C₂²	1	1	yes	characteristic
37	#41×D₃₇	C₂²	1	1	yes	characteristic
38	#11924569	C₄	1	1	yes	characteristic, complemented
39	#11924566	C₄	1	1	yes	characteristic, complemented
40	#11924565	C₄	1	2	yes	complemented
41	#11924562	C₄	1	1	yes	characteristic, complemented
42	#11924560	C₄	1	2	yes	complemented
43	#11924557	C₂²	1	2	yes	—
44	#11924555	C₂²	1	2	yes	—
45	M₁₆×D₃₇	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₃₇	C₄	1	1	yes	characteristic, complemented
47	C₂²×#3866	—	2	1	no	—
48	#41×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊D₃₇	—	2	1	no	—
51	C₂²×Dic₃₇	—	2	1	no	—
52	#10596966	C₂³	1	1	yes	characteristic
53	Q₈×D₃₇	C₂×C₄	1	1	yes	characteristic
54	#10596964	—	2	1	no	—
55	#10596964	—	2	1	no	—
56	#10596964	—	2	1	no	—
57	#10596964	C₂×C₄	1	2	yes	—
58	#10596964	C₂³	1	1	yes	characteristic
59	#10596964	C₂³	1	2	yes	—
60	D₄×D₃₇	—	2	1	no	—
61	D₄×D₃₇	C₂×C₄	1	1	yes	characteristic
62	#10596962	—	2	1	no	—
63	#10596962	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃₇	—	2	1	no	—
65	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
66	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
67	#10596957	—	2	1	no	—
68	#10596957	—	2	2	no	—
69	#10596957	—	2	2	no	—
70	#10596957	—	2	2	no	—
71	#10596957	D₄	1	2	yes	—
72	C₂×#64438	—	2	1	no	—
73	C₂×#64438	—	2	2	no	—
74	C₂×#64438	—	2	2	no	—
75	C₂×#64438	—	2	2	no	—
76	C₄×#3866	—	2	1	no	—
77	C₄×#3866	C₂³	1	1	yes	characteristic
78	C₄×#3866	C₂³	1	1	yes	characteristic
79	#10596953	—	2	1	no	—
80	#10596953	—	2	2	no	—
81	#10596953	D₄	1	1	yes	characteristic
82	#10596953	D₄	1	1	yes	characteristic
83	#10596953	C₂³	1	2	yes	—
84	#10596953	C₂³	1	1	yes	characteristic
85	#10596953	C₂³	1	1	yes	characteristic
86	D₃₇⋊C₈	—	2	1	no	—
87	D₃₇⋊C₈	—	2	2	no	—
88	D₃₇⋊C₈	C₂³	1	2	yes	—
89	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
90	D₈×C₃₇	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
92	#10596942	C₂×C₄	1	2	yes	complemented
93	#10596940	C₂×C₄	1	2	yes	complemented
94	#10596940	C₂×C₄	1	2	yes	complemented
95	#10596939	C₂×C₄	1	2	yes	complemented
96	#10596934	C₂×C₄	1	1	yes	characteristic
97	Dic₁₄₈	C₂×C₄	1	2	yes	complemented
98	#10596930	C₂×C₄	1	2	yes	complemented
99	#10596929	C₂×C₄	1	2	yes	—
100	C₈×D₃₇	C₂³	1	2	yes	—
101	C₂³×C₃₇	—	2	1	no	—
102	C₂²×D₃₇	—	2	1	no	—
103	C₂²×D₃₇	#41	1	1	yes	characteristic
104	C₂×#3866	—	2	1	no	—
105	C₂×#3866	—	2	1	no	—
106	C₂×#3866	—	4	2	no	—
107	Q₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
108	D₄×C₃₇	—	2	1	no	—
109	D₄×C₃₇	—	2	1	no	—
110	D₄×C₃₇	C₂²×C₄	1	2	yes	—
111	D₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₃₇	—	2	1	no	—
115	C₂²⋊D₃₇	—	2	2	no	—
116	C₂²⋊D₃₇	—	2	1	no	—
117	C₂²⋊D₃₇	—	2	2	no	—
118	C₂²⋊D₃₇	—	2	1	no	—
119	C₂×Dic₃₇	—	2	1	no	—
120	C₂×Dic₃₇	—	2	2	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	—	2	2	no	—
123	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
124	D₁₄₈	—	2	1	no	—
125	D₁₄₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₃₇	—	2	1	no	—
127	C₄×D₃₇	—	2	1	no	—
128	C₄×D₃₇	—	2	1	no	—
129	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
130	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
131	C₄×D₃₇	#41	1	1	yes	characteristic
132	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
133	Dic₇₄	—	2	1	no	—
134	Dic₇₄	—	2	1	no	—
135	Dic₇₄	C₂²×C₄	1	2	yes	—
136	Dic₇₄	C₂²×C₄	1	1	yes	characteristic
137	#64438	—	2	2	no	—
138	#64438	—	2	1	no	—
139	#64438	—	2	1	no	—
140	#64438	C₂×D₄	1	2	yes	—
141	#64438	C₂×D₄	1	2	yes	—
142	C₈×C₃₇	C₂²×C₄	1	2	yes	—
143	#64436	C₂²×C₄	1	2	yes	—
144	C₂²×C₃₇	—	2	2	no	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	—	8	1	no	—
147	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
148	C₂×D₃₇	—	2	1	no	—
149	C₂×D₃₇	—	4	1	no	—
150	C₂×D₃₇	—	8	1	no	—
151	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
152	C₂×D₃₇	C₂×#41	1	1	yes	characteristic
153	#3866	—	2	2	no	—
154	#3866	—	4	1	no	—
155	C₄×C₃₇	—	2	1	no	—
156	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
157	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
158	Dic₃₇	—	2	1	no	—
159	Dic₃₇	—	2	2	no	—
160	Dic₃₇	C₄×D₄	1	1	yes	characteristic
161	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
162	#2956	—	37	1	no	maximal, Sylow
163	C₂×C₃₇	—	2	1	no	—
164	C₂×C₃₇	—	4	1	no	—
165	C₂×C₃₇	—	4	2	no	—
166	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
167	D₃₇	—	2	1	no	—
168	D₃₇	—	2	1	no	—
169	D₃₇	—	4	1	no	—
170	#575	—	37	1	no	—
171	#568	—	37	1	no	—
172	#568	—	37	1	no	—
173	#444	—	37	2	no	—
174	#444	—	37	2	no	—
175	#443	—	37	2	no	—
176	#443	—	37	2	no	—
177	#430	—	37	1	no	—
178	C₂×#104	—	37	1	no	—
179	C₂×#104	—	37	1	no	—
180	C₄×M₁₆	—	37	1	no	—
181	C₃₇	#2956	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	37	1	no	—
183	C₂×#41	—	37	1	no	—
184	#137	—	37	1	no	—
185	#137	—	37	2	no	—
186	#137	—	37	1	no	—
187	AGL(1,ℤ/8)	—	37	1	no	—
188	AGL(1,ℤ/8)	—	37	2	no	—
189	AGL(1,ℤ/8)	#3866	1	1	yes	characteristic, complemented
190	#135	—	37	2	no	—
191	#135	—	37	2	no	—
192	#131	—	37	2	no	—
193	#131	—	37	2	no	—
194	#131	—	37	2	no	—
195	#131	—	37	2	no	—
196	#131	—	74	2	no	—
197	#131	—	74	1	no	—
198	#131	—	74	1	no	—
199	C₂×M₁₆	—	37	1	no	—
200	C₂×M₁₆	—	37	1	no	—
201	C₂×M₁₆	—	37	1	no	—
202	C₂×M₁₆	—	37	1	no	—
203	C₂×M₁₆	—	74	2	no	—
204	C₂×M₁₆	—	74	1	no	—
205	C₂×M₁₆	—	74	1	no	—
206	C₂×C₄²	—	37	1	no	—
207	#108	—	37	2	no	—
208	#108	—	37	2	no	—
209	#104	—	37	2	no	—
210	#104	—	37	1	no	—
211	#104	—	37	2	no	—
212	#104	—	37	1	no	—
213	#104	—	37	1	no	—
214	#104	—	37	1	no	—
215	#97	—	37	2	no	—
216	C₄×C₈	—	37	2	no	—
217	#41	—	37	1	no	—
218	#41	—	37	2	no	—
219	#41	—	37	2	no	—
220	#41	—	37	1	no	—
221	#41	—	37	1	no	—
222	#41	—	74	1	no	—
223	#41	—	74	1	no	—
224	#41	—	74	1	no	—
225	#41	—	74	1	no	—
226	#41	C₂×#3866	1	1	yes	characteristic
227	C₂×Q₈	—	37	1	no	—
228	C₂×Q₈	—	37	1	no	—
229	C₂×D₄	—	37	1	no	—
230	C₂×D₄	—	74	1	no	—
231	C₂×D₄	—	74	1	no	—
232	C₂×D₄	C₂×#3866	1	1	yes	characteristic
233	C₂²×C₄	—	37	1	no	—
234	C₂²×C₄	—	74	1	no	—
235	C₂²×C₄	—	74	1	no	—
236	C₂²×C₄	—	74	1	no	—
237	Q₁₆	—	37	2	no	—
238	Q₁₆	—	37	2	no	—
239	SD₁₆	—	37	2	no	—
240	SD₁₆	—	37	2	no	—
241	SD₁₆	—	37	2	no	—
242	SD₁₆	C₂×#3866	1	2	yes	complemented
243	D₈	—	37	2	no	—
244	D₈	C₂×#3866	1	2	yes	complemented
245	M₁₆	—	37	1	no	—
246	M₁₆	—	37	1	no	—
247	M₁₆	—	37	2	no	—
248	M₁₆	—	37	1	no	—
249	M₁₆	—	37	2	no	—
250	M₁₆	—	37	2	no	—
251	M₁₆	—	37	1	no	—
252	M₁₆	—	37	1	no	—
253	M₁₆	—	74	2	no	—
254	M₁₆	—	74	2	no	—
255	M₁₆	—	74	2	no	—
256	M₁₆	—	74	2	no	—
257	M₁₆	—	74	1	no	—
258	M₁₆	—	74	1	no	—
259	M₁₆	C₂×#3866	1	1	yes	characteristic
260	C₂×C₈	—	37	2	no	—
261	C₂×C₈	—	37	2	no	—
262	C₂×C₈	—	74	2	no	—
263	C₂×C₈	—	74	2	no	—
264	C₂×C₈	—	74	2	no	—
265	C₂×C₈	—	74	2	no	—
266	C₂×C₈	—	74	1	no	—
267	C₂×C₈	—	74	1	no	—
268	C₄²	—	37	1	no	—
269	C₄²	—	37	1	no	—
270	C₄²	—	74	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	37	1	no	—
273	C₂³	—	74	1	no	—
274	Q₈	—	37	1	no	—
275	Q₈	—	37	2	no	—
276	Q₈	—	74	1	no	—
277	Q₈	—	74	1	no	—
278	Q₈	C₂²×#3866	1	1	yes	characteristic
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	37	1	no	—
282	D₄	—	74	1	no	—
283	D₄	—	74	1	no	—
284	D₄	—	74	1	no	—
285	D₄	—	74	2	no	—
286	D₄	—	74	2	no	—
287	D₄	—	74	1	no	—
288	D₄	C₂²×#3866	1	2	yes	—
289	D₄	C₂²×#3866	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	37	1	no	—
292	C₂×C₄	—	37	1	no	—
293	C₂×C₄	—	37	1	no	—
294	C₂×C₄	—	37	1	no	—
295	C₂×C₄	—	37	1	no	—
296	C₂×C₄	—	74	1	no	—
297	C₂×C₄	—	74	1	no	—
298	C₂×C₄	—	74	1	no	—
299	C₂×C₄	—	74	2	no	—
300	C₂×C₄	—	74	2	no	—
301	C₂×C₄	—	74	1	no	—
302	C₂×C₄	—	74	1	no	—
303	C₂×C₄	—	74	1	no	—
304	C₂×C₄	—	74	1	no	—
305	C₂×C₄	—	148	2	no	—
306	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
307	C₈	—	37	2	no	—
308	C₈	—	37	2	no	—
309	C₈	—	37	2	no	—
310	C₈	—	74	2	no	—
311	C₈	—	74	1	no	—
312	C₈	—	74	1	no	—
313	C₈	C₂²×#3866	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	37	1	no	—
318	C₂²	—	37	1	no	—
319	C₂²	—	296	1	no	—
320	C₂²	—	74	1	no	—
321	C₂²	—	148	1	no	—
322	C₂²	D₄×#3866	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	37	1	no	—
325	C₄	—	37	1	no	—
326	C₄	—	74	2	no	—
327	C₄	—	74	1	no	—
328	C₄	—	74	2	no	—
329	C₄	—	148	1	no	—
330	C₄	D₄×#3866	1	1	yes	characteristic
331	C₄	C₂³×#3866	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	1	no	—
335	C₂	—	74	1	no	—
336	C₂	—	74	1	no	—
337	C₂	—	148	1	no	—
338	C₂	C₂×D₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#3866
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₄	C₂³×#3866
Fitting	AGL(1,ℤ/8)×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#14558329	C₂
cosocle	C₄×D₃₇	C₂⁴
hypercenter	AGL(1,ℤ/8)	#3866
hypocenter	C₃₇	#2956
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #14558612	C₂	semidirect
2. #14558611	C₂	semidirect
3. #14558580	C₂	semidirect
4. #14558578	C₂	semidirect
5. #14558576	C₂	semidirect
6. #14558575	C₂	semidirect
7. #14558548	C₂	semidirect
8. D₃₇⋊#104	C₂	semidirect
9. C₂×#11924600	C₂	semidirect
10. M₁₆×#3866	C₂	semidirect
12. #11924624	C₂²	semidirect
16. #11924623	C₂²	semidirect
17. #11924623	C₂²	semidirect
18. #11924623	C₂²	semidirect
26. #11924607	C₂²	semidirect
27. #11924606	C₂²	semidirect
28. #11924601	C₂²	semidirect
29. #11924601	C₂²	semidirect
30. #11924600	C₂²	semidirect
31. #11924600	C₂²	semidirect
32. #11924600	C₂²	semidirect
33. #11924600	C₂²	semidirect
34. #11924590	C₂²	semidirect
35. C₈×#3866	C₂²	semidirect
38. #11924569	C₄	semidirect
39. #11924566	C₄	semidirect
40. #11924565	C₄	semidirect
41. #11924562	C₄	semidirect
42. #11924560	C₄	semidirect
46. AGL(1,ℤ/8)×C₃₇	C₄	semidirect
89. SD₁₆×C₃₇	C₂×C₄	semidirect
90. D₈×C₃₇	C₂×C₄	semidirect
92. #10596942	C₂×C₄	semidirect
93. #10596940	C₂×C₄	semidirect
94. #10596940	C₂×C₄	semidirect
95. #10596939	C₂×C₄	semidirect
97. Dic₁₄₈	C₂×C₄	semidirect
98. #10596930	C₂×C₄	semidirect
181. C₃₇	#2956	semidirect
189. AGL(1,ℤ/8)	#3866	semidirect
242. SD₁₆	C₂×#3866	semidirect
244. D₈	C₂×#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558612	2
#14558611	2
#14558580	2
#14558578	2
#14558576	2
#14558575	2
#14558548	2
D₃₇⋊#104	2
C₂×#11924600	2
M₁₆×#3866	2
#14558329	2
#2956	37