Subgroups — #31792576

Order 17536.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578220	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578210	C₂	1	2	yes	maximal, complemented
3	#22578208	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577787	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577678	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577673	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577667	C₂	1	2	yes	maximal, complemented
8	#22577582	C₂	1	2	yes	maximal, complemented
9	#22577577	C₂	1	2	yes	maximal, complemented
10	#22577521	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#17984135	C₂²	1	2	yes	complemented
13	#17984134	C₂²	1	2	yes	—
14	#17984121	—	2	2	no	—
15	#17984121	C₂²	1	2	yes	complemented
16	#17984120	—	2	2	no	—
17	#17984118	C₂²	1	2	yes	—
18	#17984116	—	2	2	no	—
19	#17984116	C₂²	1	2	yes	—
20	#17984113	C₂²	1	2	yes	complemented
21	#17984112	—	2	2	no	—
22	#17984110	—	2	1	no	—
23	#17984110	—	2	1	no	—
24	#17984110	C₂²	1	1	yes	characteristic
25	#17984109	—	2	2	no	—
26	C₂×Dic₅₄₈	—	2	1	no	—
27	C₂×Dic₅₄₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×#14555499	C₂²	1	2	yes	complemented
29	SD₁₆×D₁₃₇	—	2	2	no	—
30	C₂×D₁₀₉₆	—	2	1	no	—
31	C₂×D₁₀₉₆	C₂²	1	1	yes	characteristic
32	#17984094	C₂²	1	2	yes	complemented
33	#17984076	C₂²	1	2	yes	complemented
34	#17984024	C₂²	1	1	yes	characteristic, complemented
35	#17984016	C₂²	1	2	yes	complemented
36	#17984006	C₂²	1	2	yes	complemented
37	#17984004	C₂²	1	2	yes	complemented
38	#135×C₁₃₇	C₂²	1	1	yes	characteristic
39	#131×C₁₃₇	C₂²	1	1	yes	characteristic
40	#131×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
41	#108×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
42	#104×C₁₃₇	C₂²	1	1	yes	characteristic
43	#104×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
45	#14555511	—	2	2	no	—
46	#14555510	—	2	2	no	—
47	#14555510	—	2	2	no	—
48	#14555510	C₂³	1	2	yes	—
49	#14555509	—	2	2	no	—
50	#14555509	—	4	2	no	—
51	C₂×Dic₂₇₄	—	2	2	no	—
52	C₂×Dic₂₇₄	C₂³	1	2	yes	—
53	Q₈×D₁₃₇	—	2	2	no	—
54	C₂×D₅₄₈	—	2	2	no	—
55	C₂×D₅₄₈	C₂³	1	2	yes	—
56	D₄×D₁₃₇	—	2	2	no	—
57	D₄×D₁₃₇	—	4	2	no	—
58	Dic₅₄₈	—	2	2	no	—
59	Dic₅₄₈	—	2	1	no	—
60	Dic₅₄₈	—	2	1	no	—
61	Dic₅₄₈	D₄	1	1	yes	characteristic
62	#14555500	—	2	2	no	—
63	#14555499	—	2	2	no	—
64	#14555499	—	2	2	no	—
65	#14555499	—	2	2	no	—
66	#14555499	—	2	2	no	—
67	#14555499	D₄	1	2	yes	—
68	#14555498	—	2	2	no	—
69	#14555497	—	2	2	no	—
70	D₁₀₉₆	—	2	2	no	—
71	D₁₀₉₆	—	2	1	no	—
72	D₁₀₉₆	—	2	1	no	—
73	D₁₀₉₆	D₄	1	1	yes	characteristic
74	#14555495	—	2	2	no	—
75	#14555494	C₂³	1	2	yes	—
76	#14555493	—	2	2	no	—
77	C₈×D₁₃₇	—	2	2	no	—
78	#14555487	—	4	2	no	—
79	#41×C₁₃₇	C₂³	1	1	yes	characteristic
80	#41×C₁₃₇	C₂³	1	1	yes	characteristic
81	Q₁₆×C₁₃₇	D₄	1	1	yes	characteristic
82	SD₁₆×C₁₃₇	D₄	1	1	yes	characteristic
83	SD₁₆×C₁₃₇	D₄	1	1	yes	characteristic
84	D₈×C₁₃₇	D₄	1	1	yes	characteristic
85	M₁₆×C₁₃₇	—	2	1	no	—
86	M₁₆×C₁₃₇	—	2	1	no	—
87	M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
88	M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₁₃₇	—	2	1	no	—
90	C₂×C₈×C₁₃₇	—	2	1	no	—
91	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
93	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
94	C₂²×D₁₃₇	—	2	2	no	—
95	C₂²×D₁₃₇	—	4	2	no	—
96	Dic₂₇₄	—	2	2	no	—
97	Dic₂₇₄	—	2	2	no	—
98	Dic₂₇₄	—	4	2	no	—
99	Dic₂₇₄	C₂×D₄	1	2	yes	—
100	D₅₄₈	—	2	2	no	—
101	D₅₄₈	—	2	2	no	—
102	D₅₄₈	—	4	2	no	—
103	D₅₄₈	C₂×D₄	1	2	yes	—
104	C₂²⋊D₁₃₇	—	2	2	no	—
105	C₂²⋊D₁₃₇	—	4	2	no	—
106	C₂²⋊D₁₃₇	—	4	2	no	—
107	C₂×Dic₁₃₇	—	2	2	no	—
108	C₂×Dic₁₃₇	—	4	2	no	—
109	C₄×D₁₃₇	—	2	2	no	—
110	C₄×D₁₃₇	—	4	2	no	—
111	C₄×D₁₃₇	—	4	2	no	—
112	#11764040	—	2	2	no	—
113	Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	D₄×C₁₃₇	—	2	1	no	—
116	D₄×C₁₃₇	—	2	1	no	—
117	D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
118	D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	—	2	1	no	—
120	C₂×C₄×C₁₃₇	—	2	1	no	—
121	C₂×C₄×C₁₃₇	—	2	1	no	—
122	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
123	C₈×C₁₃₇	—	2	1	no	—
124	C₈×C₁₃₇	—	2	1	no	—
125	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
126	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
127	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
128	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₁₃₇	—	2	1	no	—
130	C₂²×C₁₃₇	—	2	1	no	—
131	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₁₃₇	—	2	2	no	—
133	C₂×D₁₃₇	—	4	2	no	—
134	C₂×D₁₃₇	—	8	2	no	—
135	C₂×D₁₃₇	—	16	2	no	—
136	C₄×C₁₃₇	—	2	1	no	—
137	C₄×C₁₃₇	—	2	1	no	—
138	C₄×C₁₃₇	—	4	1	no	—
139	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₃₇	—	2	2	no	—
142	Dic₁₃₇	—	4	2	no	—
143	C₂×C₁₃₇	—	2	1	no	—
144	C₂×C₁₃₇	—	4	1	no	—
145	C₂×C₁₃₇	—	4	1	no	—
146	C₂×C₁₃₇	D₄²	1	1	yes	characteristic
147	D₁₃₇	—	4	2	no	—
148	D₁₃₇	—	8	2	no	—
149	C₁₃₇	#3290	1	1	yes	characteristic, complemented, Sylow
150	#3290	—	137	1	no	maximal, Sylow
151	#578	—	137	1	no	—
152	#577	—	137	2	no	—
153	#576	—	137	1	no	—
154	#495	—	137	1	no	—
155	#473	—	137	1	no	—
156	#472	—	137	1	no	—
157	#471	—	137	2	no	—
158	#455	—	137	2	no	—
159	#454	—	137	2	no	—
160	#443	—	137	1	no	—
161	#443	D₁₃₇	1	1	yes	characteristic, complemented
162	2₋¹⁺⁴	—	137	2	no	—
163	H(5,𝔽₂)	—	137	2	no	—
164	#137	—	274	2	no	—
165	#137	—	274	2	no	—
166	#137	—	137	2	no	—
167	#137	—	137	2	no	—
168	AGL(1,ℤ/8)	—	274	2	no	—
169	AGL(1,ℤ/8)	—	274	2	no	—
170	AGL(1,ℤ/8)	—	137	2	no	—
171	AGL(1,ℤ/8)	—	137	2	no	—
172	#135	—	274	1	no	—
173	#135	—	274	1	no	—
174	#135	—	274	2	no	—
175	#135	—	137	1	no	—
176	#135	C₂×D₁₃₇	1	1	yes	characteristic
177	C₂×Q₁₆	—	274	1	no	—
178	C₂×Q₁₆	—	137	1	no	—
179	C₂×SD₁₆	—	274	2	no	—
180	C₂×SD₁₆	—	137	2	no	—
181	C₂×D₈	—	274	1	no	—
182	C₂×D₈	—	137	1	no	—
183	#131	—	137	2	no	—
184	#131	C₂×D₁₃₇	1	1	yes	characteristic
185	#131	C₂×D₁₃₇	1	1	yes	characteristic, complemented
186	#124	—	137	2	no	—
187	#108	—	137	1	no	—
188	#108	C₂×D₁₃₇	1	1	yes	characteristic, complemented
189	#104	—	137	2	no	—
190	#104	C₂×D₁₃₇	1	1	yes	characteristic
191	#104	C₂×D₁₃₇	1	1	yes	characteristic, complemented
192	#101	—	137	2	no	—
193	#100	—	137	2	no	—
194	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
195	#41	—	274	2	no	—
196	#41	—	274	2	no	—
197	#41	—	274	2	no	—
198	#41	—	274	2	no	—
199	#41	—	548	2	no	—
200	#41	—	137	2	no	—
201	#41	C₂²×D₁₃₇	1	1	yes	characteristic
202	#41	C₂²×D₁₃₇	1	1	yes	characteristic
203	C₂×Q₈	—	274	2	no	—
204	C₂×Q₈	—	274	2	no	—
205	C₂×Q₈	—	137	2	no	—
206	C₂×D₄	—	274	2	no	—
207	C₂×D₄	—	274	2	no	—
208	C₂×D₄	—	548	2	no	—
209	C₂×D₄	—	137	2	no	—
210	Q₁₆	—	274	2	no	—
211	Q₁₆	—	274	2	no	—
212	Q₁₆	—	274	1	no	—
213	Q₁₆	—	274	1	no	—
214	Q₁₆	—	137	1	no	—
215	Q₁₆	D₅₄₈	1	1	yes	characteristic
216	SD₁₆	—	274	2	no	—
217	SD₁₆	—	274	2	no	—
218	SD₁₆	—	274	2	no	—
219	SD₁₆	—	274	2	no	—
220	SD₁₆	—	274	2	no	—
221	SD₁₆	—	274	2	no	—
222	SD₁₆	—	137	2	no	—
223	SD₁₆	D₅₄₈	1	1	yes	characteristic
224	SD₁₆	D₅₄₈	1	1	yes	characteristic
225	D₈	—	274	2	no	—
226	D₈	—	274	2	no	—
227	D₈	—	274	1	no	—
228	D₈	—	274	1	no	—
229	D₈	—	137	1	no	—
230	D₈	D₅₄₈	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	274	2	no	—
234	M₁₆	—	137	2	no	—
235	M₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
236	M₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	274	2	no	—
240	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
242	C₂²⋊C₄	—	548	2	no	—
243	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
244	C₂³	—	274	2	no	—
245	C₂³	—	548	2	no	—
246	Q₈	—	274	2	no	—
247	Q₈	—	274	2	no	—
248	Q₈	—	548	2	no	—
249	Q₈	—	137	2	no	—
250	Q₈	C₂×D₅₄₈	1	1	yes	characteristic
251	Q₈	C₂×D₅₄₈	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	274	2	no	—
255	D₄	—	274	2	no	—
256	D₄	—	274	2	no	—
257	D₄	—	548	2	no	—
258	D₄	—	548	2	no	—
259	D₄	—	548	2	no	—
260	D₄	—	137	2	no	—
261	D₄	C₂×D₅₄₈	1	1	yes	characteristic
262	D₄	C₂×D₅₄₈	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	274	2	no	—
267	C₂×C₄	—	274	2	no	—
268	C₂×C₄	—	548	2	no	—
269	C₂×C₄	—	548	2	no	—
270	C₂×C₄	—	548	2	no	—
271	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	274	2	no	—
275	C₈	D₄×D₁₃₇	1	1	yes	characteristic
276	C₈	D₄×D₁₃₇	1	1	yes	characteristic
277	C₈	C₂×D₅₄₈	1	1	yes	characteristic
278	C₈	C₂×D₅₄₈	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	274	2	no	—
282	C₂²	—	548	2	no	—
283	C₂²	—	1096	2	no	—
284	C₂²	—	2192	2	no	—
285	C₂²	#17984134	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	274	2	no	—
290	C₄	—	548	2	no	—
291	C₄	C₂×D₄×D₁₃₇	1	1	yes	characteristic
292	C₄	C₂²×D₅₄₈	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	548	2	no	—
297	C₂	—	1096	2	no	—
298	C₂	D₄×D₅₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅₄₈
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#443×C₁₃₇	C₂
socle	C₂×C₁₃₇	D₄²
hypercenter	#443	D₁₃₇
hypocenter	C₁₃₇	#3290
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #22578220	C₂	semidirect
2. #22578210	C₂	semidirect
3. #22578208	C₂	semidirect
4. #22577787	C₂	semidirect
5. #22577678	C₂	semidirect
6. #22577673	C₂	semidirect
7. #22577667	C₂	semidirect
8. #22577582	C₂	semidirect
9. #22577577	C₂	semidirect
10. #22577521	C₂	semidirect
11. #443×C₁₃₇	C₂	semidirect
12. #17984135	C₂²	semidirect
15. #17984121	C₂²	semidirect
20. #17984113	C₂²	semidirect
27. C₂×Dic₅₄₈	C₂²	semidirect
28. C₂×#14555499	C₂²	semidirect
32. #17984094	C₂²	semidirect
33. #17984076	C₂²	semidirect
34. #17984024	C₂²	semidirect
35. #17984016	C₂²	semidirect
36. #17984006	C₂²	semidirect
37. #17984004	C₂²	semidirect
40. #131×C₁₃₇	C₂²	semidirect
41. #108×C₁₃₇	C₂²	semidirect
43. #104×C₁₃₇	C₂²	semidirect
44. C₄×C₈×C₁₃₇	C₂²	semidirect
149. C₁₃₇	#3290	semidirect
161. #443	D₁₃₇	semidirect
185. #131	C₂×D₁₃₇	semidirect
188. #108	C₂×D₁₃₇	semidirect
191. #104	C₂×D₁₃₇	semidirect
194. C₄×C₈	C₂×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22578220	2
#22578210	2
#22578208	2
#22577787	2
#22577678	2
#22577673	2
#22577667	2
#22577582	2
#22577577	2
#22577521	2
#443×C₁₃₇	2
#3290	137