Subgroups — #31789822

Order 17536.

464 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22577482	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577480	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577474	C₂	1	2	yes	maximal, complemented
5	#22577471	C₂	1	2	yes	maximal, complemented
6	D₄×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577459	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577454	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577365	C₂	1	2	yes	maximal, complemented
10	C₂×#17984000	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₃₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
14	C₂×#14555493	C₂²	1	2	yes	complemented
15	C₂×#14555493	C₂²	1	2	yes	complemented
16	C₂²×#11764040	C₂²	1	2	yes	complemented
17	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
18	D₄×Dic₁₃₇	C₄	1	1	yes	characteristic
19	#17984050	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
21	#17984048	C₄	1	2	yes	—
22	C₄⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₁₃₇	C₄	1	2	yes	—
24	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11764040	C₂²	1	1	yes	characteristic, complemented
26	#17984017	C₂²	1	1	yes	characteristic, complemented
27	#17984017	C₂²	1	1	yes	characteristic, complemented
28	C₂²⋊#11764040	C₂²	1	2	yes	complemented
29	#17984000	C₂²	1	1	yes	characteristic
30	#17984000	C₂²	1	1	yes	characteristic
31	#17984000	C₂²	1	2	yes	—
32	#17984000	C₂²	1	2	yes	complemented
33	#17984000	C₂²	1	2	yes	complemented
34	#17983998	C₂²	1	2	yes	complemented
35	C₄×#11764040	C₂²	1	1	yes	characteristic, complemented
36	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	complemented
37	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₃₇	C₂×C₄	1	2	yes	—
42	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
44	D₄×D₁₃₇	—	2	4	no	—
45	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃₇	—	4	2	no	—
47	C₂×C₄×D₁₃₇	—	4	2	no	—
48	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
54	#14555493	—	4	2	no	—
55	#14555493	—	4	2	no	—
56	C₂×#11764040	—	4	2	no	—
57	C₂×#11764040	C₂³	1	1	yes	characteristic
58	C₂×#11764040	C₂³	1	2	yes	—
59	C₂×#11764040	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
61	#14555489	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
63	#14555487	C₂×C₄	1	2	yes	—
64	#14555487	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
70	C₂×C₈×C₁₃₇	—	4	2	no	—
71	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
73	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
76	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
77	C₂²×D₁₃₇	—	2	4	no	—
78	C₂²×D₁₃₇	—	2	4	no	—
79	C₂²×D₁₃₇	—	4	2	no	—
80	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	D₅₄₈	—	2	2	no	—
83	C₂²⋊D₁₃₇	—	2	4	no	—
84	C₂×Dic₁₃₇	—	4	2	no	—
85	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
89	C₄×D₁₃₇	—	2	2	no	—
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	—	2	2	no	—
92	C₄×D₁₃₇	—	2	2	no	—
93	C₄×D₁₃₇	—	2	1	no	—
94	C₄×D₁₃₇	—	2	1	no	—
95	C₄×D₁₃₇	—	4	2	no	—
96	C₄×D₁₃₇	—	4	2	no	—
97	#11764040	—	2	1	no	—
98	#11764040	—	2	2	no	—
99	#11764040	—	2	1	no	—
100	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
101	D₄×C₁₃₇	M₁₆	1	4	yes	complemented
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₃₇	—	2	1	no	—
109	C₈×C₁₃₇	—	2	2	no	—
110	C₈×C₁₃₇	—	2	1	no	—
111	C₂²×C₁₃₇	—	2	2	no	—
112	C₂²×C₁₃₇	—	2	2	no	—
113	C₂²×C₁₃₇	C₂×M₁₆	1	4	yes	—
114	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₃₇	—	2	1	no	—
116	C₂×D₁₃₇	—	2	1	no	—
117	C₂×D₁₃₇	—	2	2	no	—
118	C₂×D₁₃₇	—	2	2	no	—
119	C₂×D₁₃₇	—	2	2	no	—
120	C₂×D₁₃₇	—	4	8	no	—
121	C₂×D₁₃₇	#131	1	2	yes	—
122	C₄×C₁₃₇	—	2	2	no	—
123	C₄×C₁₃₇	—	2	1	no	—
124	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
125	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₁₃₇	—	2	1	no	—
128	Dic₁₃₇	—	2	2	no	—
129	Dic₁₃₇	—	2	1	no	—
130	Dic₁₃₇	—	2	1	no	—
131	C₂×C₁₃₇	—	2	4	no	—
132	C₂×C₁₃₇	#518	1	1	yes	characteristic
133	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
135	D₁₃₇	—	2	2	no	—
136	D₁₃₇	—	4	2	no	—
137	C₁₃₇	#2974	1	1	yes	characteristic, complemented, Sylow
138	#2974	—	137	1	no	maximal, Sylow
139	C₂×C₄×D₄	—	137	1	no	—
140	#436	—	137	1	no	—
141	#436	—	137	1	no	—
142	#435	—	137	2	no	—
143	#435	—	137	2	no	—
144	C₈×D₄	—	137	1	no	—
145	C₈×D₄	D₁₃₇	1	1	yes	characteristic, complemented
146	#432	—	137	1	no	—
147	#431	—	137	1	no	—
148	#407	—	137	2	no	—
149	C₂×C₂²⋊C₈	—	137	2	no	—
150	C₂²×D₄	—	137	1	no	—
151	C₂³×C₄	—	137	2	no	—
152	C₂×M₁₆	—	137	2	no	—
153	C₂×M₁₆	—	137	2	no	—
154	C₂²×C₈	—	137	2	no	—
155	C₂²×C₈	C₂×D₁₃₇	1	2	yes	complemented
156	C₄×D₄	—	137	1	no	—
157	C₄×D₄	—	137	2	no	—
158	C₄×D₄	—	137	1	no	—
159	C₄×D₄	—	137	2	no	—
160	C₄×D₄	—	137	1	no	—
161	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	137	1	no	—
163	C₂×C₂²⋊C₄	—	137	2	no	—
164	C₂×C₄²	—	137	1	no	—
165	C₄⋊C₈	—	137	1	no	—
166	C₄⋊C₈	—	137	1	no	—
167	C₄⋊C₈	—	137	1	no	—
168	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
169	C₂²⋊C₈	—	137	2	no	—
170	C₂²⋊C₈	—	137	1	no	—
171	C₂²⋊C₈	—	137	2	no	—
172	C₂²⋊C₈	—	137	1	no	—
173	C₂²⋊C₈	—	137	2	no	—
174	C₂²⋊C₈	—	137	2	no	—
175	C₂²⋊C₈	C₂×D₁₃₇	1	2	yes	complemented
176	#97	—	137	2	no	—
177	C₄×C₈	—	137	1	no	—
178	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
179	C₂⁴	—	137	2	no	—
180	C₂×D₄	—	274	4	no	—
181	C₂×D₄	—	137	2	no	—
182	C₂×D₄	—	137	1	no	—
183	C₂×D₄	C₄×D₁₃₇	1	1	yes	characteristic
184	C₂²×C₄	—	548	2	no	—
185	C₂²×C₄	—	548	2	no	—
186	C₂²×C₄	—	137	1	no	—
187	C₂²×C₄	—	137	1	no	—
188	C₂²×C₄	—	137	1	no	—
189	C₂²×C₄	—	137	2	no	—
190	C₂²×C₄	—	137	2	no	—
191	C₂²×C₄	—	137	2	no	—
192	C₂²×C₄	—	137	2	no	—
193	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
194	M₁₆	—	548	2	no	—
195	M₁₆	—	548	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	548	2	no	—
198	C₂×C₈	—	137	1	no	—
199	C₂×C₈	—	137	1	no	—
200	C₂×C₈	—	137	2	no	—
201	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
204	C₄⋊C₄	—	137	2	no	—
205	C₄⋊C₄	—	137	1	no	—
206	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	137	2	no	—
208	C₂²⋊C₄	—	137	2	no	—
209	C₂²⋊C₄	—	137	2	no	—
210	C₂²⋊C₄	C₄×D₁₃₇	1	2	yes	—
211	C₄²	—	137	1	no	—
212	C₄²	—	137	1	no	—
213	C₄²	—	137	1	no	—
214	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
215	C₂³	—	274	4	no	—
216	C₂³	—	274	4	no	—
217	C₂³	—	548	2	no	—
218	C₂³	—	137	2	no	—
219	C₂³	—	137	1	no	—
220	C₂³	C₂×C₄×D₁₃₇	1	2	yes	—
221	D₄	—	274	4	no	—
222	D₄	—	274	2	no	—
223	D₄	#14555493	1	4	yes	complemented
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	274	2	no	—
227	C₂×C₄	—	274	2	no	—
228	C₂×C₄	—	274	2	no	—
229	C₂×C₄	—	274	1	no	—
230	C₂×C₄	—	274	2	no	—
231	C₂×C₄	—	274	1	no	—
232	C₂×C₄	—	548	2	no	—
233	C₂×C₄	—	548	2	no	—
234	C₂×C₄	—	548	2	no	—
235	C₂×C₄	—	137	1	no	—
236	C₂×C₄	—	137	1	no	—
237	C₂×C₄	—	137	1	no	—
238	C₂×C₄	—	137	2	no	—
239	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	274	1	no	—
247	C₈	—	274	1	no	—
248	C₈	—	274	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	274	1	no	—
252	C₂²	—	274	1	no	—
253	C₂²	—	274	2	no	—
254	C₂²	—	274	2	no	—
255	C₂²	—	274	2	no	—
256	C₂²	—	548	8	no	—
257	C₂²	—	137	2	no	—
258	C₂²	C₂×#14555493	1	4	yes	—
259	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	274	1	no	—
263	C₄	—	274	2	no	—
264	C₄	—	274	1	no	—
265	C₄	—	274	1	no	—
266	C₄	C₂×#14555493	1	2	yes	—
267	C₄	#17984133	1	1	yes	characteristic
268	C₄	#17984134	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	—	274	2	no	—
271	C₂	—	548	2	no	—
272	C₂	#22577875	1	1	yes	characteristic
273	C₂	C₂²×#14555493	1	1	yes	characteristic
274	C₂	#131×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₈×D₄×C₁₃₇	C₂
prime radical	C₂×D₄×D₁₃₇	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₈×D₄	D₁₃₇
hypocenter	C₁₃₇	#2974
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #22577482	C₂	semidirect
3. #22577480	C₂	semidirect
4. #22577474	C₂	semidirect
5. #22577471	C₂	semidirect
6. D₄×#11764040	C₂	semidirect
7. #22577459	C₂	semidirect
8. #22577454	C₂	semidirect
9. #22577365	C₂	semidirect
10. C₂×#17984000	C₂	semidirect
11. C₈×D₄×C₁₃₇	C₂	semidirect
14. C₂×#14555493	C₂²	semidirect
15. C₂×#14555493	C₂²	semidirect
16. C₂²×#11764040	C₂²	semidirect
25. C₄⋊#11764040	C₂²	semidirect
26. #17984017	C₂²	semidirect
27. #17984017	C₂²	semidirect
28. C₂²⋊#11764040	C₂²	semidirect
32. #17984000	C₂²	semidirect
33. #17984000	C₂²	semidirect
34. #17983998	C₂²	semidirect
35. C₄×#11764040	C₂²	semidirect
36. C₂²×C₈×C₁₃₇	C₂²	semidirect
38. C₄⋊C₈×C₁₃₇	C₂²	semidirect
39. C₂²⋊C₈×C₁₃₇	C₂²	semidirect
40. C₄×C₈×C₁₃₇	C₂²	semidirect
101. D₄×C₁₃₇	M₁₆	semidirect
137. C₁₃₇	#2974	semidirect
145. C₈×D₄	D₁₃₇	semidirect
155. C₂²×C₈	C₂×D₁₃₇	semidirect
168. C₄⋊C₈	C₂×D₁₃₇	semidirect
175. C₂²⋊C₈	C₂×D₁₃₇	semidirect
178. C₄×C₈	C₂×D₁₃₇	semidirect
223. D₄	#14555493	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₁₃₇	2
#22577482	2
#22577480	2
#22577474	2
#22577471	2
D₄×#11764040	2
#22577459	2
#22577454	2
#22577365	2
C₂×#17984000	2
C₈×D₄×C₁₃₇	2
#2974	137