Subgroups — #31789624

Order 17536.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984113	C₂	1	1	yes	maximal, characteristic
2	#22577886	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577877	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577512	C₂	1	2	yes	maximal, complemented
5	#22577506	C₂	1	2	yes	maximal, complemented
6	#22577488	C₂	1	2	yes	maximal, complemented
7	#22577483	C₂	1	2	yes	maximal, complemented
8	#22577446	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#17984010	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577358	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₄₈	C₂²	1	1	yes	characteristic
13	#17984113	C₄	1	8	yes	complemented
14	C₂×#14555497	C₂²	1	2	yes	—
15	C₂×#14555495	C₂²	1	2	yes	—
16	C₂×#14555494	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₃₇	C₂²	1	2	yes	complemented
18	C₄×D₅₄₈	—	2	1	no	—
19	C₄×D₅₄₈	—	2	1	no	—
20	D₄×Dic₁₃₇	—	2	2	no	—
21	D₄×Dic₁₃₇	C₂²	1	2	yes	complemented
22	#17984050	C₂²	1	2	yes	—
23	#17984050	C₂²	1	2	yes	—
24	#17984047	—	2	1	no	—
25	#17984047	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃₇	—	2	1	no	—
27	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×D₁₃₇	—	2	2	no	—
29	#17984021	C₂²	1	2	yes	complemented
30	#17984019	C₂²	1	2	yes	complemented
31	#17984010	C₂²	1	2	yes	—
32	#17984010	C₂²	1	2	yes	—
33	#17983998	C₂²	1	2	yes	complemented
34	C₈×Dic₁₃₇	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
37	#117×C₁₃₇	C₂²	1	1	yes	characteristic
38	#103×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
39	#103×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
40	#102×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
41	#102×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₃₇	—	2	1	no	—
43	C₂×D₅₄₈	—	2	1	no	—
44	C₂×D₅₄₈	—	2	2	no	—
45	C₂×D₅₄₈	C₂×C₄	1	4	yes	complemented
46	C₂×D₅₄₈	C₂³	1	2	yes	—
47	C₂²×Dic₁₃₇	—	2	2	no	—
48	C₂²×Dic₁₃₇	—	2	1	no	—
49	C₂×C₄×D₁₃₇	—	2	2	no	—
50	C₂×C₄×D₁₃₇	—	2	2	no	—
51	#14555497	C₂×C₄	1	8	yes	complemented
52	#14555495	C₂×C₄	1	8	yes	complemented
53	#14555494	C₂×C₄	1	4	yes	complemented
54	C₂×#11764040	C₂³	1	2	yes	—
55	C₄⋊Dic₁₃₇	—	2	2	no	—
56	C₄⋊Dic₁₃₇	—	2	1	no	—
57	C₄⋊Dic₁₃₇	D₄	1	1	yes	characteristic
58	C₄⋊Dic₁₃₇	D₄	1	1	yes	characteristic
59	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
60	C₂²⋊Dic₁₃₇	—	2	2	no	—
61	C₂²⋊Dic₁₃₇	D₄	1	2	yes	—
62	#14555487	—	2	2	no	—
63	#14555487	—	2	2	no	—
64	C₄×Dic₁₃₇	—	2	2	no	—
65	C₄×Dic₁₃₇	C₂³	1	2	yes	—
66	#41×C₁₃₇	—	2	2	no	—
67	#41×C₁₃₇	C₂×C₄	1	4	yes	complemented
68	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₁₃₇	—	2	1	no	—
70	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	—	2	1	no	—
72	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
79	C₄²×C₁₃₇	—	2	1	no	—
80	C₂²×D₁₃₇	—	2	2	no	—
81	C₂²×D₁₃₇	—	2	4	no	—
82	C₂²×D₁₃₇	—	8	1	no	—
83	C₂²×D₁₃₇	—	8	1	no	—
84	D₅₄₈	—	2	4	no	—
85	D₅₄₈	—	2	2	no	—
86	D₅₄₈	C₂²×C₄	1	4	yes	—
87	C₂×Dic₁₃₇	—	2	2	no	—
88	C₂×Dic₁₃₇	—	8	2	no	—
89	C₂×Dic₁₃₇	—	8	1	no	—
90	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
91	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
92	C₄×D₁₃₇	—	8	2	no	—
93	C₄×D₁₃₇	—	8	2	no	—
94	#11764040	C₂²×C₄	1	4	yes	—
95	C₂³×C₁₃₇	—	2	1	no	—
96	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₃₇	—	2	1	no	—
98	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
99	D₄×C₁₃₇	—	2	4	no	—
100	D₄×C₁₃₇	—	2	1	no	—
101	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	1	no	—
105	C₂×C₄×C₁₃₇	—	2	1	no	—
106	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	#41	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
110	C₈×C₁₃₇	—	2	1	no	—
111	C₈×C₁₃₇	—	2	1	no	—
112	C₂²×C₁₃₇	—	2	1	no	—
113	C₂²×C₁₃₇	—	2	1	no	—
114	C₂²×C₁₃₇	—	2	2	no	—
115	C₂²×C₁₃₇	—	4	1	no	—
116	C₂²×C₁₃₇	—	4	1	no	—
117	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
118	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₃₇	—	2	4	no	—
120	C₂×D₁₃₇	—	4	2	no	—
121	C₂×D₁₃₇	—	4	2	no	—
122	C₂×D₁₃₇	—	8	4	no	—
123	C₄×C₁₃₇	—	2	2	no	—
124	C₄×C₁₃₇	—	4	1	no	—
125	C₄×C₁₃₇	—	4	1	no	—
126	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
127	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
129	Dic₁₃₇	—	2	2	no	—
130	Dic₁₃₇	—	2	2	no	—
131	Dic₁₃₇	—	4	2	no	—
132	C₂×C₁₃₇	—	2	2	no	—
133	C₂×C₁₃₇	—	4	2	no	—
134	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₃₇	#576	1	1	yes	characteristic
136	C₂×C₁₃₇	#577	1	1	yes	characteristic
137	D₁₃₇	—	4	4	no	—
138	C₁₃₇	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	137	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	—	137	1	no	—
141	#520	—	137	1	no	—
142	#518	—	137	1	no	—
143	#442	—	137	2	no	—
144	#440	—	137	2	no	—
145	C₄×SD₁₆	—	137	2	no	—
146	C₄×D₈	—	137	2	no	—
147	#428	—	137	1	no	—
148	#416	D₁₃₇	1	1	yes	characteristic, complemented
149	C₂×#102	—	137	1	no	—
150	#405	—	137	1	no	—
151	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
152	C₂²×D₄	—	137	1	no	—
153	AGL(1,ℤ/8)	—	137	8	no	—
154	C₂×SD₁₆	—	137	2	no	—
155	C₂×D₈	—	137	2	no	—
156	C₂×M₁₆	—	137	1	no	—
157	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
158	C₄×Q₈	—	137	2	no	—
159	C₄×D₄	—	274	1	no	—
160	C₄×D₄	—	274	1	no	—
161	C₄×D₄	—	274	2	no	—
162	C₄×D₄	—	137	2	no	—
163	C₄×D₄	—	137	2	no	—
164	C₄×D₄	—	137	2	no	—
165	#117	—	274	1	no	—
166	#117	—	137	1	no	—
167	#117	C₂×D₁₃₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	274	1	no	—
169	C₂×C₄⋊C₄	—	137	1	no	—
170	C₂×C₂²⋊C₄	—	274	2	no	—
171	#107	—	137	2	no	—
172	#106	—	137	2	no	—
173	#103	C₂×D₁₃₇	1	1	yes	characteristic, complemented
174	#103	C₂×D₁₃₇	1	1	yes	characteristic, complemented
175	#102	—	137	2	no	—
176	#102	—	137	2	no	—
177	#102	C₂×D₁₃₇	1	1	yes	characteristic, complemented
178	#102	C₂×D₁₃₇	1	1	yes	characteristic, complemented
179	#97	—	137	2	no	—
180	C₄×C₈	—	137	2	no	—
181	C₂⁴	—	274	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₁₃₇	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	274	1	no	—
187	C₂×D₄	—	274	2	no	—
188	C₂×D₄	—	137	4	no	—
189	C₂×D₄	—	137	2	no	—
190	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	274	2	no	—
193	C₂²×C₄	—	274	2	no	—
194	C₂²×C₄	—	274	1	no	—
195	C₂²×C₄	—	274	2	no	—
196	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
197	SD₁₆	—	137	8	no	—
198	D₈	—	137	8	no	—
199	M₁₆	—	137	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	137	2	no	—
202	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
204	C₄⋊C₄	—	274	2	no	—
205	C₄⋊C₄	—	274	1	no	—
206	C₄⋊C₄	—	137	2	no	—
207	C₄⋊C₄	—	137	1	no	—
208	C₄⋊C₄	—	137	1	no	—
209	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	274	2	no	—
213	C₂²⋊C₄	—	274	2	no	—
214	C₂²⋊C₄	—	274	2	no	—
215	C₂²⋊C₄	—	137	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	274	2	no	—
218	C₄²	—	137	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	274	2	no	—
221	C₂³	—	274	4	no	—
222	C₂³	—	1096	1	no	—
223	C₂³	—	1096	1	no	—
224	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	274	4	no	—
230	D₄	—	274	2	no	—
231	D₄	—	137	4	no	—
232	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	274	2	no	—
238	C₂×C₄	—	1096	2	no	—
239	C₂×C₄	—	1096	2	no	—
240	C₂×C₄	—	1096	2	no	—
241	C₂×C₄	—	1096	1	no	—
242	C₂×C₄	—	137	2	no	—
243	C₂×C₄	—	137	2	no	—
244	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
245	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
246	C₂×C₄	#14555509	1	2	yes	—
247	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	137	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	274	4	no	—
257	C₂²	—	548	2	no	—
258	C₂²	—	548	2	no	—
259	C₂²	—	1096	4	no	—
260	C₂²	#17984048	1	2	yes	—
261	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	274	2	no	—
266	C₄	—	274	2	no	—
267	C₄	—	548	2	no	—
268	C₄	#17984048	1	2	yes	—
269	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
270	C₄	C₂×#14555509	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	548	4	no	—
274	C₂	C₂×#17984048	1	1	yes	characteristic
275	C₂	#22578205	1	1	yes	characteristic
276	C₂	#22578210	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#416×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984113	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#416	D₁₃₇
hypocenter	C₁₃₇	#2946
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #22577886	C₂	semidirect
3. #22577877	C₂	semidirect
4. #22577512	C₂	semidirect
5. #22577506	C₂	semidirect
6. #22577488	C₂	semidirect
7. #22577483	C₂	semidirect
8. #22577446	C₂	semidirect
9. C₂×#17984010	C₂	semidirect
10. #22577358	C₂	semidirect
11. #416×C₁₃₇	C₂	semidirect
13. #17984113	C₄	semidirect
17. Q₈×Dic₁₃₇	C₂²	semidirect
21. D₄×Dic₁₃₇	C₂²	semidirect
29. #17984021	C₂²	semidirect
30. #17984019	C₂²	semidirect
33. #17983998	C₂²	semidirect
34. C₈×Dic₁₃₇	C₂²	semidirect
38. #103×C₁₃₇	C₂²	semidirect
39. #103×C₁₃₇	C₂²	semidirect
40. #102×C₁₃₇	C₂²	semidirect
41. #102×C₁₃₇	C₂²	semidirect
45. C₂×D₅₄₈	C₂×C₄	semidirect
51. #14555497	C₂×C₄	semidirect
52. #14555495	C₂×C₄	semidirect
53. #14555494	C₂×C₄	semidirect
67. #41×C₁₃₇	C₂×C₄	semidirect
138. C₁₃₇	#2946	semidirect
148. #416	D₁₃₇	semidirect
173. #103	C₂×D₁₃₇	semidirect
174. #103	C₂×D₁₃₇	semidirect
177. #102	C₂×D₁₃₇	semidirect
178. #102	C₂×D₁₃₇	semidirect
183. #41	C₄×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984113	2
#22577886	2
#22577877	2
#22577512	2
#22577506	2
#22577488	2
#22577483	2
#22577446	2
C₂×#17984010	2
#22577358	2
#416×C₁₃₇	2
#2946	137