Subgroups — C₂×#22578081 #31794288

Order 17536.

1328 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	#22578081	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984081	C₂	1	2	yes	maximal, complemented
3	C₂×#17984075	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17984070	C₂	1	2	yes	maximal, complemented
5	C₂×#17984066	C₂	1	2	yes	maximal, complemented
6	C₂×#17984056	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₂²⋊D₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×#17984046	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×D₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂×#123×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₃₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
16	#17984081	C₂²	1	16	yes	complemented
17	#17984075	C₂²	1	8	yes	complemented
18	#17984070	C₂²	1	16	yes	complemented
19	#17984066	C₂²	1	16	yes	complemented
20	#17984056	C₂²	1	8	yes	—
21	C₄×C₂²⋊D₁₃₇	C₂²	1	16	yes	complemented
22	#17984046	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	complemented
24	C₂×#14555489	C₂²	1	2	yes	complemented
25	C₂×#14555489	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₃₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₁₃₇	C₂²	1	2	yes	complemented
29	C₂×#14555487	C₂²	1	2	yes	—
30	C₂×#14555487	C₂²	1	2	yes	—
31	C₂×#14555487	C₂²	1	2	yes	—
32	C₂²⋊C₄×D₁₃₇	C₂²	1	16	yes	complemented
33	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
37	#123×C₁₃₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₃₇	—	2	8	no	—
42	C₂³×D₁₃₇	—	4	2	no	—
43	C₂³×D₁₃₇	C₂³	1	4	yes	—
44	C₂³×D₁₃₇	C₂³	1	2	yes	—
45	C₂×Dic₂₇₄	—	4	2	no	—
46	C₂×Dic₂₇₄	C₂³	1	4	yes	complemented
47	C₂×C₂²⋊D₁₃₇	—	2	8	no	—
48	C₂×C₂²⋊D₁₃₇	C₂³	1	8	yes	—
49	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
50	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
51	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
52	C₂×C₄×D₁₃₇	—	2	8	no	—
53	C₂×C₄×D₁₃₇	—	4	4	no	—
54	C₂×C₄×D₁₃₇	C₂³	1	8	yes	—
55	C₂×C₄×D₁₃₇	C₂³	1	8	yes	—
56	C₄⋊Dic₁₃₇	C₂³	1	8	yes	complemented
57	#14555489	C₂³	1	8	yes	complemented
58	#14555489	C₂³	1	8	yes	complemented
59	C₂²⋊Dic₁₃₇	C₂³	1	8	yes	—
60	C₂²⋊Dic₁₃₇	C₂³	1	4	yes	—
61	C₂²⋊Dic₁₃₇	C₂³	1	8	yes	complemented
62	#14555487	C₂³	1	8	yes	—
63	#14555487	C₂³	1	8	yes	—
64	#14555487	C₂³	1	8	yes	—
65	C₄×Dic₁₃₇	C₂³	1	4	yes	complemented
66	C₄×Dic₁₃₇	C₂³	1	8	yes	complemented
67	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₁₃₇	—	4	2	no	—
70	C₂×D₄×C₁₃₇	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₃₇	—	4	2	no	—
72	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	complemented
76	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	complemented
77	C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₁₃₇	C₂³	1	4	yes	complemented
79	C₂²×D₁₃₇	—	2	4	no	—
80	C₂²×D₁₃₇	—	2	4	no	—
81	C₂²×D₁₃₇	—	2	16	no	—
82	C₂²×D₁₃₇	—	4	4	no	—
83	C₂²×D₁₃₇	—	4	16	no	—
84	C₂²×D₁₃₇	#41	1	8	yes	—
85	C₂²×D₁₃₇	C₂⁴	1	4	yes	—
86	Dic₂₇₄	—	4	4	no	—
87	C₂²⋊D₁₃₇	—	2	16	no	—
88	C₂×Dic₁₃₇	—	2	4	no	—
89	C₂×Dic₁₃₇	—	2	4	no	—
90	C₂×Dic₁₃₇	—	2	4	no	—
91	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
92	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
93	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
94	C₄×D₁₃₇	—	2	16	no	—
95	C₄×D₁₃₇	—	4	8	no	—
96	C₂³×C₁₃₇	—	2	4	no	—
97	C₂³×C₁₃₇	—	2	4	no	—
98	C₂³×C₁₃₇	—	2	2	no	—
99	C₂³×C₁₃₇	—	2	2	no	—
100	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
101	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
102	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
103	D₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
110	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
111	C₂²×C₁₃₇	—	2	4	no	—
112	C₂²×C₁₃₇	—	2	4	no	—
113	C₂²×C₁₃₇	—	2	2	no	—
114	C₂²×C₁₃₇	—	2	2	no	—
115	C₂²×C₁₃₇	—	4	4	no	—
116	C₂²×C₁₃₇	—	4	4	no	—
117	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
118	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₃₇	—	2	16	no	—
121	C₂×D₁₃₇	—	2	4	no	—
122	C₂×D₁₃₇	—	2	4	no	—
123	C₂×D₁₃₇	—	4	16	no	—
124	C₂×D₁₃₇	C₂×#41	1	8	yes	—
125	C₄×C₁₃₇	—	2	4	no	—
126	C₄×C₁₃₇	—	2	4	no	—
127	C₄×C₁₃₇	—	2	2	no	—
128	Dic₁₃₇	—	2	4	no	—
129	Dic₁₃₇	—	2	4	no	—
130	Dic₁₃₇	—	2	4	no	—
131	C₂×C₁₃₇	—	4	2	no	—
132	C₂×C₁₃₇	—	4	2	no	—
133	C₂×C₁₃₇	#551	1	4	yes	complemented
134	C₂×C₁₃₇	C₂²×#41	1	2	yes	—
135	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	D₁₃₇	—	2	8	no	—
137	C₁₃₇	C₂×#551	1	1	yes	characteristic, complemented, Sylow
138	C₂×#551	—	137	1	no	maximal, Sylow
139	#551	—	137	16	no	—
140	C₂×#126	—	137	2	no	—
141	C₂×#124	—	137	1	no	—
142	C₂×#123	—	137	2	no	—
143	C₂×#123	D₁₃₇	1	1	yes	characteristic, complemented
144	C₂×C₂²⋊Q₈	—	137	2	no	—
145	C₂×#120	—	137	1	no	—
146	C₂×C₄×D₄	—	137	2	no	—
147	C₂×#117	—	137	2	no	—
148	C₂²×C₂²⋊C₄	—	137	2	no	—
149	C₂⁵	—	137	1	no	—
150	C₂²×Q₈	—	137	1	no	—
151	C₂²×D₄	—	137	2	no	—
152	C₂²×D₄	C₂×D₁₃₇	1	1	yes	characteristic
153	C₂³×C₄	—	137	2	no	—
154	C₂³×C₄	—	137	2	no	—
155	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic, complemented
156	#126	—	137	16	no	—
157	#124	—	137	8	no	—
158	#123	—	137	16	no	—
159	#123	C₂×D₁₃₇	1	8	yes	complemented
160	C₂²⋊Q₈	—	137	16	no	—
161	#120	—	137	8	no	—
162	C₄×D₄	—	137	16	no	—
163	#117	—	137	16	no	—
164	C₂×C₄⋊C₄	—	137	2	no	—
165	C₂×C₄⋊C₄	—	137	2	no	—
166	C₂×C₄⋊C₄	—	137	2	no	—
167	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	complemented
168	C₂×C₂²⋊C₄	—	137	2	no	—
169	C₂×C₂²⋊C₄	—	137	2	no	—
170	C₂×C₂²⋊C₄	—	137	1	no	—
171	C₂×C₂²⋊C₄	—	137	2	no	—
172	C₂×C₂²⋊C₄	—	137	2	no	—
173	C₂×C₂²⋊C₄	—	137	2	no	—
174	C₂×C₂²⋊C₄	—	137	16	no	—
175	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic, complemented
177	C₂×C₄²	—	137	1	no	—
178	C₂×C₄²	—	137	2	no	—
179	C₂⁴	—	274	8	no	—
180	C₂⁴	—	548	2	no	—
181	C₂⁴	—	137	4	no	—
182	C₂⁴	—	137	2	no	—
183	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
184	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
185	C₂×Q₈	—	548	2	no	—
186	C₂×Q₈	—	137	4	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	274	8	no	—
189	C₂×D₄	—	137	8	no	—
190	C₂×D₄	C₂²×D₁₃₇	1	4	yes	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	274	8	no	—
193	C₂²×C₄	—	548	4	no	—
194	C₂²×C₄	—	137	2	no	—
195	C₂²×C₄	—	137	2	no	—
196	C₂²×C₄	—	137	2	no	—
197	C₂²×C₄	—	137	8	no	—
198	C₂²×C₄	—	137	8	no	—
199	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	complemented
203	C₄⋊C₄	—	137	8	no	—
204	C₄⋊C₄	—	137	8	no	—
205	C₄⋊C₄	—	137	8	no	—
206	C₄⋊C₄	C₂²×D₁₃₇	1	8	yes	complemented
207	C₂²⋊C₄	—	137	8	no	—
208	C₂²⋊C₄	—	137	4	no	—
209	C₂²⋊C₄	—	137	8	no	—
210	C₂²⋊C₄	—	137	8	no	—
211	C₂²⋊C₄	—	137	8	no	—
212	C₂²⋊C₄	—	137	8	no	—
213	C₂²⋊C₄	C₂²×D₁₃₇	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₁₃₇	1	4	yes	complemented
215	C₄²	—	137	4	no	—
216	C₄²	—	137	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	274	16	no	—
222	C₂³	—	274	4	no	—
223	C₂³	—	274	4	no	—
224	C₂³	—	548	4	no	—
225	C₂³	—	548	16	no	—
226	C₂³	—	137	8	no	—
227	C₂³	—	137	4	no	—
228	C₂³	C₂³×D₁₃₇	1	2	yes	—
229	C₂³	C₂³×D₁₃₇	1	2	yes	—
230	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
231	Q₈	—	548	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	274	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	274	16	no	—
239	C₂×C₄	—	274	4	no	—
240	C₂×C₄	—	274	4	no	—
241	C₂×C₄	—	274	4	no	—
242	C₂×C₄	—	548	8	no	—
243	C₂×C₄	—	137	4	no	—
244	C₂×C₄	—	137	4	no	—
245	C₂×C₄	—	137	4	no	—
246	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
247	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
248	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	274	4	no	—
256	C₂²	—	274	4	no	—
257	C₂²	—	274	16	no	—
258	C₂²	—	548	16	no	—
259	C₂²	—	137	8	no	—
260	C₂²	#41×D₁₃₇	1	4	yes	—
261	C₂²	#17984133	1	2	yes	—
262	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	274	4	no	—
267	C₄	—	274	4	no	—
268	C₄	—	274	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	274	8	no	—
272	C₂	#22578081	1	4	yes	direct product factor
273	C₂	C₂×#41×D₁₃₇	1	2	yes	—
274	C₂	C₂×#17984133	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×#123×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂×#17984056	C₂
hypercenter	C₂×#123	D₁₃₇
hypocenter	C₁₃₇	C₂×#551
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
1. #22578081	C₂	direct (N × H)
2. C₂×#17984081	C₂	semidirect
3. C₂×#17984075	C₂	semidirect
4. C₂×#17984070	C₂	semidirect
5. C₂×#17984066	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₃₇	C₂	semidirect
8. C₂×#17984046	C₂	semidirect
9. C₂×C₂²⋊C₄×D₁₃₇	C₂	semidirect
10. C₂×#123×C₁₃₇	C₂	semidirect
12. C₂²×Dic₂₇₄	C₂²	semidirect
16. #17984081	C₂²	semidirect
17. #17984075	C₂²	semidirect
18. #17984070	C₂²	semidirect
19. #17984066	C₂²	semidirect
21. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
22. #17984046	C₂²	semidirect
23. C₂×C₄⋊Dic₁₃₇	C₂²	semidirect
24. C₂×#14555489	C₂²	semidirect
25. C₂×#14555489	C₂²	semidirect
28. C₂×C₂²⋊Dic₁₃₇	C₂²	semidirect
32. C₂²⋊C₄×D₁₃₇	C₂²	semidirect
33. C₂×C₄×Dic₁₃₇	C₂²	semidirect
34. C₂×C₄×Dic₁₃₇	C₂²	semidirect
36. C₂³×C₄×C₁₃₇	C₂²	semidirect
37. #123×C₁₃₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₁₃₇	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₁₃₇	C₂²	semidirect
46. C₂×Dic₂₇₄	C₂³	semidirect
56. C₄⋊Dic₁₃₇	C₂³	semidirect
57. #14555489	C₂³	semidirect
58. #14555489	C₂³	semidirect
61. C₂²⋊Dic₁₃₇	C₂³	semidirect
65. C₄×Dic₁₃₇	C₂³	semidirect
66. C₄×Dic₁₃₇	C₂³	semidirect
75. C₂²×C₄×C₁₃₇	C₂³	semidirect
76. C₄⋊C₄×C₁₃₇	C₂³	semidirect
78. C₂²⋊C₄×C₁₃₇	C₂³	semidirect
133. C₂×C₁₃₇	#551	semidirect
137. C₁₃₇	C₂×#551	semidirect
143. C₂×#123	D₁₃₇	semidirect
155. C₂³×C₄	C₂×D₁₃₇	semidirect
159. #123	C₂×D₁₃₇	semidirect
167. C₂×C₄⋊C₄	C₂×D₁₃₇	semidirect
176. C₂×C₂²⋊C₄	C₂×D₁₃₇	semidirect
202. C₂²×C₄	C₂²×D₁₃₇	semidirect
206. C₄⋊C₄	C₂²×D₁₃₇	semidirect
214. C₂²⋊C₄	C₂²×D₁₃₇	semidirect
272. C₂	#22578081	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578081	2
C₂×#17984081	2
C₂×#17984075	2
C₂×#17984070	2
C₂×#17984066	2
C₂×#17984056	2
C₂×C₄×C₂²⋊D₁₃₇	2
C₂×#17984046	2
C₂×C₂²⋊C₄×D₁₃₇	2
C₂×#123×C₁₃₇	2
C₂×#551	137

Subgroups — C2×#22578081 #31794288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578081 #31794288