Subgroups — #31725376

Order 17152.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864201	C₂	1	4	yes	maximal, complemented
2	#22573987	C₂	1	2	yes	maximal, complemented
3	#22570695	C₂	1	12	yes	maximal, complemented
4	#22570661	C₂	1	6	yes	maximal, complemented
5	#22570637	C₂	1	3	yes	maximal, complemented
6	C₂×#16863773	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524089	C₂²	1	4	yes	—
9	#16864201	—	2	32	no	—
10	C₂×#14524071	C₂²	1	24	yes	complemented
11	C₂×#14524065	C₂²	1	12	yes	complemented
12	C₂²×D₅₃₆	C₂²	1	12	yes	complemented
13	#16864068	—	2	4	no	—
14	D₄×D₂₆₈	—	2	12	no	—
15	#16863965	C₂²	1	12	yes	complemented
16	#16863946	C₂²	1	6	yes	complemented
17	C₂×#14524040	C₂²	1	6	yes	complemented
18	#16863785	C₂²	1	24	yes	complemented
19	#16863780	C₂²	1	12	yes	complemented
20	#16863776	C₂²	1	1	yes	characteristic, complemented
21	#16863773	C₂²	1	3	yes	complemented
22	#16863773	C₂²	1	6	yes	complemented
23	#16863773	C₂²	1	12	yes	complemented
24	#16863769	C₂²	1	6	yes	complemented
25	C₂×#131×C₆₇	C₂²	1	2	yes	—
26	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
27	#405×C₆₇	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₆₇	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	complemented
30	#14524089	—	2	32	no	—
31	C₂×#11761629	—	2	4	no	—
32	C₂×#11761628	C₂³	1	12	yes	—
33	C₂²×D₂₆₈	—	2	12	no	—
34	C₂²×D₂₆₈	C₂³	1	12	yes	—
35	C₂×D₄×D₆₇	—	2	12	no	—
36	#14524071	—	2	96	no	—
37	#14524065	—	2	48	no	—
38	C₂×Dic₂₆₈	C₂³	1	4	yes	—
39	C₂×#11761617	C₂³	1	24	yes	—
40	C₂×D₅₃₆	—	2	48	no	—
41	C₂×D₅₃₆	C₂³	1	12	yes	—
42	C₂×D₅₃₆	C₂³	1	24	yes	—
43	#14524043	C₂³	1	2	yes	—
44	#14524040	—	2	12	no	—
45	#14524040	C₂³	1	6	yes	—
46	#14524040	C₂³	1	12	yes	—
47	#14524031	C₂³	1	12	yes	—
48	#14524018	—	2	24	no	—
49	#14524017	—	2	12	no	—
50	#14524013	—	2	24	no	—
51	C₄×D₂₆₈	—	2	12	no	—
52	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
53	#131×C₆₇	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₆₇	C₂³	1	6	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
56	C₄×Q₈×C₆₇	D₄	1	2	yes	—
57	C₄×D₄×C₆₇	D₄	1	6	yes	—
58	#117×C₆₇	C₂³	1	3	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
60	#97×C₆₇	C₂³	1	6	yes	—
61	C₄×C₈×C₆₇	C₂³	1	3	yes	—
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	6	yes	—
64	C₂³×D₆₇	—	2	12	no	—
65	#11761629	—	4	16	no	—
66	#11761628	—	2	48	no	—
67	C₂×Dic₁₃₄	C₂⁴	1	4	yes	—
68	C₂×D₂₆₈	—	2	48	no	—
69	C₂×D₂₆₈	—	2	12	no	—
70	C₂×D₂₆₈	—	2	12	no	—
71	C₂×D₂₆₈	—	2	24	no	—
72	C₂×D₂₆₈	—	8	24	no	—
73	C₂×D₂₆₈	C₂⁴	1	12	yes	—
74	C₂×C₂²⋊D₆₇	—	2	12	no	—
75	D₄×D₆₇	—	4	48	no	—
76	C₂×C₄×D₆₇	—	2	12	no	—
77	Dic₂₆₈	—	2	8	no	—
78	#11761617	—	2	48	no	—
79	D₅₃₆	—	2	24	no	—
80	D₅₃₆	—	2	48	no	—
81	C₄⋊Dic₆₇	—	2	4	no	—
82	(C₂×D₆₇)⋊C₄	—	2	24	no	—
83	#41×C₆₇	C₂×D₄	1	8	yes	—
84	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
88	M₁₆×C₆₇	C₂×D₄	1	24	yes	complemented
89	C₂×C₈×C₆₇	C₂×D₄	1	24	yes	complemented
90	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
91	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
92	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
93	C₂²⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
94	C₄²×C₆₇	C₂×D₄	1	6	yes	—
95	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₄²×C₆₇	C₂⁴	1	3	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
98	C₂²×D₆₇	—	2	12	no	—
99	C₂²×D₆₇	—	4	24	no	—
100	C₂²×D₆₇	—	8	24	no	—
101	Q₈×C₆₇	#127	1	4	yes	—
102	D₄×C₆₇	#127	1	12	yes	—
103	C₂×C₄×C₆₇	—	2	12	no	—
104	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₆₇	—	4	24	no	—
110	C₂×Dic₆₇	—	2	4	no	—
111	D₂₆₈	—	2	24	no	—
112	D₂₆₈	—	4	24	no	—
113	D₂₆₈	—	4	24	no	—
114	D₂₆₈	—	8	24	no	—
115	C₄×D₆₇	—	4	24	no	—
116	Dic₁₃₄	—	2	8	no	—
117	C₈×C₆₇	C₂²×D₄	1	4	yes	—
118	C₈×C₆₇	C₂²×D₄	1	12	yes	—
119	C₂²×C₆₇	—	2	6	no	—
120	C₂²×C₆₇	C₂×#127	1	6	yes	—
121	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
122	C₂×D₆₇	—	4	12	no	—
123	C₂×D₆₇	—	4	24	no	—
124	C₂×D₆₇	—	16	24	no	—
125	C₄×C₆₇	—	2	6	no	—
126	C₄×C₆₇	C₂×#127	1	6	yes	—
127	C₄×C₆₇	#576	1	4	yes	complemented
128	C₄×C₆₇	C₂³×D₄	1	2	yes	—
129	Dic₆₇	—	4	4	no	—
130	#60647	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	2	6	no	—
132	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#576	1	2	yes	—
134	D₆₇	—	8	12	no	—
135	C₂×#576	—	67	4	no	—
136	#3498	—	67	2	no	—
137	#3155	—	67	12	no	—
138	#3151	—	67	6	no	—
139	#3147	—	67	3	no	—
140	C₂×#493	—	67	3	no	—
141	C₄×#131	D₆₇	1	1	yes	characteristic, complemented
142	C₆₇	#60647	1	1	yes	characteristic, complemented, Sylow
143	C₂×H(5,𝔽₂)	—	67	4	no	—
144	#576	—	134	32	no	—
145	C₂×AGL(1,ℤ/8)	—	67	24	no	—
146	C₂×#135	—	67	12	no	—
147	C₂²×D₈	—	67	12	no	—
148	C₂×#131	C₂×D₆₇	1	2	yes	—
149	#550	—	134	4	no	—
150	D₄²	—	134	12	no	—
151	#535	—	67	12	no	—
152	#532	—	67	6	no	—
153	C₂×#127	—	67	6	no	—
154	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
155	#496	—	67	24	no	—
156	#495	—	67	12	no	—
157	#494	—	67	1	no	—
158	#493	—	67	3	no	—
159	#493	—	67	6	no	—
160	#493	—	67	12	no	—
161	#492	—	67	6	no	—
162	#405	C₂×D₆₇	1	6	yes	complemented
163	C₄×M₁₆	C₂×D₆₇	1	3	yes	complemented
164	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	complemented
165	H(5,𝔽₂)	—	134	32	no	—
166	C₂×#41	—	67	12	no	—
167	C₂×#41	—	134	4	no	—
168	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
169	C₂²×D₄	—	67	12	no	—
170	C₂²×D₄	—	134	12	no	—
171	C₂²×D₄	—	134	12	no	—
172	AGL(1,ℤ/8)	—	134	96	no	—
173	#135	—	134	48	no	—
174	C₂×Q₁₆	—	67	4	no	—
175	C₂×SD₁₆	—	67	24	no	—
176	C₂×D₈	—	67	12	no	—
177	C₂×D₈	—	67	24	no	—
178	C₂×D₈	—	134	48	no	—
179	#131	D₂₆₈	1	16	yes	complemented
180	C₂×M₁₆	C₂²×D₆₇	1	6	yes	—
181	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
182	C₄⋊Q₈	—	67	2	no	—
183	#127	—	67	6	no	—
184	#127	—	67	12	no	—
185	#127	—	134	12	no	—
186	#124	—	67	12	no	—
187	#121	—	134	12	no	—
188	#121	—	134	24	no	—
189	#120	—	134	24	no	—
190	C₄×Q₈	D₂₆₈	1	2	yes	—
191	C₄×D₄	—	134	12	no	—
192	C₄×D₄	D₂₆₈	1	6	yes	—
193	#117	C₂²×D₆₇	1	3	yes	—
194	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
195	#97	C₂²×D₆₇	1	6	yes	—
196	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₆₇	1	3	yes	—
198	C₄×C₈	C₂²×D₆₇	1	6	yes	—
199	C₂⁴	—	134	12	no	—
200	#41	—	268	16	no	—
201	#41	—	134	48	no	—
202	#41	C₂×D₂₆₈	1	8	yes	—
203	C₂×Q₈	—	67	4	no	—
204	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
205	C₂×D₄	—	268	48	no	—
206	C₂×D₄	—	536	24	no	—
207	C₂×D₄	—	67	12	no	—
208	C₂×D₄	—	134	12	no	—
209	C₂×D₄	—	134	12	no	—
210	C₂×D₄	—	134	12	no	—
211	C₂×D₄	—	134	48	no	—
212	C₂×D₄	—	134	24	no	—
213	C₂×D₄	C₂×D₂₆₈	1	3	yes	—
214	C₂²×C₄	—	134	12	no	—
215	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
216	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
217	Q₁₆	—	134	8	no	—
218	SD₁₆	—	134	48	no	—
219	D₈	—	134	24	no	—
220	D₈	—	134	48	no	—
221	M₁₆	C₂×D₂₆₈	1	24	yes	complemented
222	C₂×C₈	C₂×D₂₆₈	1	24	yes	complemented
223	C₂×C₈	C₂³×D₆₇	1	2	yes	—
224	C₂×C₈	C₂³×D₆₇	1	6	yes	—
225	C₄⋊C₄	—	134	4	no	—
226	C₄⋊C₄	C₂×D₂₆₈	1	6	yes	—
227	C₂²⋊C₄	—	134	24	no	—
228	C₂²⋊C₄	C₂×D₂₆₈	1	6	yes	—
229	C₄²	C₂×D₂₆₈	1	6	yes	—
230	C₄²	C₂³×D₆₇	1	1	yes	characteristic
231	C₄²	C₂³×D₆₇	1	3	yes	—
232	C₂³	—	268	24	no	—
233	C₂³	—	536	24	no	—
234	C₂³	—	134	12	no	—
235	C₂³	C₂²×D₂₆₈	1	3	yes	—
236	Q₈	—	134	8	no	—
237	Q₈	#14524040	1	4	yes	—
238	D₄	—	268	24	no	—
239	D₄	—	268	24	no	—
240	D₄	—	268	24	no	—
241	D₄	—	536	24	no	—
242	D₄	—	134	24	no	—
243	D₄	#14524040	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	268	24	no	—
246	C₂×C₄	—	134	4	no	—
247	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
248	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
249	C₂×C₄	C₂²×D₂₆₈	1	12	yes	—
250	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
251	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
252	C₈	C₂²×D₂₆₈	1	4	yes	—
253	C₈	C₂²×D₂₆₈	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	268	24	no	—
256	C₂²	—	268	12	no	—
257	C₂²	—	1072	24	no	—
258	C₂²	C₂×#14524040	1	6	yes	—
259	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	268	4	no	—
262	C₄	C₂×#14524040	1	6	yes	—
263	C₄	#16864201	1	4	yes	complemented
264	C₄	C₂³×D₂₆₈	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	536	12	no	—
267	C₂	C₂²×#14524040	1	1	yes	characteristic
268	C₂	C₂×#16864201	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×D₄
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#60647
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16864201	C₂	semidirect
2. #22573987	C₂	semidirect
3. #22570695	C₂	semidirect
4. #22570661	C₂	semidirect
5. #22570637	C₂	semidirect
6. C₂×#16863773	C₂	semidirect
7. C₄×#131×C₆₇	C₂	semidirect
10. C₂×#14524071	C₂²	semidirect
11. C₂×#14524065	C₂²	semidirect
12. C₂²×D₅₃₆	C₂²	semidirect
15. #16863965	C₂²	semidirect
16. #16863946	C₂²	semidirect
17. C₂×#14524040	C₂²	semidirect
18. #16863785	C₂²	semidirect
19. #16863780	C₂²	semidirect
20. #16863776	C₂²	semidirect
21. #16863773	C₂²	semidirect
22. #16863773	C₂²	semidirect
23. #16863773	C₂²	semidirect
24. #16863769	C₂²	semidirect
27. #405×C₆₇	C₂²	semidirect
28. C₄×M₁₆×C₆₇	C₂²	semidirect
29. C₂×C₄×C₈×C₆₇	C₂²	semidirect
53. #131×C₆₇	D₄	semidirect
88. M₁₆×C₆₇	C₂×D₄	semidirect
89. C₂×C₈×C₆₇	C₂×D₄	semidirect
127. C₄×C₆₇	#576	semidirect
141. C₄×#131	D₆₇	semidirect
142. C₆₇	#60647	semidirect
162. #405	C₂×D₆₇	semidirect
163. C₄×M₁₆	C₂×D₆₇	semidirect
164. C₂×C₄×C₈	C₂×D₆₇	semidirect
179. #131	D₂₆₈	semidirect
221. M₁₆	C₂×D₂₆₈	semidirect
222. C₂×C₈	C₂×D₂₆₈	semidirect
263. C₄	#16864201	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864201	2
#22573987	2
#22570695	2
#22570661	2
#22570637	2
C₂×#16863773	2
C₄×#131×C₆₇	2
#60647	67