Subgroups — #22573408

Order 8576.

1540 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#16863878	C₂	1	2	yes	maximal, complemented
3	#16863875	C₂	1	6	yes	maximal, complemented
4	#16863872	C₂	1	2	yes	maximal, complemented
5	#16863868	C₂	1	6	yes	maximal, complemented
6	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#11761628	C₂	1	3	yes	maximal, complemented
8	#16863864	C₂	1	3	yes	maximal, complemented
9	C₂×#14524000	C₂	1	3	yes	maximal, complemented
10	#117×D₆₇	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761629	C₂²	1	1	yes	characteristic
13	C₂×#11761628	C₂²	1	3	yes	—
14	C₂×#11761627	C₂²	1	3	yes	—
15	#41×D₆₇	C₄	1	16	yes	complemented
16	C₂×Q₈×D₆₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₆₇	C₂²	1	3	yes	—
18	C₂²×C₄×D₆₇	C₂²	1	3	yes	—
19	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
20	C₄×Dic₁₃₄	C₂²	1	6	yes	complemented
21	#14524008	C₂²	1	6	yes	complemented
22	C₄×D₂₆₈	C₂²	1	6	yes	complemented
23	D₄×Dic₆₇	C₂²	1	6	yes	complemented
24	#14524005	C₂²	1	6	yes	complemented
25	C₄×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
26	(C₂²⋊D₆₇)⋊C₄	C₂²	1	12	yes	complemented
27	#14524002	C₂²	1	3	yes	complemented
28	#14524001	C₂²	1	6	yes	—
29	#14524000	C₂²	1	3	yes	—
30	#14524000	C₂²	1	1	yes	characteristic
31	#14524000	C₂²	1	6	yes	—
32	#14524000	C₂²	1	3	yes	complemented
33	#14524000	C₂²	1	3	yes	complemented
34	#14524000	C₂²	1	6	yes	complemented
35	#14523999	C₂²	1	6	yes	complemented
36	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	complemented
37	C₄⋊C₄×D₆₇	C₂²	1	6	yes	complemented
38	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
39	C₂²⋊C₄×D₆₇	C₂²	1	6	yes	—
40	C₂×C₄×Dic₆₇	C₂²	1	3	yes	complemented
41	C₄²×D₆₇	C₂²	1	6	yes	complemented
42	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
44	C₄×D₄×C₆₇	C₂²	1	6	yes	complemented
45	#117×C₆₇	C₂²	1	3	yes	complemented
46	C₂×C₄²×C₆₇	C₂²	1	3	yes	complemented
47	C₂³×D₆₇	C₂³	1	3	yes	—
48	#11761629	C₂×C₄	1	8	yes	complemented
49	#11761628	C₂×C₄	1	24	yes	complemented
50	#11761627	C₂×C₄	1	24	yes	complemented
51	C₂×Dic₁₃₄	C₂³	1	3	yes	—
52	Q₈×D₆₇	C₂×C₄	1	8	yes	complemented
53	C₂×D₂₆₈	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₆₇	C₂³	1	6	yes	—
55	D₄×D₆₇	C₂×C₄	1	24	yes	complemented
56	C₂²×Dic₆₇	C₂³	1	6	yes	—
57	C₂²×Dic₆₇	C₂³	1	3	yes	—
58	C₂×C₄×D₆₇	C₂×C₄	1	24	yes	complemented
59	C₂×C₄×D₆₇	C₂³	1	12	yes	—
60	C₂×C₄×D₆₇	C₂³	1	3	yes	—
61	C₂×C₄×D₆₇	C₂³	1	3	yes	—
62	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₆₇	C₂³	1	3	yes	—
64	C₄⋊Dic₆₇	C₂³	1	6	yes	—
65	Dic₆₇⋊C₄	C₂³	1	6	yes	—
66	Dic₆₇⋊C₄	C₂³	1	2	yes	—
67	Dic₆₇⋊C₄	C₂³	1	6	yes	—
68	Dic₆₇⋊C₄	C₂³	1	6	yes	—
69	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
70	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
72	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
74	C₄×Dic₆₇	C₂³	1	3	yes	—
75	C₄×Dic₆₇	C₂³	1	12	yes	—
76	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
77	C₄×Dic₆₇	C₂³	1	6	yes	—
78	#41×C₆₇	C₂×C₄	1	8	yes	complemented
79	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₆₇	C₂³	1	3	yes	—
81	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
82	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
83	C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
84	C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
85	C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₄²×C₆₇	C₂³	1	3	yes	—
87	C₄²×C₆₇	C₂³	1	6	yes	—
88	C₂³×C₆₇	C₂⁴	1	3	yes	—
89	C₂²×D₆₇	—	4	6	no	—
90	C₂²×D₆₇	C₂²×C₄	1	12	yes	—
91	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
92	C₂²×D₆₇	C₂⁴	1	3	yes	—
93	Q₈×C₆₇	C₂²×C₄	1	4	yes	—
94	D₄×C₆₇	C₂²×C₄	1	12	yes	—
95	C₂×C₄×C₆₇	—	4	6	no	—
96	C₂×C₄×C₆₇	C₂²×C₄	1	12	yes	—
97	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
98	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
99	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
100	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₆₇	C₂²×C₄	1	24	yes	—
102	C₂×Dic₆₇	—	4	6	no	—
103	C₂×Dic₆₇	C₂²×C₄	1	12	yes	—
104	C₂×Dic₆₇	C₂⁴	1	2	yes	—
105	C₂×Dic₆₇	C₂⁴	1	6	yes	—
106	C₂×Dic₆₇	C₂⁴	1	3	yes	—
107	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
108	D₂₆₈	C₂²×C₄	1	12	yes	—
109	C₄×D₆₇	—	4	12	no	—
110	C₄×D₆₇	C₂²×C₄	1	4	yes	—
111	C₄×D₆₇	C₂²×C₄	1	12	yes	—
112	C₄×D₆₇	C₂²×C₄	1	12	yes	—
113	C₄×D₆₇	C₂²×C₄	1	12	yes	—
114	Dic₁₃₄	C₂²×C₄	1	12	yes	—
115	C₂²×C₆₇	—	2	6	no	—
116	C₂²×C₆₇	C₂³×C₄	1	6	yes	—
117	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
118	C₂×D₆₇	—	2	6	no	—
119	C₂×D₆₇	—	2	2	no	—
120	C₂×D₆₇	—	4	12	no	—
121	C₂×D₆₇	C₂³×C₄	1	2	yes	—
122	C₂×D₆₇	C₂³×C₄	1	6	yes	—
123	C₄×C₆₇	—	2	6	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	C₂³×C₄	1	6	yes	—
126	C₄×C₆₇	C₂³×C₄	1	2	yes	—
127	Dic₆₇	—	2	6	no	—
128	Dic₆₇	—	2	2	no	—
129	Dic₆₇	C₂³×C₄	1	6	yes	—
130	Dic₆₇	C₂³×C₄	1	2	yes	—
131	C₂×C₆₇	—	2	6	no	—
132	C₂×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
133	C₂×C₆₇	#585	1	2	yes	—
134	D₆₇	—	2	2	no	—
135	D₆₇	—	2	6	no	—
136	#3428	—	67	1	no	maximal, Sylow
137	C₆₇	#3428	1	1	yes	characteristic, complemented, Sylow
138	C₂²×#41	—	67	1	no	—
139	#520	—	67	2	no	—
140	#520	—	67	6	no	—
141	#519	—	67	2	no	—
142	#518	—	67	6	no	—
143	C₄×#41	—	67	1	no	—
144	C₄×#41	—	67	3	no	—
145	C₄×#41	—	67	3	no	—
146	C₄×#41	D₆₇	1	1	yes	characteristic, complemented
147	C₂×#117	—	67	3	no	—
148	C₂×#117	—	67	3	no	—
149	C₂×#41	—	67	16	no	—
150	C₂×#41	—	67	1	no	—
151	C₂×#41	—	67	3	no	—
152	C₂×#41	—	67	3	no	—
153	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
154	C₂²×Q₈	—	67	1	no	—
155	C₂²×D₄	—	67	3	no	—
156	C₂³×C₄	—	67	3	no	—
157	C₄×Q₈	—	67	2	no	—
158	C₄×Q₈	—	67	6	no	—
159	C₄×Q₈	—	67	6	no	—
160	C₄×Q₈	C₂×D₆₇	1	2	yes	complemented
161	C₄×D₄	—	67	6	no	—
162	C₄×D₄	—	67	12	no	—
163	C₄×D₄	—	67	6	no	—
164	C₄×D₄	—	67	12	no	—
165	C₄×D₄	—	67	6	no	—
166	C₄×D₄	C₂×D₆₇	1	6	yes	complemented
167	#117	—	67	3	no	—
168	#117	—	67	6	no	—
169	#117	—	67	6	no	—
170	#117	—	67	3	no	—
171	#117	—	67	1	no	—
172	#117	—	67	3	no	—
173	#117	—	67	3	no	—
174	#117	—	67	6	no	—
175	#117	—	67	6	no	—
176	#117	C₂×D₆₇	1	3	yes	complemented
177	C₂×C₄⋊C₄	—	67	6	no	—
178	C₂×C₄⋊C₄	—	67	6	no	—
179	C₂×C₂²⋊C₄	—	67	6	no	—
180	C₂×C₂²⋊C₄	—	67	6	no	—
181	C₂×C₄²	—	67	3	no	—
182	C₂×C₄²	—	67	6	no	—
183	C₂×C₄²	C₂×D₆₇	1	3	yes	complemented
184	C₂⁴	—	67	3	no	—
185	#41	—	67	8	no	—
186	#41	—	67	24	no	—
187	#41	—	67	24	no	—
188	#41	C₄×D₆₇	1	8	yes	complemented
189	C₂×Q₈	—	67	8	no	—
190	C₂×Q₈	—	67	3	no	—
191	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
192	C₂×D₄	—	67	6	no	—
193	C₂×D₄	—	67	24	no	—
194	C₂×D₄	—	67	3	no	—
195	C₂×D₄	C₂²×D₆₇	1	3	yes	—
196	C₂²×C₄	—	67	6	no	—
197	C₂²×C₄	—	67	12	no	—
198	C₂²×C₄	—	67	3	no	—
199	C₂²×C₄	—	67	24	no	—
200	C₂²×C₄	—	67	3	no	—
201	C₂²×C₄	—	67	3	no	—
202	C₂²×C₄	—	67	3	no	—
203	C₂²×C₄	—	67	1	no	—
204	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
205	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
206	C₄⋊C₄	—	67	6	no	—
207	C₄⋊C₄	—	67	2	no	—
208	C₄⋊C₄	—	67	6	no	—
209	C₄⋊C₄	—	67	6	no	—
210	C₄⋊C₄	—	67	6	no	—
211	C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
212	C₂²⋊C₄	—	67	6	no	—
213	C₂²⋊C₄	—	67	6	no	—
214	C₂²⋊C₄	—	67	2	no	—
215	C₂²⋊C₄	—	67	6	no	—
216	C₂²⋊C₄	—	67	6	no	—
217	C₂²⋊C₄	C₂²×D₆₇	1	6	yes	—
218	C₄²	—	67	12	no	—
219	C₄²	—	67	6	no	—
220	C₄²	—	67	1	no	—
221	C₄²	—	67	3	no	—
222	C₄²	C₂²×D₆₇	1	6	yes	—
223	C₄²	C₂²×D₆₇	1	3	yes	—
224	C₄²	C₂²×D₆₇	1	1	yes	characteristic
225	C₂³	—	268	6	no	—
226	C₂³	—	67	12	no	—
227	C₂³	—	67	1	no	—
228	C₂³	—	67	3	no	—
229	C₂³	C₂³×D₆₇	1	3	yes	—
230	Q₈	—	67	12	no	—
231	Q₈	C₂×C₄×D₆₇	1	4	yes	—
232	D₄	—	67	24	no	—
233	D₄	—	67	12	no	—
234	D₄	C₂×C₄×D₆₇	1	12	yes	—
235	C₂×C₄	—	4	6	no	—
236	C₂×C₄	—	268	12	no	—
237	C₂×C₄	—	268	6	no	—
238	C₂×C₄	—	67	12	no	—
239	C₂×C₄	—	67	3	no	—
240	C₂×C₄	—	67	1	no	—
241	C₂×C₄	—	67	4	no	—
242	C₂×C₄	—	67	12	no	—
243	C₂×C₄	—	67	12	no	—
244	C₂×C₄	—	67	12	no	—
245	C₂×C₄	—	67	6	no	—
246	C₂×C₄	—	67	2	no	—
247	C₂×C₄	C₂×C₄×D₆₇	1	12	yes	—
248	C₂×C₄	C₂³×D₆₇	1	6	yes	—
249	C₂×C₄	C₂³×D₆₇	1	2	yes	—
250	C₂×C₄	C₂³×D₆₇	1	3	yes	—
251	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
252	C₂²	—	2	6	no	—
253	C₂²	—	268	12	no	—
254	C₂²	—	67	2	no	—
255	C₂²	—	67	6	no	—
256	C₂²	—	134	6	no	—
257	C₂²	—	134	2	no	—
258	C₂²	C₂²×C₄×D₆₇	1	6	yes	—
259	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	6	no	—
262	C₄	—	67	2	no	—
263	C₄	—	67	6	no	—
264	C₄	—	134	6	no	—
265	C₄	—	134	2	no	—
266	C₄	C₂²×C₄×D₆₇	1	2	yes	—
267	C₄	C₂²×C₄×D₆₇	1	6	yes	—
268	C₂	—	2	6	no	—
269	C₂	—	134	2	no	—
270	C₂	—	134	6	no	—
271	C₂	C₂³×C₄×D₆₇	1	1	yes	characteristic
272	C₂	#16864232	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₆₇
commutator	C₂×C₆₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₆₇
Fitting	C₄×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂⁵
hypercenter	C₄×#41	D₆₇
hypocenter	C₆₇	#3428
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #16863878	C₂	semidirect
3. #16863875	C₂	semidirect
4. #16863872	C₂	semidirect
5. #16863868	C₂	semidirect
6. #41×Dic₆₇	C₂	semidirect
7. C₄×#11761628	C₂	semidirect
8. #16863864	C₂	semidirect
9. C₂×#14524000	C₂	semidirect
10. #117×D₆₇	C₂	semidirect
11. C₄×#41×C₆₇	C₂	semidirect
15. #41×D₆₇	C₄	semidirect
19. Q₈×Dic₆₇	C₂²	semidirect
20. C₄×Dic₁₃₄	C₂²	semidirect
21. #14524008	C₂²	semidirect
22. C₄×D₂₆₈	C₂²	semidirect
23. D₄×Dic₆₇	C₂²	semidirect
24. #14524005	C₂²	semidirect
25. C₄×C₂²⋊D₆₇	C₂²	semidirect
26. (C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
27. #14524002	C₂²	semidirect
32. #14524000	C₂²	semidirect
33. #14524000	C₂²	semidirect
34. #14524000	C₂²	semidirect
35. #14523999	C₂²	semidirect
36. C₂×Dic₆₇⋊C₄	C₂²	semidirect
37. C₄⋊C₄×D₆₇	C₂²	semidirect
40. C₂×C₄×Dic₆₇	C₂²	semidirect
41. C₄²×D₆₇	C₂²	semidirect
43. C₄×Q₈×C₆₇	C₂²	semidirect
44. C₄×D₄×C₆₇	C₂²	semidirect
45. #117×C₆₇	C₂²	semidirect
46. C₂×C₄²×C₆₇	C₂²	semidirect
48. #11761629	C₂×C₄	semidirect
49. #11761628	C₂×C₄	semidirect
50. #11761627	C₂×C₄	semidirect
52. Q₈×D₆₇	C₂×C₄	semidirect
55. D₄×D₆₇	C₂×C₄	semidirect
58. C₂×C₄×D₆₇	C₂×C₄	semidirect
78. #41×C₆₇	C₂×C₄	semidirect
137. C₆₇	#3428	semidirect
146. C₄×#41	D₆₇	semidirect
160. C₄×Q₈	C₂×D₆₇	semidirect
166. C₄×D₄	C₂×D₆₇	semidirect
176. #117	C₂×D₆₇	semidirect
183. C₂×C₄²	C₂×D₆₇	semidirect
188. #41	C₄×D₆₇	semidirect