Subgroups — #22569055

Order 8576.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524073	C₂	1	1	yes	maximal, characteristic
2	#16863879	C₂	1	1	yes	maximal, characteristic
3	#16863873	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863500	C₂	1	2	yes	maximal, complemented
5	#16863493	C₂	1	2	yes	maximal, complemented
6	#16863488	C₂	1	2	yes	maximal, complemented
7	#16863487	C₂	1	2	yes	maximal, complemented
8	#16863439	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523968	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863351	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	#14524073	C₄	1	8	yes	complemented
14	C₂×#11761618	C₂²	1	2	yes	—
15	C₂×#11761616	C₂²	1	2	yes	—
16	C₂×#11761612	C₂²	1	1	yes	characteristic
17	Q₈×Dic₆₇	C₂²	1	2	yes	—
18	C₄×Dic₁₃₄	—	2	1	no	—
19	C₄×Dic₁₃₄	—	2	1	no	—
20	#14524008	C₂²	1	2	yes	—
21	#14524008	C₂²	1	2	yes	—
22	D₄×Dic₆₇	—	2	2	no	—
23	D₄×Dic₆₇	C₂²	1	2	yes	—
24	#14524002	—	2	1	no	—
25	#14524002	C₂²	1	1	yes	characteristic
26	#14523999	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	1	no	—
28	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
29	#14523976	C₂²	1	2	yes	—
30	#14523974	C₂²	1	2	yes	—
31	#14523968	C₂²	1	2	yes	—
32	#14523968	C₂²	1	2	yes	—
33	#14523953	C₂²	1	2	yes	—
34	C₈×Dic₆₇	C₂²	1	2	yes	—
35	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#117×C₆₇	C₂²	1	1	yes	characteristic
38	#103×C₆₇	C₂²	1	1	yes	characteristic
39	#103×C₆₇	C₂²	1	1	yes	characteristic
40	#102×C₆₇	C₂²	1	1	yes	characteristic
41	#102×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×Dic₁₃₄	—	2	1	no	—
43	C₂×Dic₁₃₄	—	2	2	no	—
44	C₂×Dic₁₃₄	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₃₄	C₂³	1	2	yes	—
46	C₂²×Dic₆₇	—	2	1	no	—
47	C₂²×Dic₆₇	—	2	1	no	—
48	C₂²×Dic₆₇	—	2	2	no	—
49	#11761618	C₂×C₄	1	8	yes	complemented
50	#11761616	C₂×C₄	1	8	yes	complemented
51	#11761612	C₂×C₄	1	4	yes	complemented
52	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₄⋊Dic₆₇	—	2	1	no	—
54	C₄⋊Dic₆₇	—	2	2	no	—
55	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
56	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
57	C₄⋊Dic₆₇	C₂³	1	2	yes	—
58	Dic₆₇⋊C₄	—	2	2	no	—
59	Dic₆₇⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₆₇	—	2	2	no	—
61	C₂²⋊Dic₆₇	D₄	1	2	yes	—
62	C₄×Dic₆₇	—	2	2	no	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₄×Dic₆₇	—	2	2	no	—
65	C₄×Dic₆₇	C₂³	1	2	yes	—
66	#41×C₆₇	—	2	2	no	—
67	#41×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₆₇	—	2	1	no	—
70	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₆₇	—	2	1	no	—
79	C₄²×C₆₇	—	2	1	no	—
80	C₂³×C₆₇	—	2	1	no	—
81	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₆₇	—	2	1	no	—
83	Q₈×C₆₇	C₂²×C₄	1	2	yes	—
84	D₄×C₆₇	—	2	4	no	—
85	D₄×C₆₇	—	2	1	no	—
86	D₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#41	1	2	yes	—
94	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₆₇	—	2	4	no	—
96	C₂×Dic₆₇	—	2	2	no	—
97	C₂×Dic₆₇	—	2	2	no	—
98	C₂×Dic₆₇	—	8	2	no	—
99	C₂×Dic₆₇	—	8	1	no	—
100	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
101	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
102	Dic₁₃₄	—	2	4	no	—
103	Dic₁₃₄	—	2	2	no	—
104	Dic₁₃₄	C₂²×C₄	1	4	yes	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	1	no	—
112	C₂²×C₆₇	—	4	1	no	—
113	C₂²×C₆₇	C₄×D₄	1	2	yes	—
114	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	4	1	no	—
117	C₄×C₆₇	—	4	1	no	—
118	C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
121	Dic₆₇	—	2	4	no	—
122	Dic₆₇	—	2	2	no	—
123	Dic₆₇	—	2	2	no	—
124	Dic₆₇	—	4	2	no	—
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	—	4	2	no	—
127	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₆₇	#577	1	1	yes	characteristic
129	C₂×C₆₇	#578	1	1	yes	characteristic
130	#2947	—	67	1	no	maximal, Sylow
131	C₆₇	#2947	1	1	yes	characteristic, complemented, Sylow
132	C₂×#137	—	67	1	no	—
133	#520	—	67	1	no	—
134	#519	—	67	1	no	—
135	#441	—	67	2	no	—
136	#440	—	67	2	no	—
137	C₄×Q₁₆	—	67	2	no	—
138	C₄×SD₁₆	—	67	2	no	—
139	#428	—	67	1	no	—
140	#416	D₆₇	1	1	yes	characteristic
141	C₂×#103	—	67	1	no	—
142	#405	—	67	1	no	—
143	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
144	C₂²×Q₈	—	67	1	no	—
145	#137	—	67	8	no	—
146	C₂×Q₁₆	—	67	2	no	—
147	C₂×SD₁₆	—	67	2	no	—
148	C₂×M₁₆	—	67	1	no	—
149	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
150	C₄×Q₈	—	67	2	no	—
151	C₄×Q₈	—	67	2	no	—
152	C₄×Q₈	—	67	2	no	—
153	C₄×Q₈	—	134	1	no	—
154	C₄×Q₈	—	134	1	no	—
155	C₄×D₄	—	67	2	no	—
156	C₄×D₄	—	134	2	no	—
157	#117	—	67	1	no	—
158	#117	—	134	2	no	—
159	#117	—	134	1	no	—
160	#117	C₂×D₆₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	67	1	no	—
162	C₂×C₄⋊C₄	—	134	1	no	—
163	#107	—	67	2	no	—
164	#106	—	67	2	no	—
165	#103	—	67	2	no	—
166	#103	—	67	2	no	—
167	#103	C₂×D₆₇	1	1	yes	characteristic
168	#103	C₂×D₆₇	1	1	yes	characteristic
169	#102	C₂×D₆₇	1	1	yes	characteristic
170	#102	C₂×D₆₇	1	1	yes	characteristic
171	#97	—	67	2	no	—
172	C₄×C₈	—	67	2	no	—
173	#41	—	2	2	no	—
174	#41	C₄×D₆₇	1	4	yes	—
175	C₂×Q₈	—	67	4	no	—
176	C₂×Q₈	—	67	2	no	—
177	C₂×Q₈	—	134	1	no	—
178	C₂×Q₈	—	134	2	no	—
179	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	134	2	no	—
184	C₂²×C₄	—	134	1	no	—
185	C₂²×C₄	—	134	1	no	—
186	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
187	Q₁₆	—	67	8	no	—
188	SD₁₆	—	67	8	no	—
189	M₁₆	—	67	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	67	2	no	—
192	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₄⋊C₄	—	67	1	no	—
195	C₄⋊C₄	—	67	1	no	—
196	C₄⋊C₄	—	67	2	no	—
197	C₄⋊C₄	—	134	2	no	—
198	C₄⋊C₄	—	134	2	no	—
199	C₄⋊C₄	—	134	2	no	—
200	C₄⋊C₄	—	134	1	no	—
201	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	67	2	no	—
205	C₂²⋊C₄	—	134	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	67	2	no	—
208	C₄²	—	134	2	no	—
209	C₄²	—	134	2	no	—
210	C₄²	—	134	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	D₄×D₆₇	1	1	yes	characteristic
213	Q₈	—	2	1	no	—
214	Q₈	—	67	4	no	—
215	Q₈	—	134	4	no	—
216	Q₈	—	134	2	no	—
217	Q₈	C₂×C₄×D₆₇	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂×C₄×D₆₇	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	536	1	no	—
226	C₂×C₄	—	536	2	no	—
227	C₂×C₄	—	67	2	no	—
228	C₂×C₄	—	67	2	no	—
229	C₂×C₄	—	134	2	no	—
230	C₂×C₄	—	134	4	no	—
231	C₂×C₄	—	134	2	no	—
232	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
233	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
234	C₂×C₄	#11761627	1	2	yes	—
235	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	67	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
245	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	268	2	no	—
250	C₄	—	134	4	no	—
251	C₄	—	134	2	no	—
252	C₄	—	134	2	no	—
253	C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
254	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
255	C₄	C₂×#11761627	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
259	C₂	#16864203	1	1	yes	characteristic
260	C₂	#16864210	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#416×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#416	D₆₇
hypocenter	C₆₇	#2947
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
3. #16863873	C₂	semidirect
4. #16863500	C₂	semidirect
5. #16863493	C₂	semidirect
6. #16863488	C₂	semidirect
7. #16863487	C₂	semidirect
8. #16863439	C₂	semidirect
9. C₂×#14523968	C₂	semidirect
10. #16863351	C₂	semidirect
13. #14524073	C₄	semidirect
44. C₂×Dic₁₃₄	C₂×C₄	semidirect
49. #11761618	C₂×C₄	semidirect
50. #11761616	C₂×C₄	semidirect
51. #11761612	C₂×C₄	semidirect
131. C₆₇	#2947	semidirect