Subgroups — #22564631

Order 8576.

468 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×#14524036	C₂	1	2	yes	maximal
2	#16863336	C₂	1	2	yes	maximal, complemented
3	#16863321	C₂	1	2	yes	maximal
4	#16863294	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863294	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863281	C₂	1	2	yes	maximal, complemented
7	#16863266	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863256	C₂	1	1	yes	maximal, characteristic
9	#16863256	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#14524036	—	2	8	no	—
13	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
19	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
24	C₆₇⋊#95	C₂²	1	2	yes	—
25	C₆₇⋊#95	C₂²	1	2	yes	—
26	C₆₇⋊#95	C₂²	1	2	yes	—
27	C₆₇⋊#95	C₂²	1	2	yes	—
28	C₆₇⋊#95	C₂²	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#95×C₆₇	C₂²	1	1	yes	characteristic
35	#95×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×Dic₆₇	C₂³	1	2	yes	—
37	C₂²×Dic₆₇	C₂³	1	2	yes	—
38	C₂²×Dic₆₇	C₂³	1	2	yes	—
39	C₂²×Dic₆₇	C₂³	1	2	yes	—
40	C₄⋊Dic₆₇	—	2	4	no	—
41	C₄⋊Dic₆₇	—	2	2	no	—
42	C₄⋊Dic₆₇	—	2	2	no	—
43	C₄⋊Dic₆₇	—	2	4	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	Dic₆₇⋊C₄	—	2	4	no	—
47	Dic₆₇⋊C₄	—	2	4	no	—
48	C₂²⋊Dic₆₇	—	2	4	no	—
49	C₂²⋊Dic₆₇	—	2	4	no	—
50	C₄×Dic₆₇	—	2	4	no	—
51	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	—	2	2	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂²⋊C₄×C₆₇	D₄	1	4	yes	—
63	C₄²×C₆₇	—	2	2	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	—	2	1	no	—
66	C₂³×C₆₇	—	2	1	no	—
67	C₂³×C₆₇	—	2	1	no	—
68	C₂³×C₆₇	—	2	1	no	—
69	C₂³×C₆₇	—	2	1	no	—
70	C₂³×C₆₇	—	2	1	no	—
71	C₂³×C₆₇	C₂×D₄	1	2	yes	—
72	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₆₇	#41	1	2	yes	—
89	C₂×C₄×C₆₇	#41	1	2	yes	—
90	C₂×C₄×C₆₇	#41	1	2	yes	—
91	C₂×C₄×C₆₇	#41	1	2	yes	—
92	C₂×Dic₆₇	—	2	2	no	—
93	C₂×Dic₆₇	—	2	2	no	—
94	C₂×Dic₆₇	—	2	2	no	—
95	C₂×Dic₆₇	—	2	2	no	—
96	C₂×Dic₆₇	—	2	2	no	—
97	C₂×Dic₆₇	—	2	2	no	—
98	C₂×Dic₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	2	2	no	—
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	—	2	2	no	—
107	C₂²×C₆₇	—	4	1	no	—
108	C₂²×C₆₇	—	4	1	no	—
109	C₂²×C₆₇	—	4	1	no	—
110	C₂²×C₆₇	—	4	1	no	—
111	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	4	1	no	—
122	C₄×C₆₇	—	4	1	no	—
123	C₄×C₆₇	#126	1	4	yes	complemented
124	Dic₆₇	—	4	2	no	—
125	Dic₆₇	—	4	2	no	—
126	Dic₆₇	—	4	2	no	—
127	Dic₆₇	—	4	2	no	—
128	C₂×C₆₇	—	4	2	no	—
129	C₂×C₆₇	C₂×#126	1	1	yes	characteristic
130	C₂×C₆₇	#532	1	1	yes	characteristic
131	C₂×C₆₇	#541	1	1	yes	characteristic
132	C₂×C₆₇	#547	1	1	yes	characteristic
133	C₂×C₆₇	#547	1	1	yes	characteristic
134	C₂×C₆₇	#552	1	1	yes	characteristic
135	C₂×C₆₇	#556	1	1	yes	characteristic
136	#2496	—	67	1	no	maximal, Sylow
137	C₆₇	#2496	1	1	yes	characteristic, complemented, Sylow
138	C₂×#126	—	67	2	no	—
139	#400	—	67	2	no	—
140	#397	—	67	2	no	—
141	#389	—	67	1	no	—
142	#389	—	67	1	no	—
143	#387	—	67	2	no	—
144	#384	—	67	1	no	—
145	#380	—	67	1	no	—
146	#380	—	67	1	no	—
147	C₄×C₄⋊C₄	—	67	1	no	—
148	C₄×C₂²⋊C₄	D₆₇	1	1	yes	characteristic
149	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
150	#126	—	134	8	no	—
151	C₂×C₄⋊C₄	—	67	2	no	—
152	C₂×C₄⋊C₄	—	67	2	no	—
153	C₂×C₄⋊C₄	—	67	2	no	—
154	C₂×C₄⋊C₄	—	67	1	no	—
155	C₂×C₄⋊C₄	—	67	1	no	—
156	C₂×C₄⋊C₄	—	67	1	no	—
157	C₂×C₄⋊C₄	—	67	1	no	—
158	C₂×C₄⋊C₄	—	67	2	no	—
159	C₂×C₂²⋊C₄	—	67	2	no	—
160	C₂×C₂²⋊C₄	—	67	2	no	—
161	C₂×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
163	C₂×C₄²	—	67	2	no	—
164	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
166	#95	—	67	2	no	—
167	#95	—	67	2	no	—
168	#95	—	67	2	no	—
169	#95	—	67	2	no	—
170	#95	—	67	2	no	—
171	#95	C₂×D₆₇	1	1	yes	characteristic
172	#95	C₂×D₆₇	1	1	yes	characteristic
173	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	67	2	no	—
178	C₂²×C₄	—	67	2	no	—
179	C₂²×C₄	—	67	2	no	—
180	C₂²×C₄	—	67	2	no	—
181	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
187	C₄⋊C₄	—	134	4	no	—
188	C₄⋊C₄	—	134	4	no	—
189	C₄⋊C₄	—	134	4	no	—
190	C₄⋊C₄	—	134	2	no	—
191	C₄⋊C₄	—	134	2	no	—
192	C₄⋊C₄	—	134	4	no	—
193	C₄⋊C₄	—	134	2	no	—
194	C₄⋊C₄	—	134	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	134	4	no	—
197	C₂²⋊C₄	—	134	4	no	—
198	C₂²⋊C₄	D₂₆₈	1	4	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	134	4	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	C₂×D₂₆₈	1	2	yes	—
209	C₂³	C₂³×D₆₇	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	134	2	no	—
224	C₂×C₄	—	134	2	no	—
225	C₂×C₄	—	134	2	no	—
226	C₂×C₄	—	134	2	no	—
227	C₂×C₄	—	134	2	no	—
228	C₂×C₄	—	134	2	no	—
229	C₂×C₄	—	134	2	no	—
230	C₂×C₄	—	134	2	no	—
231	C₂×C₄	—	134	2	no	—
232	C₂×C₄	—	134	2	no	—
233	C₂×C₄	—	134	2	no	—
234	C₂×C₄	—	134	2	no	—
235	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
236	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
237	C₂×C₄	#11761627	1	2	yes	—
238	C₂×C₄	#11761627	1	2	yes	—
239	C₂×C₄	#11761628	1	2	yes	—
240	C₂×C₄	#11761628	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
249	C₂²	C₂×#11761627	1	1	yes	characteristic
250	C₂²	C₂×#11761627	1	1	yes	characteristic
251	C₂²	C₂×#11761628	1	1	yes	characteristic
252	C₂²	C₂×#11761628	1	1	yes	characteristic
253	C₂²	#14524090	1	1	yes	characteristic
254	C₂²	#14524090	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	268	2	no	—
261	C₄	—	268	2	no	—
262	C₄	—	268	2	no	—
263	C₄	—	268	2	no	—
264	C₄	#14524036	1	4	yes	complemented
265	C₂	—	4	2	no	—
266	C₂	C₂×#14524036	1	1	yes	characteristic
267	C₂	#16863946	1	1	yes	characteristic
268	C₂	#16864008	1	1	yes	characteristic
269	C₂	#16864050	1	1	yes	characteristic
270	C₂	#16864050	1	1	yes	characteristic
271	C₂	#16864086	1	1	yes	characteristic
272	C₂	#16864116	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₆₇	C₂
hypercenter	C₄×C₂²⋊C₄	D₆₇
hypocenter	C₆₇	#2496
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #16863336	C₂	semidirect
4. #16863294	C₂	semidirect
5. #16863294	C₂	semidirect
6. #16863281	C₂	semidirect
7. #16863266	C₂	semidirect
10. C₄×C₄⋊Dic₆₇	C₂	semidirect
123. C₄×C₆₇	#126	semidirect
137. C₆₇	#2496	semidirect
264. C₄	#14524036	semidirect