Subgroups — #30749041

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569143	C₂	1	1	yes	maximal, characteristic
2	#22560373	C₂	1	1	yes	maximal, characteristic
3	#22558774	C₂	1	2	yes	maximal
4	#22558068	C₂	1	2	yes	maximal
5	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863526	—	2	1	no	—
8	#16863526	C₄	1	2	yes	complemented
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	complemented
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523975	—	2	1	no	—
13	#16863422	—	2	1	no	—
14	#16863420	C₂²	1	1	yes	characteristic
15	#16863351	—	2	1	no	—
16	#16863262	C₂²	1	2	yes	—
17	#16863134	—	2	4	no	—
18	#431×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#335×C₆₇	—	2	1	no	—
21	#334×C₆₇	—	2	1	no	—
22	C₂×#11761612	—	2	1	no	—
23	#14524043	Q₈	1	1	yes	characteristic
24	#14524043	Q₈	1	1	yes	characteristic
25	#14524035	—	2	1	no	—
26	#14524022	—	2	1	no	—
27	#14524022	D₄	1	2	yes	—
28	C₄×Dic₁₃₄	—	2	1	no	—
29	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
30	#14524002	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	#14523977	—	4	2	no	—
33	#14523975	—	4	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₆₇⋊C₈	—	2	2	no	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	#14523953	—	2	2	no	—
39	C₈×Dic₆₇	—	2	2	no	—
40	C₆₇⋊#95	—	2	2	no	—
41	C₆₇⋊#95	—	2	2	no	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₇	—	2	2	no	—
46	C₂²⋊C₈×C₆₇	—	2	2	no	—
47	C₄×C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×Dic₁₃₄	—	2	1	no	—
53	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	#11761612	—	4	1	no	—
56	#11761612	—	4	1	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₄⋊Dic₆₇	—	2	1	no	—
59	C₄⋊Dic₆₇	—	2	1	no	—
60	C₄⋊Dic₆₇	—	2	1	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
64	Dic₆₇⋊C₄	—	2	2	no	—
65	Dic₆₇⋊C₄	—	2	2	no	—
66	C₂²⋊Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
79	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₂³×C₆₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	—	4	1	no	—
87	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
97	C₂×Dic₆₇	—	2	2	no	—
98	C₂×Dic₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	8	2	no	—
100	C₂×Dic₆₇	—	8	2	no	—
101	C₂×Dic₆₇	—	8	2	no	—
102	Dic₁₃₄	—	4	2	no	—
103	Dic₁₃₄	—	4	2	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	8	2	no	—
107	C₈×C₆₇	#103	1	2	yes	—
108	C₈×C₆₇	#103	1	2	yes	—
109	C₆₇⋊C₈	—	4	2	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	#380	1	1	yes	characteristic
113	C₂²×C₆₇	#460	1	1	yes	characteristic
114	C₂²×C₆₇	#462	1	1	yes	characteristic
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
120	C₄×C₆₇	#417	1	1	yes	characteristic
121	C₄×C₆₇	#475	1	1	yes	characteristic
122	C₄×C₆₇	#477	1	1	yes	characteristic
123	Dic₆₇	—	4	2	no	—
124	Dic₆₇	—	4	2	no	—
125	Dic₆₇	—	16	2	no	—
126	#8675	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	1	no	—
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	#1803	1	1	yes	characteristic
130	C₂×C₆₇	#1812	1	1	yes	characteristic
131	C₂×C₆₇	#1814	1	1	yes	characteristic
132	#2960	—	67	1	no	—
133	#1846	—	67	1	no	—
134	#1564	D₆₇	1	1	yes	characteristic
135	#1518	—	67	2	no	—
136	#1286	—	67	2	no	—
137	C₆₇	#8675	1	1	yes	characteristic, complemented, Sylow
138	#533	—	67	1	no	—
139	#446	—	67	2	no	—
140	#446	—	134	1	no	—
141	C₈×Q₈	—	67	2	no	—
142	C₈×Q₈	—	134	1	no	—
143	#431	C₂×D₆₇	1	2	yes	—
144	C₂×#107	—	134	1	no	—
145	C₂×#106	—	134	1	no	—
146	#424	—	134	1	no	—
147	#423	—	67	1	no	—
148	#405	—	134	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#382	—	67	2	no	—
151	#335	—	2	1	no	—
152	#334	—	2	1	no	—
153	#328	—	134	4	no	—
154	C₂×M₁₆	—	134	1	no	—
155	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
156	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
157	C₄⋊Q₈	—	67	1	no	—
158	C₄⋊Q₈	—	67	1	no	—
159	#125	—	134	1	no	—
160	C₂²⋊Q₈	—	67	2	no	—
161	C₂²⋊Q₈	—	134	1	no	—
162	C₄×Q₈	—	67	2	no	—
163	C₄×Q₈	—	134	1	no	—
164	#117	—	134	1	no	—
165	C₂×C₄⋊C₄	—	134	2	no	—
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	#107	—	268	2	no	—
168	#106	—	268	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	67	2	no	—
171	C₄⋊C₈	—	134	2	no	—
172	C₄⋊C₈	—	134	2	no	—
173	C₄⋊C₈	—	134	1	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	#97	—	134	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	134	2	no	—
178	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
179	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	#95	—	134	2	no	—
183	#95	—	134	2	no	—
184	C₂×Q₈	—	67	2	no	—
185	C₂×Q₈	—	134	1	no	—
186	C₂²×C₄	—	268	2	no	—
187	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
189	C₂²×C₄	#11761629	1	1	yes	characteristic
190	M₁₆	—	268	1	no	—
191	M₁₆	—	268	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	134	2	no	—
196	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
197	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	#11761617	1	2	yes	—
201	C₂×C₈	Dic₂₆₈	1	2	yes	—
202	C₄⋊C₄	—	268	2	no	—
203	C₄⋊C₄	—	268	2	no	—
204	C₄⋊C₄	—	67	2	no	—
205	C₄⋊C₄	—	134	2	no	—
206	C₄⋊C₄	—	134	2	no	—
207	C₄⋊C₄	—	134	1	no	—
208	C₄⋊C₄	—	134	1	no	—
209	C₄⋊C₄	—	134	1	no	—
210	C₂²⋊C₄	—	134	2	no	—
211	C₄²	—	134	2	no	—
212	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
214	C₄²	Q₈×D₆₇	1	1	yes	characteristic
215	C₄²	#11761627	1	1	yes	characteristic
216	C₂³	#14524018	1	1	yes	characteristic
217	Q₈	—	268	2	no	—
218	Q₈	—	268	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	536	2	no	—
222	C₂×C₄	—	536	2	no	—
223	C₂×C₄	—	536	2	no	—
224	C₂×C₄	—	134	2	no	—
225	C₂×C₄	—	134	2	no	—
226	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
228	C₂×C₄	#14524001	1	1	yes	characteristic
229	C₂×C₄	#14524014	1	1	yes	characteristic
230	C₂×C₄	#14524019	1	1	yes	characteristic
231	C₂×C₄	#14524023	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
234	C₂×C₄	#14524071	1	1	yes	characteristic
235	C₂×C₄	#14524076	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	268	2	no	—
240	C₈	#14523968	1	2	yes	—
241	C₈	#14523968	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#16863253	1	1	yes	characteristic
245	C₂²	#16863597	1	1	yes	characteristic
246	C₂²	#16863607	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	268	2	no	—
252	C₄	—	268	2	no	—
253	C₄	—	1072	2	no	—
254	C₄	C₂×#14523968	1	1	yes	characteristic
255	C₄	#16863398	1	1	yes	characteristic
256	C₄	#16863682	1	1	yes	characteristic
257	C₄	#16863692	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#22560215	1	1	yes	characteristic
261	C₂	#22560252	1	1	yes	characteristic
262	C₂	#22560264	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#8675
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
8. #16863526	C₄	semidirect
10. C₈×Dic₁₃₄	C₄	semidirect
137. C₆₇	#8675	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
#22560373	2
#22558774	2
#22558068	2
C₆₇×#1564	2
#8675	67