Subgroups — #30760091

Order 17152.

454 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	#22561133	C₂	1	2	yes	maximal
2	#22560927	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×Dic₆₇⋊C₈	—	2	1	no	—
9	C₂×#14523967	—	2	2	no	—
10	C₂×#14523967	—	2	2	no	—
11	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863150	—	2	1	no	—
13	#16863142	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	—	2	4	no	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	1	no	—
21	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₇	—	2	1	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	Dic₆₇⋊C₈	—	4	2	no	—
28	#14523967	—	4	4	no	—
29	#14523967	—	4	4	no	—
30	C₆₇⋊#95	—	2	1	no	—
31	C₂²×Q₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₇	—	2	2	no	—
35	C₄⋊Q₈×C₆₇	—	2	2	no	—
36	C₄⋊Q₈×C₆₇	—	2	2	no	—
37	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	#107×C₆₇	—	2	4	no	—
42	#107×C₆₇	—	2	4	no	—
43	C₄×C₈×C₆₇	—	2	2	no	—
44	C₂²×Dic₆₇	—	2	1	no	—
45	C₂²×Dic₆₇	—	2	1	no	—
46	C₂×C₆₇⋊C₈	—	4	1	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	1	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	1	no	—
52	C₄⋊Dic₆₇	—	2	1	no	—
53	C₄⋊Dic₆₇	—	2	2	no	—
54	C₄⋊Dic₆₇	D₈	1	2	yes	—
55	C₄×Dic₆₇	—	4	1	no	—
56	C₄×Dic₆₇	—	4	1	no	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	4	2	no	—
60	C₂×Q₈×C₆₇	—	4	2	no	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂×Q₈×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	no	—
69	C₂×C₈×C₆₇	—	2	1	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
72	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
73	C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	—	4	2	no	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	#41	1	2	yes	—
79	C₂³×C₆₇	#121	1	1	yes	characteristic
80	Q₈×C₆₇	—	4	4	no	—
81	Q₈×C₆₇	—	4	4	no	—
82	Q₈×C₆₇	—	8	2	no	—
83	Q₈×C₆₇	—	8	2	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	4	4	no	—
87	C₂×C₄×C₆₇	—	8	2	no	—
88	C₂×C₄×C₆₇	—	8	2	no	—
89	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#121	1	2	yes	—
91	C₂×C₄×C₆₇	#123	1	2	yes	—
92	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
97	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	4	1	no	—
102	C₂×Dic₆₇	—	4	1	no	—
103	C₂×Dic₆₇	—	4	1	no	—
104	C₂×Dic₆₇	—	4	1	no	—
105	C₂×Dic₆₇	—	4	1	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	—	2	2	no	—
108	C₆₇⋊C₈	—	8	1	no	—
109	C₆₇⋊C₈	—	8	2	no	—
110	C₂²×C₆₇	#394	1	1	yes	characteristic
111	C₂²×C₆₇	#461	1	1	yes	characteristic
112	C₂²×C₆₇	#462	1	1	yes	characteristic
113	C₂²×C₆₇	#466	1	1	yes	characteristic
114	C₂²×C₆₇	#469	1	1	yes	characteristic
115	C₂²×C₆₇	#473	1	1	yes	characteristic
116	C₂²×C₆₇	#473	1	1	yes	characteristic
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	8	4	no	—
119	C₄×C₆₇	#448	1	1	yes	characteristic
120	C₄×C₆₇	#451	1	1	yes	characteristic
121	C₄×C₆₇	#459	1	2	yes	—
122	C₄×C₆₇	#463	1	2	yes	—
123	C₄×C₆₇	#487	1	1	yes	characteristic
124	C₄×C₆₇	#491	1	1	yes	characteristic
125	Dic₆₇	—	4	2	no	—
126	Dic₆₇	—	8	1	no	—
127	#10545	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	#1673	1	1	yes	characteristic
129	C₂×C₆₇	#1674	1	1	yes	characteristic
130	C₂×C₆₇	#1693	1	1	yes	characteristic
131	C₂×C₆₇	#1694	1	1	yes	characteristic
132	C₂×C₆₇	#2043	1	1	yes	characteristic
133	C₂×C₆₇	#2047	1	1	yes	characteristic
134	C₂×C₆₇	#2054	1	1	yes	characteristic
135	C₂×#500	D₆₇	1	1	yes	characteristic
136	#1981	—	67	2	no	—
137	#1939	—	67	1	no	—
138	C₂×#332	—	67	2	no	—
139	#1277	—	67	1	no	—
140	C₆₇	#10545	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
142	#500	—	2	4	no	—
143	C₂×#107	—	2	1	no	—
144	C₂×#107	—	2	1	no	—
145	C₂×C₄⋊C₈	—	67	1	no	—
146	C₂×C₄⋊C₈	—	67	2	no	—
147	C₂×C₄⋊C₈	—	134	1	no	—
148	C₂×#103	—	134	2	no	—
149	C₂×#103	—	134	2	no	—
150	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	67	1	no	—
152	#336	—	134	1	no	—
153	#332	—	134	4	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×C₈	—	134	2	no	—
156	C₂²×C₈	—	134	1	no	—
157	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	C₂²⋊D₆₇	1	4	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	67	1	no	—
166	C₂×C₄⋊C₄	—	67	1	no	—
167	C₂×C₄²	—	134	1	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	#107	—	2	4	no	—
170	#107	—	2	4	no	—
171	C₄⋊C₈	—	268	2	no	—
172	C₄⋊C₈	—	134	2	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	#103	—	268	4	no	—
176	#103	—	268	4	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	134	1	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	134	1	no	—
187	C₂²×C₄	—	134	1	no	—
188	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²×C₄	#11761627	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	268	1	no	—
196	C₂×C₈	—	268	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	—	268	1	no	—
199	C₂×C₈	—	268	2	no	—
200	C₂×C₈	—	268	1	no	—
201	C₂×C₈	#11761615	1	2	yes	—
202	C₂×C₈	#11761618	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	67	2	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₄⋊C₄	—	134	1	no	—
210	C₄²	—	268	1	no	—
211	C₄²	—	268	1	no	—
212	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
213	C₄²	#11761627	1	2	yes	—
214	C₂³	#14524014	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	268	1	no	—
225	C₂×C₄	—	268	1	no	—
226	C₂×C₄	—	268	1	no	—
227	C₂×C₄	—	268	1	no	—
228	C₂×C₄	—	268	1	no	—
229	C₂×C₄	—	134	2	no	—
230	C₂×C₄	#14524012	1	1	yes	characteristic
231	C₂×C₄	#14524015	1	2	yes	—
232	C₂×C₄	#14524024	1	2	yes	—
233	C₂×C₄	#14524030	1	1	yes	characteristic
234	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
235	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
236	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
237	C₂×C₄	#14524062	1	1	yes	characteristic
238	C₂×C₄	#14524067	1	2	yes	—
239	C₂×C₄	#14524074	1	1	yes	characteristic
240	C₂×C₄	#14524074	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	536	1	no	—
244	C₈	—	536	2	no	—
245	C₂²	#16863310	1	1	yes	characteristic
246	C₂²	#16863598	1	1	yes	characteristic
247	C₂²	#16863603	1	1	yes	characteristic
248	C₂²	#16863629	1	1	yes	characteristic
249	C₂²	#16863646	1	1	yes	characteristic
250	C₂²	#16863668	1	1	yes	characteristic
251	C₂²	#16863668	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	—	268	2	no	—
255	C₄	—	536	1	no	—
256	C₄	#16863535	1	1	yes	characteristic
257	C₄	#16863552	1	1	yes	characteristic
258	C₄	#16863589	1	2	yes	—
259	C₄	#16863609	1	2	yes	—
260	C₄	#16863746	1	1	yes	characteristic
261	C₄	#16863764	1	1	yes	characteristic
262	C₂	#22559595	1	1	yes	characteristic
263	C₂	#22559600	1	1	yes	characteristic
264	C₂	#22559719	1	1	yes	characteristic
265	C₂	#22559724	1	1	yes	characteristic
266	C₂	#22561452	1	1	yes	characteristic
267	C₂	#22561474	1	1	yes	characteristic
268	C₂	#22561517	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#10545
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
140. C₆₇	#10545	semidirect

Maximal subgroups

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

subgroup	index
#22561133	2
#22560927	2
C₂×#16863142	2
#22558044	2
C₂×#500×C₆₇	2
#10545	67