Subgroups — #30874662

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22563164	C₂	1	2	yes	maximal
3	#22560976	C₂	1	4	yes	maximal
4	#22560967	C₂	1	2	yes	maximal
5	#22560370	C₂	1	2	yes	maximal
6	#22560366	C₂	1	1	yes	maximal, characteristic
7	#106×Dic₆₇	C₂	1	2	yes	maximal
8	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₃₄	C₄	1	8	yes	complemented
11	C₂×#14523976	C₂²	1	4	yes	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	C₂×#14523975	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	#16863302	—	2	2	no	—
18	#16863283	—	2	4	no	—
19	#16863266	C₂²	1	2	yes	—
20	#16863261	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₆₇	—	2	2	no	—
23	#16863158	C₂²	1	4	yes	—
24	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	#340×C₆₇	C₂²	1	2	yes	—
29	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₃₄	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
37	C₂×Dic₆₇⋊C₄	D₄	1	2	yes	—
38	C₂×C₄×Dic₆₇	—	2	4	no	—
39	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
40	#14523976	—	4	4	no	—
41	#14523975	—	2	4	no	—
42	#14523975	—	4	2	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
45	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
46	C₆₇⋊#95	—	2	4	no	—
47	C₆₇⋊#95	—	2	4	no	—
48	C₆₇⋊#95	—	2	4	no	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#106×C₆₇	—	2	2	no	—
56	#106×C₆₇	Q₈	1	4	yes	complemented
57	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
58	#95×C₆₇	—	2	2	no	—
59	C₂×Dic₁₃₄	—	2	4	no	—
60	C₂×Dic₁₃₄	C₄⋊C₄	1	4	yes	—
61	C₂²×Dic₆₇	—	2	4	no	—
62	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
63	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	4	4	no	—
70	C₄⋊Dic₆₇	—	8	2	no	—
71	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
72	Dic₆₇⋊C₄	C₄⋊C₄	1	8	yes	—
73	C₄×Dic₆₇	—	8	4	no	—
74	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
75	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₆₇	—	2	1	no	—
82	C₂×C₈×C₆₇	—	2	1	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₆₇	#41	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
91	C₄²×C₆₇	—	4	4	no	—
92	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	#135	1	4	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₆₇	—	2	4	no	—
112	C₂×Dic₆₇	—	2	4	no	—
113	C₂×Dic₆₇	—	4	4	no	—
114	C₂×Dic₆₇	—	4	4	no	—
115	C₂×Dic₆₇	—	4	4	no	—
116	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
118	Dic₁₃₄	—	2	8	no	—
119	C₈×C₆₇	—	2	2	no	—
120	C₈×C₆₇	C₄×Q₈	1	4	yes	—
121	C₆₇⋊C₈	—	2	4	no	—
122	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
124	C₂²×C₆₇	#550	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#135	1	2	yes	—
126	C₂²×C₆₇	#577	1	2	yes	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	4	4	no	—
129	C₄×C₆₇	—	8	2	no	—
130	C₄×C₆₇	#427	1	4	yes	—
131	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₆₇	#520	1	1	yes	characteristic
133	C₄×C₆₇	#557	1	1	yes	characteristic
134	C₄×C₆₇	Q₈²	1	1	yes	characteristic
135	Dic₆₇	—	2	4	no	—
136	Dic₆₇	—	2	4	no	—
137	Dic₆₇	—	8	4	no	—
138	#22725	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
141	C₂×C₆₇	#2912	1	1	yes	characteristic
142	C₂×C₆₇	#3381	1	2	yes	—
143	C₂×C₆₇	#3394	1	2	yes	—
144	C₂×C₈×Q₈	—	67	1	no	—
145	#2355	—	67	2	no	—
146	#1949	—	67	4	no	—
147	#1947	—	67	2	no	—
148	#1846	—	67	2	no	—
149	#1844	—	67	1	no	—
150	C₄×#106	—	67	2	no	—
151	C₄×#106	D₆₇	1	1	yes	characteristic
152	C₆₇	#22725	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄×Q₈	—	67	1	no	—
154	C₈×Q₈	—	67	8	no	—
155	C₂×#107	—	67	4	no	—
156	C₂×#106	—	67	2	no	—
157	C₂×#106	—	67	2	no	—
158	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
159	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	67	2	no	—
161	C₂×C₄⋊C₈	—	67	1	no	—
162	C₂×C₄×C₈	—	67	2	no	—
163	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
164	#391	—	134	2	no	—
165	#387	—	134	4	no	—
166	#384	—	67	2	no	—
167	#382	—	67	4	no	—
168	C₄×C₄⋊C₄	—	67	4	no	—
169	C₄×C₄⋊C₄	—	134	2	no	—
170	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
171	#340	—	67	4	no	—
172	#340	C₂×D₆₇	1	2	yes	—
173	C₂²×Q₈	—	67	1	no	—
174	C₂²×C₈	—	67	2	no	—
175	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	C₄×Q₈	—	67	4	no	—
178	C₄×Q₈	—	67	4	no	—
179	C₂×C₄⋊C₄	—	67	2	no	—
180	C₂×C₄⋊C₄	—	67	1	no	—
181	C₂×C₄⋊C₄	—	67	4	no	—
182	C₂×C₄⋊C₄	—	134	2	no	—
183	C₂×C₄⋊C₄	—	134	2	no	—
184	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	67	2	no	—
188	C₂×C₄²	—	134	4	no	—
189	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
190	#107	—	268	4	no	—
191	#106	—	2	2	no	—
192	#106	—	268	2	no	—
193	#106	—	134	4	no	—
194	#106	Dic₁₃₄	1	4	yes	complemented
195	C₄⋊C₈	—	67	8	no	—
196	C₄⋊C₈	—	67	4	no	—
197	C₄×C₈	—	67	8	no	—
198	C₄×C₈	C₄×D₆₇	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	134	4	no	—
201	#95	—	134	4	no	—
202	#95	—	134	4	no	—
203	C₂×Q₈	—	67	4	no	—
204	C₂×Q₈	—	134	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	67	2	no	—
208	C₂²×C₄	—	67	2	no	—
209	C₂²×C₄	—	134	4	no	—
210	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	67	4	no	—
216	C₂×C₈	—	134	2	no	—
217	C₂×C₈	—	134	2	no	—
218	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
220	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
221	C₂×C₈	#11761628	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	268	4	no	—
226	C₄⋊C₄	—	536	2	no	—
227	C₄⋊C₄	—	67	8	no	—
228	C₄⋊C₄	—	67	4	no	—
229	C₄⋊C₄	—	134	4	no	—
230	C₄⋊C₄	—	134	4	no	—
231	C₄⋊C₄	C₂×Dic₁₃₄	1	4	yes	—
232	C₄²	—	4	4	no	—
233	C₄²	—	536	4	no	—
234	C₄²	—	67	4	no	—
235	C₄²	—	67	4	no	—
236	C₄²	C₂×C₄×D₆₇	1	2	yes	—
237	C₄²	Q₈×D₆₇	1	2	yes	—
238	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
239	Q₈	—	134	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	268	4	no	—
248	C₂×C₄	—	268	4	no	—
249	C₂×C₄	—	268	4	no	—
250	C₂×C₄	—	67	4	no	—
251	C₂×C₄	—	67	4	no	—
252	C₂×C₄	—	134	4	no	—
253	C₂×C₄	—	134	4	no	—
254	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
255	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
256	C₂×C₄	#14524064	1	4	yes	—
257	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
260	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
261	C₂×C₄	#14524089	1	1	yes	characteristic
262	C₂×C₄	#14524090	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	134	4	no	—
265	C₈	C₄×Dic₁₃₄	1	4	yes	—
266	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
267	C₂²	D₄×Dic₁₃₄	1	1	yes	characteristic
268	C₂²	#16864067	1	1	yes	characteristic
269	C₂²	C₂×#14524064	1	2	yes	—
270	C₂²	#16864203	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	8	2	no	—
274	C₄	—	536	4	no	—
275	C₄	—	134	4	no	—
276	C₄	—	134	4	no	—
277	C₄	#16863431	1	4	yes	—
278	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
279	C₄	#16863876	1	1	yes	characteristic
280	C₄	#16864124	1	1	yes	characteristic
281	C₄	Q₈×Dic₁₃₄	1	1	yes	characteristic
282	C₂	C₄⋊C₄×Dic₁₃₄	1	1	yes	characteristic
283	C₂	C₂×#16863431	1	1	yes	characteristic
284	C₂	#22568801	1	1	yes	characteristic
285	C₂	#22573082	1	2	yes	—
286	C₂	#22573203	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#106×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#106	D₆₇
hypocenter	C₆₇	#22725
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₃₄	C₄	semidirect
56. #106×C₆₇	Q₈	semidirect
152. C₆₇	#22725	semidirect
194. #106	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22563164	2
#22560976	2
#22560967	2
#22560370	2
#22560366	2
#106×Dic₆₇	2
C₄×#106×C₆₇	2
#22725	67