Subgroups — #30749592

Order 17152.

382 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	#22560360	C₂	1	1	yes	maximal, characteristic
2	#22560325	C₂	1	1	yes	maximal, characteristic
3	#22559954	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863148	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863146	C₂	1	1	yes	maximal, characteristic
7	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₇⋊C₈	—	2	2	no	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
13	#16863150	—	2	2	no	—
14	#16863150	C₄	1	1	yes	characteristic
15	#16863150	C₄	1	1	yes	characteristic
16	#16863150	C₄	1	1	yes	characteristic
17	#16863150	C₄	1	1	yes	characteristic
18	#16863149	—	2	2	no	—
19	#16863148	—	2	2	no	—
20	#16863146	—	2	2	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#334×C₆₇	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	1	no	—
26	C₂²×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
32	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	Dic₆₇⋊C₈	—	4	4	no	—
38	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
39	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₇	—	2	1	no	—
46	C₄⋊C₈×C₆₇	—	2	1	no	—
47	C₄⋊C₈×C₆₇	—	2	1	no	—
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₄×C₈×C₆₇	Q₈	1	2	yes	—
51	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
52	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₆₇⋊C₈	—	2	1	no	—
54	C₂×C₆₇⋊C₈	—	2	1	no	—
55	C₂×C₆₇⋊C₈	—	2	1	no	—
56	C₂×C₆₇⋊C₈	—	2	1	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₆₇	—	2	1	no	—
64	C₄×Dic₆₇	—	2	1	no	—
65	C₄×Dic₆₇	—	2	1	no	—
66	C₄×Dic₆₇	—	2	1	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	2	2	no	—
69	C₄×Dic₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	—	2	1	no	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	—	4	1	no	—
79	C₂×C₈×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
99	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₆₇	#137	1	2	yes	—
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	2	2	no	—
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	#108	1	4	yes	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	—	8	1	no	—
110	C₈×C₆₇	—	8	1	no	—
111	C₆₇⋊C₈	—	2	2	no	—
112	C₆₇⋊C₈	—	2	2	no	—
113	C₆₇⋊C₈	—	8	1	no	—
114	C₆₇⋊C₈	—	8	1	no	—
115	C₂²×C₆₇	#384	1	1	yes	characteristic
116	C₂²×C₆₇	#424	1	1	yes	characteristic
117	C₂²×C₆₇	#424	1	1	yes	characteristic
118	C₂²×C₆₇	#428	1	2	yes	—
119	C₂²×C₆₇	C₂×#108	1	2	yes	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	#424	1	1	yes	characteristic
123	C₄×C₆₇	#424	1	1	yes	characteristic
124	C₄×C₆₇	#424	1	1	yes	characteristic
125	C₄×C₆₇	#424	1	1	yes	characteristic
126	C₄×C₆₇	#496	1	1	yes	characteristic
127	C₄×C₆₇	#497	1	1	yes	characteristic
128	C₄×C₆₇	#501	1	2	yes	—
129	Dic₆₇	—	2	4	no	—
130	Dic₆₇	—	4	2	no	—
131	#8812	—	67	1	no	maximal, Sylow
132	C₂×C₆₇	#1573	1	2	yes	—
133	C₂×C₆₇	#1574	1	2	yes	—
134	C₂×C₆₇	#1843	1	1	yes	characteristic
135	C₂×C₆₇	#1847	1	1	yes	characteristic
136	C₂×C₆₇	#1848	1	1	yes	characteristic
137	#1842	—	67	1	no	—
138	#1831	—	67	1	no	—
139	#1744	—	67	1	no	—
140	C₂×#335	—	67	1	no	—
141	C₂×#335	—	67	1	no	—
142	C₂×#334	—	67	1	no	—
143	C₂×#334	D₆₇	1	1	yes	characteristic
144	C₆₇	#8812	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄⋊C₈	—	67	1	no	—
146	C₂×C₄⋊C₈	—	67	1	no	—
147	C₂×C₄⋊C₈	—	134	2	no	—
148	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
150	C₂×C₄×C₈	—	67	1	no	—
151	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
152	C₄³	—	67	1	no	—
153	#336	—	67	1	no	—
154	#336	—	67	1	no	—
155	#336	—	67	1	no	—
156	#336	—	67	1	no	—
157	#336	—	134	2	no	—
158	#335	—	134	2	no	—
159	#335	—	134	2	no	—
160	#334	—	2	2	no	—
161	#334	—	134	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	67	1	no	—
165	C₂²×C₈	—	67	1	no	—
166	C₂²×C₈	—	134	1	no	—
167	C₂²×C₈	—	134	1	no	—
168	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
170	C₂×C₄²	—	67	1	no	—
171	C₂×C₄²	—	67	1	no	—
172	C₂×C₄²	—	67	2	no	—
173	C₂×C₄²	—	67	2	no	—
174	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	268	4	no	—
180	C₄⋊C₈	—	134	1	no	—
181	C₄⋊C₈	—	134	1	no	—
182	C₄⋊C₈	—	134	1	no	—
183	C₄⋊C₈	—	134	1	no	—
184	C₄×C₈	—	67	2	no	—
185	C₄×C₈	—	67	2	no	—
186	C₄×C₈	Dic₁₃₄	1	2	yes	—
187	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
188	C₂²×C₄	—	67	2	no	—
189	C₂²×C₄	—	67	2	no	—
190	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
192	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	268	1	no	—
202	C₂×C₈	—	268	1	no	—
203	C₂×C₈	—	268	2	no	—
204	C₂×C₈	—	268	2	no	—
205	C₂×C₈	—	67	2	no	—
206	C₂×C₈	—	67	2	no	—
207	C₂×C₈	—	134	1	no	—
208	C₂×C₈	—	134	1	no	—
209	C₂×C₈	—	134	1	no	—
210	C₂×C₈	—	134	1	no	—
211	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
212	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
213	C₄²	—	134	4	no	—
214	C₄²	—	134	1	no	—
215	C₄²	—	134	1	no	—
216	C₄²	—	134	1	no	—
217	C₄²	—	134	1	no	—
218	C₄²	—	134	2	no	—
219	C₄²	—	134	2	no	—
220	C₄²	D₄×D₆₇	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
222	C₄²	Q₈×D₆₇	1	1	yes	characteristic
223	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
224	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	67	4	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	C₄⋊C₄×D₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
234	C₂×C₄	#14524039	1	1	yes	characteristic
235	C₂×C₄	#14524041	1	1	yes	characteristic
236	C₂×C₄	#14524042	1	1	yes	characteristic
237	C₂×C₄	#14524042	1	1	yes	characteristic
238	C₂×C₄	#14524049	1	1	yes	characteristic
239	C₂×C₄	#14524049	1	1	yes	characteristic
240	C₂×C₄	#14524051	1	1	yes	characteristic
241	C₂×C₄	#14524051	1	1	yes	characteristic
242	C₂×C₄	#14524070	1	2	yes	—
243	C₂×C₄	#14524072	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	8	1	no	—
247	C₈	—	8	1	no	—
248	C₈	—	536	1	no	—
249	C₈	—	536	1	no	—
250	C₈	—	134	2	no	—
251	C₈	—	134	2	no	—
252	C₂²	#16863265	1	1	yes	characteristic
253	C₂²	#16863421	1	1	yes	characteristic
254	C₂²	#16863421	1	1	yes	characteristic
255	C₂²	#16863435	1	2	yes	—
256	C₂²	#108×D₆₇	1	2	yes	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	268	2	no	—
260	C₄	—	134	4	no	—
261	C₄	#16863421	1	1	yes	characteristic
262	C₄	#16863421	1	1	yes	characteristic
263	C₄	#16863422	1	1	yes	characteristic
264	C₄	#16863424	1	1	yes	characteristic
265	C₄	#16863782	1	1	yes	characteristic
266	C₄	#16863787	1	1	yes	characteristic
267	C₄	#16863803	1	2	yes	—
268	C₂	#22558991	1	2	yes	—
269	C₂	#22558996	1	2	yes	—
270	C₂	#22560362	1	1	yes	characteristic
271	C₂	#22560374	1	1	yes	characteristic
272	C₂	#22560379	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#8812
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
144. C₆₇	#8812	semidirect

Maximal subgroups

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

subgroup	index
#22560360	2
#22560325	2
#22559954	2
C₂×#16863149	2
C₂×#16863148	2
C₂×#16863146	2
C₂×#334×C₆₇	2
#8812	67