Subgroups — #30745944

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#22562616	C₂	1	1	yes	maximal, characteristic
3	#22558093	C₂	1	4	yes	maximal, complemented
4	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	—	2	1	no	—
6	C₂×#14524002	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	#16863266	C₂²	1	2	yes	—
12	#16863262	—	2	1	no	—
13	#16863256	—	2	1	no	—
14	#16863256	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₆₇	—	2	1	no	—
16	C₄×C₄⋊Dic₆₇	C₄	1	2	yes	—
17	C₂²×M₁₆×C₆₇	—	2	1	no	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#423×C₆₇	—	2	2	no	—
20	#423×C₆₇	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	4	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	4	4	no	—
26	#14524002	—	4	2	no	—
27	#14524002	—	4	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
36	C₆₇⋊#95	—	2	2	no	—
37	C₆₇⋊#95	—	2	2	no	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	—	4	2	no	—
43	C₂×M₁₆×C₆₇	—	4	2	no	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	—	2	4	no	—
54	C₄⋊C₈×C₆₇	—	2	2	no	—
55	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
56	C₂²×Dic₆₇	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	4	8	no	—
59	C₂×C₆₇⋊C₈	—	4	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₄⋊Dic₆₇	—	4	2	no	—
64	C₄⋊Dic₆₇	—	4	4	no	—
65	C₂²⋊Dic₆₇	—	4	4	no	—
66	C₄×Dic₆₇	—	4	4	no	—
67	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	D₈	1	2	yes	—
81	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
83	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	M₁₆×C₆₇	—	4	8	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	C₄²	1	2	yes	—
92	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
95	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
96	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
97	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	#95	1	2	yes	—
104	C₂³×C₆₇	#102	1	2	yes	—
105	C₂³×C₆₇	#103	1	2	yes	—
106	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	#95	1	2	yes	—
116	C₂×C₄×C₆₇	#95	1	2	yes	—
117	C₂×C₄×C₆₇	#95	1	2	yes	—
118	C₂×C₄×C₆₇	#102	1	2	yes	—
119	C₂×C₄×C₆₇	#102	1	4	yes	—
120	C₂×C₄×C₆₇	#103	1	2	yes	—
121	C₂×C₄×C₆₇	#103	1	4	yes	—
122	C₂×C₄×C₆₇	#106	1	4	yes	—
123	C₂×C₄×C₆₇	#107	1	4	yes	—
124	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₆₇	#137	1	2	yes	—
136	C₂×Dic₆₇	—	4	4	no	—
137	C₂×Dic₆₇	—	4	4	no	—
138	C₂×Dic₆₇	—	4	4	no	—
139	C₈×C₆₇	—	4	4	no	—
140	C₆₇⋊C₈	—	4	4	no	—
141	C₂²×C₆₇	—	2	4	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	#340	1	4	yes	—
146	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
147	C₂²×C₆₇	#413	1	2	yes	—
148	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
150	C₂²×C₆₇	#417	1	2	yes	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	#340	1	4	yes	—
156	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
157	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
158	C₄×C₆₇	#418	1	1	yes	characteristic
159	C₄×C₆₇	#419	1	1	yes	characteristic
160	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
161	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
162	C₄×C₆₇	#428	1	2	yes	—
163	Dic₆₇	—	8	4	no	—
164	#8045	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	2	4	no	—
166	C₂×C₆₇	#1547	1	2	yes	—
167	C₂×C₆₇	#1550	1	2	yes	—
168	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
169	C₂×C₆₇	#1736	1	1	yes	characteristic
170	C₂×C₆₇	#1742	1	1	yes	characteristic
171	C₂×#423	D₆₇	1	1	yes	characteristic
172	C₂²×C₄⋊C₈	—	67	1	no	—
173	#2285	—	67	1	no	—
174	#1296	—	67	4	no	—
175	C₆₇	#8045	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	2	1	no	—
177	C₂³×C₈	—	134	1	no	—
178	C₂×#117	—	134	1	no	—
179	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
180	#423	—	2	2	no	—
181	#423	Dic₆₇	1	4	yes	complemented
182	C₂×C₄⋊C₈	—	2	1	no	—
183	C₂×C₄⋊C₈	—	67	4	no	—
184	C₂×C₄⋊C₈	—	67	2	no	—
185	C₂×C₄⋊C₈	—	134	1	no	—
186	C₂×C₄⋊C₈	—	134	2	no	—
187	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
188	#384	—	67	2	no	—
189	#382	—	134	1	no	—
190	#380	—	67	2	no	—
191	#380	—	134	1	no	—
192	C₄×C₄⋊C₄	—	67	2	no	—
193	C₄×C₄⋊C₄	—	134	1	no	—
194	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
195	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	4	2	no	—
199	C₂×M₁₆	—	4	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	268	4	no	—
202	C₂²×C₈	—	134	4	no	—
203	C₂²×C₈	—	134	2	no	—
204	#117	—	268	2	no	—
205	#117	—	268	2	no	—
206	C₂×C₄⋊C₄	—	67	4	no	—
207	C₂×C₄⋊C₄	—	134	2	no	—
208	C₂×C₄⋊C₄	—	134	2	no	—
209	C₂×C₂²⋊C₄	—	134	2	no	—
210	C₂×C₄²	—	134	2	no	—
211	C₂×C₄²	Dic₁₃₄	1	2	yes	—
212	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₆₇	1	2	yes	—
214	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	67	4	no	—
222	C₄⋊C₈	—	134	4	no	—
223	C₄⋊C₈	—	134	2	no	—
224	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
225	#95	—	134	2	no	—
226	#95	—	134	2	no	—
227	C₂⁴	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	134	4	no	—
232	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
233	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
236	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
238	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
239	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
240	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
241	C₂²×C₄	#11761613	1	2	yes	—
242	C₂²×C₄	#11761615	1	2	yes	—
243	C₂²×C₄	#11761616	1	2	yes	—
244	C₂²×C₄	#11761618	1	2	yes	—
245	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
248	M₁₆	—	4	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	268	8	no	—
253	C₂×C₈	—	268	4	no	—
254	C₂×C₈	—	134	4	no	—
255	C₄⋊C₄	—	268	4	no	—
256	C₄⋊C₄	—	268	2	no	—
257	C₄⋊C₄	—	268	2	no	—
258	C₄⋊C₄	—	268	2	no	—
259	C₄⋊C₄	—	268	2	no	—
260	C₂²⋊C₄	—	268	4	no	—
261	C₄²	—	268	4	no	—
262	C₄²	C₄×Dic₆₇	1	2	yes	—
263	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
264	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
265	C₄²	Dic₆₇⋊C₄	1	2	yes	—
266	C₄²	Dic₆₇⋊C₄	1	2	yes	—
267	C₄²	C₄⋊Dic₆₇	1	2	yes	—
268	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
269	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
270	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
271	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	C₆₇⋊#95	1	2	yes	—
275	C₂³	#14523963	1	2	yes	—
276	C₂³	#14523966	1	2	yes	—
277	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	268	4	no	—
287	C₂×C₄	—	268	4	no	—
288	C₂×C₄	—	268	4	no	—
289	C₂×C₄	C₆₇⋊#95	1	2	yes	—
290	C₂×C₄	C₆₇⋊#95	1	2	yes	—
291	C₂×C₄	C₆₇⋊#95	1	2	yes	—
292	C₂×C₄	#14523963	1	2	yes	—
293	C₂×C₄	#14523964	1	4	yes	—
294	C₂×C₄	#14523966	1	2	yes	—
295	C₂×C₄	#14523967	1	4	yes	—
296	C₂×C₄	#14523974	1	4	yes	—
297	C₂×C₄	#14523976	1	4	yes	—
298	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
299	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
307	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
308	C₂×C₄	#14524071	1	2	yes	—
309	C₂×C₄	#14524076	1	2	yes	—
310	C₈	—	4	4	no	—
311	C₈	—	268	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#16863158	1	4	yes	—
317	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
318	C₂²	#16863378	1	2	yes	—
319	C₂²	C₂×#14523963	1	1	yes	characteristic
320	C₂²	C₂×#14523966	1	1	yes	characteristic
321	C₂²	#16863398	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	536	4	no	—
327	C₄	#16863158	1	4	yes	—
328	C₄	C₂×#14523964	1	1	yes	characteristic
329	C₄	C₂×#14523967	1	1	yes	characteristic
330	C₄	#16863402	1	1	yes	characteristic
331	C₄	#16863406	1	1	yes	characteristic
332	C₄	C₂×#14523974	1	1	yes	characteristic
333	C₄	C₂×#14523976	1	1	yes	characteristic
334	C₄	#16863438	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	#22558887	1	2	yes	—
337	C₂	#22558900	1	2	yes	—
338	C₂	C₂×#16863158	1	1	yes	characteristic
339	C₂	#22559929	1	1	yes	characteristic
340	C₂	#22559947	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	#8045
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #22558093	C₂	semidirect
9. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
20. #423×C₆₇	C₄	semidirect
175. C₆₇	#8045	semidirect
181. #423	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
#22562616	2
#22558093	2
C₂×#423×C₆₇	2
#8045	67