Subgroups — #30745372

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22558050	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	—	2	1	no	—
6	#16863449	—	2	2	no	—
7	#16863449	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	—	2	1	no	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	C₄	1	2	yes	—
18	#382×C₆₇	—	2	1	no	—
19	#380×C₆₇	—	2	1	no	—
20	#380×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	4	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	4	2	no	—
27	C₂²×C₆₇⋊C₈	—	4	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
32	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
33	C₄×C₆₇⋊C₈	—	2	2	no	—
34	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	#117×C₆₇	—	4	2	no	—
39	#117×C₆₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	#95×C₆₇	—	2	2	no	—
54	#95×C₆₇	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	4	4	no	—
58	C₂×C₆₇⋊C₈	—	4	8	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	D₈	1	2	yes	—
75	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄⋊C₄×C₆₇	—	4	4	no	—
83	C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₂²⋊C₄×C₆₇	—	4	4	no	—
85	C₄²×C₆₇	—	4	2	no	—
86	C₄²×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	C₄²	1	2	yes	—
88	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
97	C₂³×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	#95	1	2	yes	—
100	C₂³×C₆₇	#102	1	2	yes	—
101	C₂³×C₆₇	#103	1	2	yes	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	#95	1	2	yes	—
114	C₂×C₄×C₆₇	#95	1	2	yes	—
115	C₂×C₄×C₆₇	#95	1	2	yes	—
116	C₂×C₄×C₆₇	#102	1	4	yes	—
117	C₂×C₄×C₆₇	#102	1	2	yes	—
118	C₂×C₄×C₆₇	#103	1	4	yes	—
119	C₂×C₄×C₆₇	#103	1	2	yes	—
120	C₂×C₄×C₆₇	#106	1	4	yes	—
121	C₂×C₄×C₆₇	#107	1	4	yes	—
122	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#131	1	4	yes	—
129	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
133	C₆₇⋊C₈	—	4	4	no	—
134	C₆₇⋊C₈	—	4	4	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	4	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	#340	1	4	yes	—
140	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
141	C₂²×C₆₇	#408	1	2	yes	—
142	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
144	C₂²×C₆₇	#421	1	2	yes	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	8	4	no	—
150	C₄×C₆₇	#340	1	4	yes	—
151	C₄×C₆₇	#405	1	2	yes	—
152	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
154	C₄×C₆₇	#424	1	2	yes	—
155	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
156	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
157	#7893	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	4	no	—
159	C₂×C₆₇	#1480	1	2	yes	—
160	C₂×C₆₇	#1481	1	2	yes	—
161	C₂×C₆₇	#1729	1	1	yes	characteristic
162	C₂×C₆₇	#1733	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
164	C₂×#431	—	67	1	no	—
165	C₂²×C₄⋊C₈	—	67	1	no	—
166	#2285	D₆₇	1	1	yes	characteristic
167	#1279	—	67	4	no	—
168	C₆₇	#7893	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	134	1	no	—
170	C₂×#117	—	2	1	no	—
171	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
172	#431	—	67	4	no	—
173	#431	—	134	2	no	—
174	C₂×C₄⋊C₈	—	67	4	no	—
175	C₂×C₄⋊C₈	—	67	2	no	—
176	C₂×C₄⋊C₈	—	134	2	no	—
177	C₂×C₄⋊C₈	—	134	1	no	—
178	C₂×C₄⋊C₈	—	134	1	no	—
179	C₂×C₂²⋊C₈	—	134	1	no	—
180	C₂×C₄×C₈	—	67	2	no	—
181	#384	Dic₆₇	1	2	yes	—
182	#382	—	2	1	no	—
183	#380	—	2	1	no	—
184	#380	Dic₆₇	1	2	yes	—
185	C₄×C₄⋊C₄	—	2	1	no	—
186	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
187	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²×C₈	—	268	2	no	—
191	C₂²×C₈	—	268	2	no	—
192	C₂²×C₈	—	134	4	no	—
193	C₂²×C₈	—	134	2	no	—
194	C₂²×C₈	—	134	2	no	—
195	#117	—	4	2	no	—
196	#117	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	Dic₁₃₄	1	2	yes	—
203	C₂×C₄²	C₄×D₆₇	1	2	yes	—
204	C₂×C₄²	C₄×D₆₇	1	2	yes	—
205	C₂×C₄²	D₂₆₈	1	2	yes	—
206	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
210	C₄⋊C₈	—	67	4	no	—
211	C₄⋊C₈	—	134	4	no	—
212	C₄⋊C₈	—	134	4	no	—
213	C₄⋊C₈	—	134	2	no	—
214	C₂²⋊C₈	—	134	4	no	—
215	C₄×C₈	—	67	4	no	—
216	C₄×C₈	—	134	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
230	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
231	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
232	C₂²×C₄	#11761613	1	2	yes	—
233	C₂²×C₄	#11761615	1	2	yes	—
234	C₂²×C₄	#11761616	1	2	yes	—
235	C₂²×C₄	#11761618	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
239	C₂×C₈	—	268	4	no	—
240	C₂×C₈	—	268	8	no	—
241	C₂×C₈	—	268	2	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	—	134	4	no	—
244	C₂×C₈	—	134	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₄²	—	4	2	no	—
250	C₄²	—	4	2	no	—
251	C₄²	C₄×Dic₆₇	1	2	yes	—
252	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
253	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
254	C₄²	Dic₆₇⋊C₄	1	2	yes	—
255	C₄²	Dic₆₇⋊C₄	1	2	yes	—
256	C₄²	C₄⋊Dic₆₇	1	2	yes	—
257	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
260	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₆₇⋊#95	1	2	yes	—
264	C₂³	#14523964	1	2	yes	—
265	C₂³	#14523967	1	2	yes	—
266	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₆₇⋊#95	1	2	yes	—
278	C₂×C₄	C₆₇⋊#95	1	2	yes	—
279	C₂×C₄	C₆₇⋊#95	1	2	yes	—
280	C₂×C₄	#14523963	1	4	yes	—
281	C₂×C₄	#14523964	1	2	yes	—
282	C₂×C₄	#14523966	1	4	yes	—
283	C₂×C₄	#14523967	1	2	yes	—
284	C₂×C₄	#14523974	1	4	yes	—
285	C₂×C₄	#14523976	1	4	yes	—
286	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
292	C₂×C₄	#14524050	1	4	yes	—
293	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
294	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
295	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
297	C₈	—	268	4	no	—
298	C₈	—	268	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	#16863158	1	4	yes	—
304	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
305	C₂²	#16863363	1	2	yes	—
306	C₂²	C₂×#14523964	1	1	yes	characteristic
307	C₂²	C₂×#14523967	1	1	yes	characteristic
308	C₂²	#16863413	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	4	no	—
314	C₄	#16863158	1	4	yes	—
315	C₄	#16863352	1	2	yes	—
316	C₄	C₂×#14523963	1	1	yes	characteristic
317	C₄	C₂×#14523966	1	1	yes	characteristic
318	C₄	#16863423	1	2	yes	—
319	C₄	C₂×#14523974	1	1	yes	characteristic
320	C₄	C₂×#14523976	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	#22558608	1	2	yes	—
323	C₂	#22558612	1	2	yes	—
324	C₂	#22559906	1	1	yes	characteristic
325	C₂	#22559918	1	1	yes	characteristic
326	C₂	C₂×#16863158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2285	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2285	D₆₇
hypocenter	C₆₇	#7893
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #22558050	C₂	semidirect
7. #16863449	C₄	semidirect
11. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
168. C₆₇	#7893	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
C₂²×C₄⋊(C₆₇⋊C₈)	2
#22558050	2
C₆₇×#2285	2
#7893	67