Subgroups — #30751467

Order 17152.

644 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#22560189	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863145	C₂	1	2	yes	maximal, complemented
4	#22558092	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
12	#16863151	—	2	1	no	—
13	#16863145	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#391×C₆₇	C₄	1	2	yes	—
18	#389×C₆₇	—	2	1	no	—
19	#389×C₆₇	—	2	1	no	—
20	#384×C₆₇	C₂²	1	1	yes	characteristic
21	#380×C₆₇	—	2	1	no	—
22	#380×C₆₇	—	2	1	no	—
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	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
33	C₂²⋊(C₆₇⋊C₈)	—	4	2	no	—
34	C₂²×Q₈×C₆₇	C₂×C₄	1	2	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	C₄⋊Q₈×C₆₇	—	2	2	no	—
39	C₄⋊Q₈×C₆₇	—	2	2	no	—
40	C₂²⋊Q₈×C₆₇	—	4	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#95×C₆₇	—	2	2	no	—
55	#95×C₆₇	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	4	8	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×Q₈×C₆₇	—	2	4	no	—
65	C₂×Q₈×C₆₇	—	4	2	no	—
66	C₂×Q₈×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	4	1	no	—
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
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	Q₁₆	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	—	4	2	no	—
89	C₄⋊C₄×C₆₇	—	4	4	no	—
90	C₄⋊C₄×C₆₇	—	4	1	no	—
91	C₄⋊C₄×C₆₇	—	4	2	no	—
92	C₄⋊C₄×C₆₇	—	4	1	no	—
93	C₄⋊C₄×C₆₇	—	4	2	no	—
94	C₄⋊C₄×C₆₇	—	4	1	no	—
95	C₄⋊C₄×C₆₇	—	4	1	no	—
96	C₂²⋊C₄×C₆₇	—	4	4	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₄²×C₆₇	#41	1	2	yes	—
103	C₂³×C₆₇	—	2	1	no	—
104	C₂³×C₆₇	—	2	1	no	—
105	C₂³×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	—	2	1	no	—
107	C₂³×C₆₇	#100	1	2	yes	—
108	C₂³×C₆₇	#103	1	2	yes	—
109	C₂³×C₆₇	#103	1	2	yes	—
110	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	Q₈×C₆₇	—	8	4	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	#103	1	2	yes	—
135	C₂×C₄×C₆₇	#103	1	2	yes	—
136	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#121	1	2	yes	—
139	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#135	1	2	yes	—
145	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
147	C₆₇⋊C₈	—	4	4	no	—
148	C₆₇⋊C₈	—	8	2	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	#333	1	4	yes	complemented
156	C₂²×C₆₇	#386	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
158	C₂²×C₆₇	#413	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
161	C₂²×C₆₇	#416	1	1	yes	characteristic
162	C₂²×C₆₇	#417	1	1	yes	characteristic
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	#450	1	1	yes	characteristic
170	C₄×C₆₇	#451	1	1	yes	characteristic
171	C₄×C₆₇	#465	1	2	yes	—
172	C₄×C₆₇	#467	1	2	yes	—
173	C₄×C₆₇	#483	1	1	yes	characteristic
174	C₄×C₆₇	#484	1	1	yes	characteristic
175	#9127	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	4	no	—
177	C₂×C₆₇	C₂×#333	1	1	yes	characteristic
178	C₂×C₆₇	#1542	1	1	yes	characteristic
179	C₂×C₆₇	#1550	1	1	yes	characteristic
180	C₂×C₆₇	#1551	1	1	yes	characteristic
181	C₂×C₆₇	#1872	1	1	yes	characteristic
182	C₂×C₆₇	#1892	1	1	yes	characteristic
183	C₂×C₆₇	#1900	1	1	yes	characteristic
184	C₂²×C₄⋊C₈	—	67	1	no	—
185	#2857	D₆₇	1	1	yes	characteristic
186	#1797	—	67	1	no	—
187	C₂×#333	—	67	2	no	—
188	#1296	—	67	2	no	—
189	C₆₇	#9127	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₈	—	134	1	no	—
191	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
192	C₂×C₂²⋊Q₈	Dic₆₇	1	2	yes	—
193	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄⋊C₈	—	67	2	no	—
195	C₂×C₄⋊C₈	—	67	2	no	—
196	C₂×C₄⋊C₈	—	134	4	no	—
197	C₂×C₄⋊C₈	—	134	1	no	—
198	C₂×C₂²⋊C₈	—	134	1	no	—
199	#391	Dic₆₇	1	2	yes	—
200	#389	—	2	1	no	—
201	#389	—	2	1	no	—
202	#384	C₂×D₆₇	1	1	yes	characteristic
203	#380	—	2	1	no	—
204	#380	—	2	1	no	—
205	#336	—	134	1	no	—
206	#333	—	134	4	no	—
207	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
208	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
209	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂²×C₈	—	268	2	no	—
212	C₂²×C₈	—	268	2	no	—
213	C₂²×C₈	—	134	2	no	—
214	C₂²×C₈	—	134	4	no	—
215	C₂²×C₈	—	134	2	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	4	4	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
232	C₄⋊C₈	—	67	4	no	—
233	C₄⋊C₈	—	134	4	no	—
234	C₄⋊C₈	—	134	2	no	—
235	C₄⋊C₈	—	134	2	no	—
236	C₄⋊C₈	—	134	2	no	—
237	C₂²⋊C₈	—	268	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
258	C₂²×C₄	#11761615	1	2	yes	—
259	C₂²×C₄	#11761615	1	2	yes	—
260	C₂²×C₄	#11761618	1	2	yes	—
261	C₂²×C₄	#11761618	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
265	C₂×C₈	—	268	8	no	—
266	C₂×C₈	—	268	2	no	—
267	C₂×C₈	—	268	2	no	—
268	C₂×C₈	—	268	2	no	—
269	C₂×C₈	—	268	2	no	—
270	C₂×C₈	—	268	2	no	—
271	C₂×C₈	—	134	4	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	D₄×D₆₇	1	2	yes	—
286	C₄²	#11761627	1	2	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	#14523960	1	2	yes	—
292	C₂³	#14523967	1	2	yes	—
293	C₂³	#14523967	1	2	yes	—
294	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
295	Q₈	—	8	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	#14523967	1	2	yes	—
319	C₂×C₄	#14523967	1	2	yes	—
320	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
321	C₂×C₄	#14524011	1	1	yes	characteristic
322	C₂×C₄	#14524015	1	2	yes	—
323	C₂×C₄	#14524024	1	1	yes	characteristic
324	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
325	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
326	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
327	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
328	C₂×C₄	#14524063	1	2	yes	—
329	C₂×C₄	#14524068	1	1	yes	characteristic
330	C₂×C₄	#14524073	1	1	yes	characteristic
331	C₈	—	268	4	no	—
332	C₈	—	536	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#16863145	1	4	yes	complemented
340	C₂²	#16863276	1	1	yes	characteristic
341	C₂²	C₂×#14523960	1	1	yes	characteristic
342	C₂²	#16863380	1	1	yes	characteristic
343	C₂²	C₂×#14523967	1	1	yes	characteristic
344	C₂²	C₂×#14523967	1	1	yes	characteristic
345	C₂²	#16863392	1	1	yes	characteristic
346	C₂²	#16863397	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	2	no	—
353	C₄	#16863546	1	1	yes	characteristic
354	C₄	#16863551	1	1	yes	characteristic
355	C₄	#16863627	1	2	yes	—
356	C₄	#16863635	1	2	yes	—
357	C₄	#16863724	1	1	yes	characteristic
358	C₄	#16863729	1	1	yes	characteristic
359	C₂	—	2	4	no	—
360	C₂	C₂×#16863145	1	1	yes	characteristic
361	C₂	#22558866	1	1	yes	characteristic
362	C₂	#22558899	1	1	yes	characteristic
363	C₂	#22558904	1	1	yes	characteristic
364	C₂	#22560517	1	1	yes	characteristic
365	C₂	#22560633	1	1	yes	characteristic
366	C₂	#22560673	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2857	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2857	D₆₇
hypocenter	C₆₇	#9127
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. C₂×#16863145	C₂	semidirect
4. #22558092	C₂	semidirect
155. C₂²×C₆₇	#333	semidirect
189. C₆₇	#9127	semidirect
339. C₂²	#16863145	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
#22560189	2
C₂×#16863145	2
#22558092	2
C₆₇×#2857	2
#9127	67