Subgroups — #30745921

Order 17152.

706 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	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×#431×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×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#431×C₆₇	—	2	2	no	—
19	#431×C₆₇	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	4	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	4	2	no	—
26	C₂²×C₆₇⋊C₈	—	4	2	no	—
27	#14524002	—	4	2	no	—
28	#14524002	—	4	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
37	C₆₇⋊#95	—	2	2	no	—
38	C₆₇⋊#95	—	2	2	no	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	—	2	2	no	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
53	C₂²⋊C₈×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	4	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	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	4	4	no	—
63	C₄⋊Dic₆₇	—	4	4	no	—
64	C₂²⋊Dic₆₇	—	4	4	no	—
65	C₄×Dic₆₇	—	4	4	no	—
66	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	D₈	1	2	yes	—
79	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
81	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	4	4	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	#95	1	2	yes	—
111	C₂×C₄×C₆₇	#95	1	2	yes	—
112	C₂×C₄×C₆₇	#95	1	2	yes	—
113	C₂×C₄×C₆₇	#102	1	2	yes	—
114	C₂×C₄×C₆₇	#102	1	4	yes	—
115	C₂×C₄×C₆₇	#103	1	2	yes	—
116	C₂×C₄×C₆₇	#103	1	4	yes	—
117	C₂×C₄×C₆₇	#106	1	4	yes	—
118	C₂×C₄×C₆₇	#107	1	4	yes	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#135	1	4	yes	—
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	4	4	no	—
132	C₂×Dic₆₇	—	4	4	no	—
133	C₈×C₆₇	—	4	4	no	—
134	C₆₇⋊C₈	—	4	4	no	—
135	C₂²×C₆₇	—	2	4	no	—
136	C₂²×C₆₇	—	2	4	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	#340	1	4	yes	—
140	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
141	C₂²×C₆₇	#413	1	2	yes	—
142	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
144	C₂²×C₆₇	#416	1	2	yes	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	#340	1	4	yes	—
150	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
152	C₄×C₆₇	#416	1	2	yes	—
153	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
154	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
155	C₄×C₆₇	#427	1	2	yes	—
156	Dic₆₇	—	8	4	no	—
157	#8039	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	4	no	—
159	C₂×C₆₇	#1540	1	2	yes	—
160	C₂×C₆₇	#1542	1	2	yes	—
161	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
162	C₂×C₆₇	#1735	1	1	yes	characteristic
163	C₂×C₆₇	#1742	1	1	yes	characteristic
164	C₂×#431	D₆₇	1	1	yes	characteristic
165	C₂²×C₄⋊C₈	—	67	1	no	—
166	#2285	—	67	1	no	—
167	#1296	—	67	4	no	—
168	C₆₇	#8039	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	134	1	no	—
170	C₂×#117	—	134	1	no	—
171	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
172	#431	—	2	2	no	—
173	#431	Dic₆₇	1	4	yes	complemented
174	C₂×C₄⋊C₈	—	67	4	no	—
175	C₂×C₄⋊C₈	—	67	2	no	—
176	C₂×C₄⋊C₈	—	134	1	no	—
177	C₂×C₄⋊C₈	—	134	2	no	—
178	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	C₂×C₄×C₈	—	2	1	no	—
181	#384	—	67	2	no	—
182	#382	—	134	1	no	—
183	#380	—	67	2	no	—
184	#380	—	134	1	no	—
185	C₄×C₄⋊C₄	—	67	2	no	—
186	C₄×C₄⋊C₄	—	134	1	no	—
187	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
188	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
189	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	268	2	no	—
192	C₂²×C₈	—	268	2	no	—
193	C₂²×C₈	—	134	4	no	—
194	C₂²×C₈	—	134	2	no	—
195	#117	—	268	2	no	—
196	#117	—	268	2	no	—
197	C₂×C₄⋊C₄	—	67	4	no	—
198	C₂×C₄⋊C₄	—	134	2	no	—
199	C₂×C₄⋊C₄	—	134	2	no	—
200	C₂×C₂²⋊C₄	—	134	2	no	—
201	C₂×C₄²	—	134	2	no	—
202	C₂×C₄²	Dic₁₃₄	1	2	yes	—
203	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₆₇	1	2	yes	—
205	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	67	4	no	—
212	C₄⋊C₈	—	134	4	no	—
213	C₄⋊C₈	—	134	2	no	—
214	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	#95	—	134	2	no	—
218	#95	—	134	2	no	—
219	C₂⁴	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	134	4	no	—
223	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
227	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
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	D₅₃₆	1	2	yes	—
233	C₂²×C₄	#11761617	1	2	yes	—
234	C₂²×C₄	#11761617	1	2	yes	—
235	C₂²×C₄	Dic₂₆₈	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	—	268	8	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	—	134	4	no	—
245	C₄⋊C₄	—	268	4	no	—
246	C₄⋊C₄	—	268	2	no	—
247	C₄⋊C₄	—	268	2	no	—
248	C₄⋊C₄	—	268	4	no	—
249	C₂²⋊C₄	—	268	4	no	—
250	C₄²	—	268	4	no	—
251	C₄²	C₄×Dic₆₇	1	2	yes	—
252	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
253	C₄²	C₂²⋊Dic₆₇	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	C₂²×Dic₆₇	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₆₇	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	#14523965	1	2	yes	—
265	C₂³	#14523968	1	2	yes	—
266	C₂³	C₂×(C₂×D₆₇)⋊C₄	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	—	268	4	no	—
275	C₂×C₄	—	268	4	no	—
276	C₂×C₄	—	268	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	#14523965	1	4	yes	—
281	C₂×C₄	#14523965	1	2	yes	—
282	C₂×C₄	#14523968	1	4	yes	—
283	C₂×C₄	#14523968	1	2	yes	—
284	C₂×C₄	#14523975	1	4	yes	—
285	C₂×C₄	#14523977	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×C₂²⋊Dic₆₇	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	C₂×D₅₃₆	1	1	yes	characteristic
293	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
294	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
295	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
296	C₂×C₄	#14524063	1	4	yes	—
297	C₈	—	4	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	#16863159	1	4	yes	—
304	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
305	C₂²	#16863378	1	2	yes	—
306	C₂²	C₂×#14523965	1	1	yes	characteristic
307	C₂²	C₂×#14523968	1	1	yes	characteristic
308	C₂²	#16863390	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	536	4	no	—
314	C₄	#16863159	1	4	yes	—
315	C₄	C₂×#14523965	1	1	yes	characteristic
316	C₄	C₂×#14523968	1	1	yes	characteristic
317	C₄	#16863392	1	2	yes	—
318	C₄	C₂×#14523975	1	1	yes	characteristic
319	C₄	C₂×#14523977	1	1	yes	characteristic
320	C₄	#16863432	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	#22558856	1	2	yes	—
323	C₂	#22558864	1	2	yes	—
324	C₂	C₂×#16863159	1	1	yes	characteristic
325	C₂	#22559925	1	1	yes	characteristic
326	C₂	#22559947	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#8039
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

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
19. #431×C₆₇	C₄	semidirect
168. C₆₇	#8039	semidirect
173. #431	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×#431×C₆₇	2
#8039	67