Subgroups — #30874271

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₆₇	C₂	1	2	yes	maximal
2	#22563130	C₂	1	2	yes	maximal
3	#22560943	C₂	1	2	yes	maximal
4	#22560479	C₂	1	4	yes	maximal
5	#22560373	C₂	1	1	yes	maximal, characteristic
6	#22560224	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₆₇	C₂²	1	4	yes	—
10	Q₁₆×Dic₆₇	—	2	8	no	—
11	C₂×#14523977	C₂²	1	2	yes	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14523967	C₂²	1	4	yes	—
15	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
16	#16863295	—	2	4	no	—
17	#16863295	C₂²	1	4	yes	—
18	#16863262	—	2	2	no	—
19	#16863262	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₆₇	—	2	2	no	—
21	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863158	C₂²	1	4	yes	—
23	#16863150	C₂²	1	2	yes	—
24	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₆₇	C₄	1	8	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₆₇	—	2	8	no	—
32	Q₈×Dic₆₇	—	4	4	no	—
33	C₂×C₄⋊Dic₆₇	—	2	4	no	—
34	C₂×C₄⋊Dic₆₇	—	2	4	no	—
35	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
36	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
37	C₂×C₄×Dic₆₇	—	2	4	no	—
38	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
39	#14523977	—	2	4	no	—
40	#14523975	—	4	2	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
42	#14523967	—	2	8	no	—
43	C₈×Dic₆₇	—	2	4	no	—
44	C₆₇⋊#95	—	2	4	no	—
45	C₆₇⋊#95	—	2	2	no	—
46	C₆₇⋊#95	—	2	2	no	—
47	C₆₇⋊#95	D₄	1	1	yes	characteristic
48	C₆₇⋊#95	D₄	1	1	yes	characteristic
49	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
52	C₂×Q₁₆×C₆₇	Q₈	1	4	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	C₂×C₄	1	4	yes	—
62	#103×C₆₇	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
64	C₂²×Dic₆₇	—	2	4	no	—
65	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
66	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	2	4	no	—
72	C₄⋊Dic₆₇	—	2	4	no	—
73	C₄⋊Dic₆₇	—	4	8	no	—
74	C₄⋊Dic₆₇	—	4	8	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₄×Dic₆₇	—	2	2	no	—
77	C₄×Dic₆₇	—	4	8	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
83	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
96	C₄⋊C₄×C₆₇	—	2	8	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
99	C₄²×C₆₇	—	2	8	no	—
100	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₆₇	#41	1	2	yes	—
102	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	—	2	4	no	—
105	Q₈×C₆₇	—	2	4	no	—
106	Q₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	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	4	no	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₆₇	—	2	2	no	—
123	C₂×Dic₆₇	—	2	2	no	—
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	2	2	no	—
126	C₂×Dic₆₇	—	2	2	no	—
127	C₂×Dic₆₇	—	4	4	no	—
128	C₂×Dic₆₇	—	4	4	no	—
129	C₂×Dic₆₇	—	4	4	no	—
130	C₂×Dic₆₇	#135	1	4	yes	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
133	C₆₇⋊C₈	—	4	2	no	—
134	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₆₇	D₄²	1	1	yes	characteristic
136	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#135	1	2	yes	—
138	C₂²×C₆₇	#578	1	2	yes	—
139	C₄×C₆₇	—	2	8	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	4	4	no	—
143	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₆₇	#520	1	1	yes	characteristic
145	C₄×C₆₇	#547	1	1	yes	characteristic
146	C₄×C₆₇	#554	1	1	yes	characteristic
147	Dic₆₇	—	2	4	no	—
148	Dic₆₇	—	4	2	no	—
149	Dic₆₇	—	8	4	no	—
150	#22685	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
153	C₂×C₆₇	#2950	1	1	yes	characteristic
154	C₂×C₆₇	#3285	1	2	yes	—
155	C₂×C₆₇	#3383	1	2	yes	—
156	C₂×C₄×Q₁₆	—	67	2	no	—
157	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
158	#2351	—	67	2	no	—
159	#1942	—	67	2	no	—
160	#1865	—	67	4	no	—
161	#1846	—	67	1	no	—
162	#1805	—	67	2	no	—
163	C₈×C₄⋊C₄	—	67	1	no	—
164	C₆₇	#22685	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	—	67	4	no	—
167	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
168	C₄×Q₁₆	—	134	8	no	—
169	C₄×Q₁₆	Dic₆₇	1	8	yes	complemented
170	C₂×#107	—	67	2	no	—
171	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#106	—	67	2	no	—
173	C₂×C₄⋊C₈	—	67	1	no	—
174	C₂×#103	—	67	4	no	—
175	C₂×#103	C₂×D₆₇	1	2	yes	—
176	C₂×C₄×C₈	—	67	2	no	—
177	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
178	#389	—	67	4	no	—
179	#389	—	134	4	no	—
180	#382	—	67	2	no	—
181	#382	—	134	2	no	—
182	C₄×C₄⋊C₄	—	67	1	no	—
183	C₄×C₄⋊C₄	—	134	2	no	—
184	#340	—	67	4	no	—
185	#336	—	67	2	no	—
186	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
187	C₂×Q₁₆	Dic₁₃₄	1	4	yes	—
188	C₂×Q₁₆	D₂₆₈	1	4	yes	—
189	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
190	C₂²×C₈	—	67	2	no	—
191	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	C₄×Q₈	—	2	4	no	—
194	C₄×Q₈	—	268	4	no	—
195	C₄×Q₈	—	134	8	no	—
196	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	67	2	no	—
199	C₂×C₄⋊C₄	—	67	4	no	—
200	C₂×C₄⋊C₄	—	134	4	no	—
201	C₂×C₄⋊C₄	—	134	4	no	—
202	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	67	2	no	—
205	C₂×C₄²	—	134	4	no	—
206	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
207	#107	—	134	4	no	—
208	#107	C₂×Dic₆₇	1	4	yes	—
209	#106	—	268	2	no	—
210	C₄⋊C₈	—	134	2	no	—
211	#103	—	134	8	no	—
212	#103	C₂×Dic₆₇	1	8	yes	—
213	C₄×C₈	—	134	4	no	—
214	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
215	#95	—	67	1	no	—
216	#95	—	67	1	no	—
217	#95	—	134	4	no	—
218	#95	—	134	2	no	—
219	#95	—	134	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
224	C₂×Q₈	C₂×D₂₆₈	1	4	yes	—
225	C₂×Q₈	C₂×Dic₁₃₄	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	67	2	no	—
229	C₂²×C₄	—	67	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
234	Q₁₆	C₄⋊Dic₆₇	1	16	yes	complemented
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	—	134	2	no	—
239	C₂×C₈	—	134	2	no	—
240	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
241	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
242	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
243	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	268	8	no	—
247	C₄⋊C₄	—	268	8	no	—
248	C₄⋊C₄	—	134	4	no	—
249	C₄⋊C₄	—	134	4	no	—
250	C₄⋊C₄	—	134	4	no	—
251	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
252	C₄²	—	2	8	no	—
253	C₄²	—	268	8	no	—
254	C₄²	—	134	2	no	—
255	C₄²	—	134	2	no	—
256	C₄²	C₂²×Dic₆₇	1	2	yes	—
257	C₄²	#11761627	1	2	yes	—
258	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
259	Q₈	—	2	4	no	—
260	Q₈	—	2	4	no	—
261	Q₈	—	2	4	no	—
262	Q₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	268	4	no	—
271	C₂×C₄	—	268	4	no	—
272	C₂×C₄	—	268	4	no	—
273	C₂×C₄	—	67	4	no	—
274	C₂×C₄	—	134	2	no	—
275	C₂×C₄	—	134	2	no	—
276	C₂×C₄	—	134	2	no	—
277	C₂×C₄	—	134	2	no	—
278	C₂×C₄	—	134	2	no	—
279	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
280	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
281	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
285	C₂×C₄	#14524089	1	1	yes	characteristic
286	C₂×C₄	#14524090	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	268	2	no	—
289	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
290	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
291	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
292	C₂²	D₄×Dic₁₃₄	1	1	yes	characteristic
293	C₂²	#135×D₆₇	1	2	yes	—
294	C₂²	#16864209	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	268	2	no	—
300	C₄	—	536	4	no	—
301	C₄	—	134	4	no	—
302	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
303	C₄	#16863879	1	1	yes	characteristic
304	C₄	#16864050	1	1	yes	characteristic
305	C₄	#16864103	1	1	yes	characteristic
306	C₂	D₄×C₄⋊Dic₆₇	1	1	yes	characteristic
307	C₂	#135×Dic₆₇	1	1	yes	characteristic
308	C₂	#22569077	1	1	yes	characteristic
309	C₂	#22571951	1	2	yes	—
310	C₂	#22573100	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#22685
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×Q₁₆×C₆₇	C₄	semidirect
89. Q₁₆×C₆₇	C₄⋊C₄	semidirect
164. C₆₇	#22685	semidirect
169. C₄×Q₁₆	Dic₆₇	semidirect
234. Q₁₆	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₆₇	2
#22563130	2
#22560943	2
#22560479	2
#22560373	2
#22560224	2
C₈×C₄⋊Dic₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
#22685	67