Subgroups — #30763915

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863422	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863164	C₂	1	2	yes	maximal
4	#22559929	C₂	1	2	yes	maximal
5	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂×#14524002	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×#14523975	—	2	4	no	—
10	C₂×#14523975	—	2	2	no	—
11	#16863422	—	2	2	no	—
12	#16863422	C₄	1	4	yes	complemented
13	C₂×Dic₆₇⋊C₈	—	2	2	no	—
14	#16863164	—	2	4	no	—
15	#16863159	—	2	4	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#341×C₆₇	—	2	2	no	—
19	C₂³×Dic₆₇	—	2	2	no	—
20	C₂×#11761612	—	2	1	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	C₂×C₄	1	2	yes	—
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	#14524002	—	2	1	no	—
26	#14524002	—	2	1	no	—
27	#14524002	C₂×C₄	1	2	yes	—
28	#14524002	Q₈	1	2	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	D₄	1	4	yes	—
35	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
36	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄×Dic₆₇	—	2	1	no	—
38	#14523975	—	2	8	no	—
39	Dic₆₇⋊C₈	—	2	8	no	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	D₄	1	2	yes	—
52	C₂²×C₈×C₆₇	D₄	1	2	yes	—
53	C₂²×Dic₆₇	—	2	2	no	—
54	C₂²×Dic₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	4	2	no	—
56	C₂²×Dic₆₇	—	4	2	no	—
57	C₂²×Dic₆₇	—	8	4	no	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	2	no	—
61	C₂×C₆₇⋊C₈	—	4	1	no	—
62	C₂×C₆₇⋊C₈	—	4	1	no	—
63	C₄⋊Dic₆₇	—	2	2	no	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	—	2	4	no	—
66	C₄⋊Dic₆₇	—	4	2	no	—
67	C₄⋊Dic₆₇	—	4	2	no	—
68	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
69	C₂²⋊Dic₆₇	—	2	2	no	—
70	C₂²⋊Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	—	2	2	no	—
72	C₄×Dic₆₇	—	2	2	no	—
73	C₂⁴×C₆₇	#41	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	M₁₆×C₆₇	—	2	4	no	—
85	M₁₆×C₆₇	—	2	4	no	—
86	M₁₆×C₆₇	—	4	2	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	—	4	2	no	—
95	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	#117	1	1	yes	characteristic
98	C₂³×C₆₇	#121	1	1	yes	characteristic
99	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₆₇	#123	1	2	yes	—
101	C₂³×C₆₇	#125	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#121	1	2	yes	—
109	C₂×C₄×C₆₇	#121	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#125	1	2	yes	—
117	C₂×C₄×C₆₇	#126	1	2	yes	—
118	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×Dic₆₇	—	2	4	no	—
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	—	8	4	no	—
124	C₂×Dic₆₇	—	8	4	no	—
125	C₂×Dic₆₇	—	8	4	no	—
126	C₈×C₆₇	—	2	4	no	—
127	C₈×C₆₇	—	4	4	no	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	#380	1	1	yes	characteristic
135	C₂²×C₆₇	#387	1	2	yes	—
136	C₂²×C₆₇	#394	1	1	yes	characteristic
137	C₂²×C₆₇	#398	1	1	yes	characteristic
138	C₂²×C₆₇	#398	1	2	yes	—
139	C₂²×C₆₇	#471	1	4	yes	—
140	C₄×C₆₇	#386	1	1	yes	characteristic
141	C₄×C₆₇	#389	1	2	yes	—
142	C₄×C₆₇	#391	1	1	yes	characteristic
143	C₄×C₆₇	#395	1	1	yes	characteristic
144	C₄×C₆₇	#396	1	2	yes	—
145	C₄×C₆₇	#400	1	1	yes	characteristic
146	Dic₆₇	—	4	4	no	—
147	Dic₆₇	—	8	4	no	—
148	#11238	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	#1441	1	1	yes	characteristic
152	C₂×C₆₇	#1815	1	2	yes	—
153	C₂×C₆₇	#2046	1	2	yes	—
154	C₂×C₆₇	#2090	1	2	yes	—
155	C₂²×#106	—	67	1	no	—
156	C₂×#424	—	67	1	no	—
157	C₂×#343	—	67	2	no	—
158	C₂×#341	D₆₇	1	1	yes	characteristic
159	#1736	—	67	2	no	—
160	C₆₇	#11238	1	1	yes	characteristic, complemented, Sylow
161	C₂²×M₁₆	—	67	1	no	—
162	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
163	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#117	—	67	1	no	—
165	C₂²×C₄⋊C₄	—	67	2	no	—
166	C₂×#106	—	134	4	no	—
167	C₂×#106	—	134	2	no	—
168	#424	—	67	4	no	—
169	#424	—	134	2	no	—
170	C₂×C₄⋊C₈	—	134	2	no	—
171	#343	—	134	4	no	—
172	#341	—	2	2	no	—
173	#340	—	134	4	no	—
174	C₂³×C₄	—	134	2	no	—
175	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	67	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	1	no	—
185	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	134	1	no	—
190	C₂²×C₈	C₄×D₆₇	1	2	yes	—
191	C₂²×C₈	D₂₆₈	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
193	#117	—	67	2	no	—
194	#117	—	67	2	no	—
195	#117	—	134	1	no	—
196	#117	—	134	1	no	—
197	C₂×C₄⋊C₄	—	67	2	no	—
198	C₂×C₄⋊C₄	—	67	4	no	—
199	C₂×C₄⋊C₄	—	134	2	no	—
200	C₂×C₄⋊C₄	—	134	2	no	—
201	C₂×C₄⋊C₄	—	134	2	no	—
202	C₂×C₄⋊C₄	—	134	2	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₂²⋊C₄	—	134	1	no	—
205	C₂×C₄²	—	134	1	no	—
206	#106	—	134	8	no	—
207	C₄⋊C₈	—	134	8	no	—
208	C₂⁴	#11761629	1	1	yes	characteristic
209	C₂²×C₄	—	268	2	no	—
210	C₂²×C₄	—	268	2	no	—
211	C₂²×C₄	—	536	4	no	—
212	C₂²×C₄	—	134	2	no	—
213	C₂²×C₄	—	134	4	no	—
214	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₆₇	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
219	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
221	C₂²×C₄	#11761627	1	2	yes	—
222	C₂²×C₄	#11761627	1	1	yes	characteristic
223	C₂²×C₄	#11761628	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	268	2	no	—
229	M₁₆	—	134	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	268	1	no	—
238	C₂×C₈	—	268	1	no	—
239	C₂×C₈	—	134	2	no	—
240	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
241	C₄⋊C₄	—	268	2	no	—
242	C₄⋊C₄	—	268	2	no	—
243	C₄⋊C₄	—	67	4	no	—
244	C₄⋊C₄	—	134	4	no	—
245	C₄⋊C₄	—	134	4	no	—
246	C₄⋊C₄	—	134	2	no	—
247	C₂²⋊C₄	—	134	2	no	—
248	C₂²⋊C₄	—	134	2	no	—
249	C₄²	—	134	2	no	—
250	C₄²	—	134	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	#14524001	1	1	yes	characteristic
253	C₂³	#14524018	1	1	yes	characteristic
254	C₂³	#14524019	1	1	yes	characteristic
255	C₂³	#14524028	1	2	yes	—
256	C₂³	#14524034	1	2	yes	—
257	C₂×C₄	—	268	4	no	—
258	C₂×C₄	—	268	4	no	—
259	C₂×C₄	—	536	4	no	—
260	C₂×C₄	—	536	4	no	—
261	C₂×C₄	—	536	4	no	—
262	C₂×C₄	—	134	4	no	—
263	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
266	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
267	C₂×C₄	#14524013	1	1	yes	characteristic
268	C₂×C₄	#14524014	1	1	yes	characteristic
269	C₂×C₄	#14524015	1	2	yes	—
270	C₂×C₄	#14524016	1	2	yes	—
271	C₂×C₄	#14524020	1	2	yes	—
272	C₂×C₄	#14524020	1	2	yes	—
273	C₂×C₄	#14524022	1	2	yes	—
274	C₂×C₄	#14524023	1	1	yes	characteristic
275	C₂×C₄	#14524027	1	1	yes	characteristic
276	C₂×C₄	#14524031	1	1	yes	characteristic
277	C₂×C₄	#14524033	1	2	yes	—
278	C₂×C₄	#14524036	1	2	yes	—
279	C₂×C₄	#14524042	1	2	yes	—
280	C₂×C₄	#14524043	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	4	no	—
283	C₈	—	268	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#16863253	1	1	yes	characteristic
290	C₂²	#16863283	1	2	yes	—
291	C₂²	#16863314	1	1	yes	characteristic
292	C₂²	#16863328	1	1	yes	characteristic
293	C₂²	#16863329	1	2	yes	—
294	C₂²	#16863660	1	4	yes	—
295	C₄	—	268	4	no	—
296	C₄	—	536	4	no	—
297	C₄	#16863273	1	1	yes	characteristic
298	C₄	#16863293	1	2	yes	—
299	C₄	#16863302	1	1	yes	characteristic
300	C₄	#16863317	1	1	yes	characteristic
301	C₄	#16863319	1	2	yes	—
302	C₄	#16863336	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#22558463	1	1	yes	characteristic
306	C₂	#22560265	1	2	yes	—
307	C₂	#22561473	1	2	yes	—
308	C₂	#22561721	1	2	yes	—

	subgroup	G/H
center	C₂³	#14524018
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#341×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#341	D₆₇
hypocenter	C₆₇	#11238
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
12. #16863422	C₄	semidirect
160. C₆₇	#11238	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523975	2
C₂×#16863422	2
C₂×#16863164	2
#22559929	2
C₂×#341×C₆₇	2
#11238	67