Subgroups — #30826489

Order 17152.

640 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22569287	C₂	1	2	yes	maximal, complemented
3	#22559122	C₂	1	2	yes	maximal, complemented
4	#22559093	C₂	1	2	yes	maximal, complemented
5	#22559032	C₂	1	2	yes	maximal, complemented
6	#22559003	C₂	1	2	yes	maximal, complemented
7	#22558677	C₂	1	2	yes	maximal, complemented
8	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863946	C₂²	1	2	yes	—
11	#16863801	C₄	1	1	yes	characteristic
12	#16863773	C₄	1	1	yes	characteristic
13	#16863743	C₄	1	2	yes	—
14	#16863632	C₄	1	2	yes	—
15	#16863632	C₄	1	2	yes	—
16	C₄×D₅₃₆	C₂²	1	2	yes	—
17	C₄×D₅₃₆	C₂²	1	2	yes	—
18	#16863473	C₂²	1	4	yes	—
19	#16863469	—	2	4	no	—
20	#16863461	C₂²	1	4	yes	—
21	#16863449	C₂²	1	1	yes	characteristic
22	#16863449	C₂²	1	1	yes	characteristic
23	#16863361	—	2	4	no	—
24	#16863123	C₂²	1	4	yes	—
25	#16863123	C₂²	1	4	yes	—
26	#16863117	C₂²	1	1	yes	characteristic
27	#16863117	C₂²	1	1	yes	characteristic
28	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	#432×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#335×C₆₇	C₂²	1	2	yes	—
33	#334×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761628	—	2	2	no	—
35	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
36	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
37	C₂×#11761611	—	2	4	no	—
38	C₂×C₈×D₆₇	—	2	4	no	—
39	#14524043	C₂×C₄	1	2	yes	—
40	#14524040	C₂×C₄	1	2	yes	—
41	#14524031	—	2	2	no	—
42	#14524017	—	2	2	no	—
43	#14524017	C₂×C₄	1	4	yes	—
44	C₄×D₂₆₈	—	2	2	no	—
45	C₄×D₂₆₈	C₂³	1	4	yes	—
46	#14523977	C₂×C₄	1	2	yes	—
47	#14523977	C₂×C₄	1	2	yes	—
48	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
50	#14523965	C₂×C₄	1	4	yes	—
51	#14523965	C₂×C₄	1	4	yes	—
52	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
53	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
54	(C₂×D₆₇)⋊C₈	—	2	4	no	—
55	(C₂×D₆₇)⋊C₈	—	2	4	no	—
56	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
57	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
59	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
64	C₂²⋊C₈×C₆₇	—	2	2	no	—
65	#97×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	#11761628	—	8	4	no	—
71	C₂×Dic₁₃₄	—	2	2	no	—
72	C₂×D₂₆₈	—	2	2	no	—
73	C₂×D₂₆₈	—	2	2	no	—
74	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
75	C₂×C₂²⋊D₆₇	—	2	4	no	—
76	C₂×C₄×D₆₇	—	2	4	no	—
77	D₅₃₆	—	2	4	no	—
78	D₅₃₆	—	2	4	no	—
79	#11761611	—	8	4	no	—
80	C₂×C₆₇⋊C₈	—	2	1	no	—
81	C₂×C₆₇⋊C₈	—	2	1	no	—
82	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
83	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
84	C₈×D₆₇	—	8	4	no	—
85	C₄⋊Dic₆₇	—	2	2	no	—
86	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
87	(C₂×D₆₇)⋊C₄	—	2	4	no	—
88	(C₂×D₆₇)⋊C₄	—	2	4	no	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₇	#41	1	1	yes	characteristic
102	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	4	1	no	—
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂²⋊D₆₇	—	8	4	no	—
116	C₂×Dic₆₇	—	2	4	no	—
117	D₂₆₈	—	2	4	no	—
118	D₂₆₈	—	2	4	no	—
119	D₂₆₈	—	8	2	no	—
120	D₂₆₈	—	8	2	no	—
121	C₄×D₆₇	—	4	4	no	—
122	C₄×D₆₇	—	4	4	no	—
123	Dic₁₃₄	—	8	2	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	4	4	no	—
127	C₈×C₆₇	#131	1	2	yes	—
128	C₈×C₆₇	#131	1	2	yes	—
129	C₆₇⋊C₈	—	2	2	no	—
130	C₆₇⋊C₈	—	2	2	no	—
131	C₆₇⋊C₈	—	4	1	no	—
132	C₆₇⋊C₈	—	4	1	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×D₆₇	—	4	4	no	—
138	C₂×D₆₇	—	4	4	no	—
139	C₂×D₆₇	—	4	4	no	—
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
145	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
146	C₄×C₆₇	#576	1	2	yes	—
147	Dic₆₇	—	4	4	no	—
148	#17970	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	4	1	no	—
150	C₂×C₆₇	#2931	1	1	yes	characteristic
151	C₂×C₆₇	#2948	1	1	yes	characteristic
152	C₂×C₆₇	#2957	1	1	yes	characteristic
153	D₆₇	—	8	4	no	—
154	#3169	—	67	1	no	—
155	#2981	—	67	2	no	—
156	#1594	—	67	2	no	—
157	#1589	—	67	2	no	—
158	#1580	—	67	2	no	—
159	C₈×D₈	—	67	2	no	—
160	#1568	D₆₇	1	1	yes	characteristic, complemented
161	#1496	—	67	2	no	—
162	#1450	—	67	1	no	—
163	C₆₇	#17970	1	1	yes	characteristic, complemented, Sylow
164	#532	—	67	2	no	—
165	#500	—	67	1	no	—
166	#493	—	67	1	no	—
167	#486	—	67	2	no	—
168	#466	—	67	2	no	—
169	#466	—	67	2	no	—
170	C₄×D₈	—	67	2	no	—
171	C₄×D₈	—	67	2	no	—
172	#436	—	67	4	no	—
173	#435	—	134	4	no	—
174	C₈×D₄	—	67	4	no	—
175	#432	C₂×D₆₇	1	2	yes	—
176	#431	—	67	1	no	—
177	#431	—	67	1	no	—
178	#408	—	134	4	no	—
179	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
180	#335	C₂×D₆₇	1	2	yes	—
181	#334	C₂×D₆₇	1	2	yes	—
182	D₄⋊C₈	—	67	4	no	—
183	D₄⋊C₈	—	67	4	no	—
184	#322	—	67	1	no	—
185	#322	—	67	1	no	—
186	C₈²	—	67	1	no	—
187	C₈²	—	67	1	no	—
188	C₂×#41	—	134	2	no	—
189	C₂×D₈	—	67	2	no	—
190	C₂×D₈	—	67	2	no	—
191	C₂×M₁₆	—	134	4	no	—
192	C₂²×C₈	—	134	4	no	—
193	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
195	C₄⋊Q₈	—	67	2	no	—
196	#127	—	67	2	no	—
197	#124	—	134	2	no	—
198	#121	—	67	4	no	—
199	#121	—	134	2	no	—
200	C₄×D₄	—	67	4	no	—
201	C₄×D₄	—	134	2	no	—
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	#107	—	67	2	no	—
204	#107	—	67	2	no	—
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	134	1	no	—
207	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
208	#102	—	67	4	no	—
209	#102	—	67	4	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	67	2	no	—
212	C₂²⋊C₈	—	134	4	no	—
213	C₂²⋊C₈	—	134	4	no	—
214	C₂²⋊C₈	—	134	1	no	—
215	#97	—	2	2	no	—
216	C₄×C₈	—	67	1	no	—
217	C₄×C₈	—	67	1	no	—
218	C₄×C₈	—	67	1	no	—
219	C₄×C₈	—	67	1	no	—
220	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
221	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
224	#41	—	536	4	no	—
225	C₂×Q₈	—	134	2	no	—
226	C₂×D₄	—	67	4	no	—
227	C₂×D₄	—	134	4	no	—
228	C₂×D₄	—	134	2	no	—
229	C₂×D₄	—	134	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	#11761629	1	1	yes	characteristic
234	D₈	—	134	4	no	—
235	D₈	—	134	4	no	—
236	M₁₆	—	536	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	536	4	no	—
241	C₂×C₈	—	67	2	no	—
242	C₂×C₈	—	67	2	no	—
243	C₂×C₈	—	134	1	no	—
244	C₂×C₈	—	134	1	no	—
245	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₄⋊C₄	—	67	4	no	—
250	C₄⋊C₄	—	134	2	no	—
251	C₂²⋊C₄	—	134	4	no	—
252	C₂²⋊C₄	—	134	4	no	—
253	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₄²	D₄×D₆₇	1	1	yes	characteristic
255	C₄²	#11761629	1	1	yes	characteristic
256	C₄²	C₂³×D₆₇	1	1	yes	characteristic
257	C₂³	—	134	4	no	—
258	C₂³	#14524005	1	1	yes	characteristic
259	Q₈	—	536	2	no	—
260	D₄	—	536	2	no	—
261	D₄	—	536	2	no	—
262	D₄	—	536	4	no	—
263	D₄	—	134	4	no	—
264	D₄	—	134	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	268	4	no	—
271	C₂×C₄	—	134	4	no	—
272	C₂×C₄	#14524005	1	1	yes	characteristic
273	C₂×C₄	#14524005	1	1	yes	characteristic
274	C₂×C₄	#14524005	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	4	4	no	—
281	C₈	—	268	1	no	—
282	C₈	—	268	1	no	—
283	C₈	—	134	2	no	—
284	C₈	—	134	2	no	—
285	C₈	#14524051	1	2	yes	—
286	C₈	#14524051	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	268	4	no	—
291	C₂²	—	268	4	no	—
292	C₂²	—	268	4	no	—
293	C₂²	C₂×#14524005	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	268	4	no	—
299	C₄	C₂×#14524051	1	1	yes	characteristic
300	C₄	C₂×#14524051	1	1	yes	characteristic
301	C₄	#16864198	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	536	4	no	—
304	C₂	#22568920	1	1	yes	characteristic
305	C₂	#22569060	1	1	yes	characteristic
306	C₂	#22569125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1568	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570788	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#17970
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. #22569287	C₂	semidirect
3. #22559122	C₂	semidirect
4. #22559093	C₂	semidirect
5. #22559032	C₂	semidirect
6. #22559003	C₂	semidirect
7. #22558677	C₂	semidirect
8. #22558492	C₂	semidirect
9. C₆₇×#1568	C₂	semidirect
160. #1568	D₆₇	semidirect
163. C₆₇	#17970	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22569287	2
#22559122	2
#22559093	2
#22559032	2
#22559003	2
#22558677	2
#22558492	2
C₆₇×#1568	2
#17970	67