Subgroups — #30746408

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863473	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523959	C₂	1	1	yes	maximal, characteristic
3	#22558529	C₂	1	1	yes	maximal, characteristic
4	#22558499	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558059	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761611	C₄	1	2	yes	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	#16863473	—	2	1	no	—
11	#16863473	—	2	1	no	—
12	#16863473	—	2	1	no	—
13	#16863473	—	2	1	no	—
14	#16863449	C₂²	1	1	yes	characteristic
15	C₂×#14523959	C₄	1	2	yes	—
16	C₂×(C₂×D₆₇)⋊C₈	C₄	1	2	yes	—
17	C₄×#11761612	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	#16863164	C₄	1	2	yes	—
20	#16863119	C₄	1	4	yes	—
21	#16863118	—	2	1	no	—
22	#16863117	—	2	1	no	—
23	#16863117	—	2	1	no	—
24	C₈×C₆₇⋊C₈	—	2	1	no	—
25	#431×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₆₇	C₂×C₄	1	2	yes	—
30	C₂×#11761612	C₂×C₄	1	2	yes	—
31	C₂×#11761611	—	2	2	no	—
32	C₂×#11761611	—	2	2	no	—
33	C₂×#11761611	—	4	4	no	—
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₆₈	—	2	1	no	—
37	C₄×D₂₆₈	—	2	1	no	—
38	C₄×D₂₆₈	—	2	1	no	—
39	C₄×D₂₆₈	—	2	1	no	—
40	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	#14523959	—	2	4	no	—
44	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
45	(C₂×D₆₇)⋊C₈	—	2	2	no	—
46	(C₂×D₆₇)⋊C₈	—	2	2	no	—
47	#14523954	—	2	1	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
53	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	—	2	1	no	—
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
61	#97×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₄×C₈×C₆₇	—	2	1	no	—
64	C₂³×D₆₇	C₄²	1	2	yes	—
65	C₂×D₂₆₈	—	2	1	no	—
66	C₂×D₂₆₈	—	2	1	no	—
67	C₂×D₂₆₈	—	2	2	no	—
68	C₂×D₂₆₈	—	2	2	no	—
69	C₂²×Dic₆₇	C₄²	1	2	yes	—
70	C₂×C₄×D₆₇	—	2	2	no	—
71	C₂×C₄×D₆₇	—	2	2	no	—
72	C₂×C₄×D₆₇	—	4	2	no	—
73	C₂×C₄×D₆₇	—	4	2	no	—
74	#11761612	—	2	2	no	—
75	#11761612	—	2	2	no	—
76	#11761611	—	4	4	no	—
77	#11761611	—	4	4	no	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	—	2	2	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	4	1	no	—
82	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊Dic₆₇	—	2	1	no	—
87	C₄⋊Dic₆₇	—	2	1	no	—
88	(C₂×D₆₇)⋊C₄	—	2	2	no	—
89	(C₂×D₆₇)⋊C₄	—	2	2	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	4	2	no	—
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₇	#41	1	1	yes	characteristic
102	C₄²×C₆₇	#41	1	1	yes	characteristic
103	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂²×D₆₇	—	2	2	no	—
105	C₂²×D₆₇	—	2	2	no	—
106	C₂²×D₆₇	—	2	2	no	—
107	C₂²×D₆₇	—	4	2	no	—
108	C₂²×D₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₆₇	#131	1	2	yes	—
120	C₂×Dic₆₇	—	2	2	no	—
121	C₂×Dic₆₇	—	4	2	no	—
122	D₂₆₈	—	4	2	no	—
123	D₂₆₈	—	4	2	no	—
124	C₄×D₆₇	—	4	4	no	—
125	C₄×D₆₇	—	4	2	no	—
126	C₄×D₆₇	—	4	2	no	—
127	C₈×C₆₇	—	2	4	no	—
128	C₈×C₆₇	—	4	2	no	—
129	C₈×C₆₇	—	4	2	no	—
130	C₆₇⋊C₈	—	2	1	no	—
131	C₆₇⋊C₈	—	2	1	no	—
132	C₆₇⋊C₈	—	2	1	no	—
133	C₆₇⋊C₈	—	4	2	no	—
134	C₆₇⋊C₈	—	4	2	no	—
135	C₆₇⋊C₈	#100	1	2	yes	complemented
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
140	C₂²×C₆₇	#405	1	1	yes	characteristic
141	C₂×D₆₇	—	4	2	no	—
142	C₂×D₆₇	—	4	2	no	—
143	C₂×D₆₇	—	4	2	no	—
144	C₂×D₆₇	—	8	2	no	—
145	C₂×D₆₇	—	8	2	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
150	C₄×C₆₇	#413	1	1	yes	characteristic
151	C₄×C₆₇	#435	1	2	yes	—
152	C₄×C₆₇	#436	1	2	yes	—
153	Dic₆₇	—	4	2	no	—
154	#8166	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	#1753	1	1	yes	characteristic
158	C₂×C₆₇	C₄×#100	1	1	yes	characteristic
159	C₂×C₆₇	#1757	1	1	yes	characteristic
160	D₆₇	—	8	2	no	—
161	C₂×#436	—	67	1	no	—
162	C₄×#100	—	67	1	no	—
163	#1460	—	67	1	no	—
164	#1451	—	67	1	no	—
165	C₈×M₁₆	—	67	1	no	—
166	#1283	—	67	1	no	—
167	#1283	D₆₇	1	1	yes	characteristic, complemented
168	C₆₇	#8166	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	67	2	no	—
170	C₂×C₄×D₄	—	67	1	no	—
171	#436	—	134	1	no	—
172	#436	—	134	1	no	—
173	#436	—	134	1	no	—
174	#436	—	134	1	no	—
175	#431	—	67	1	no	—
176	#431	C₂×D₆₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
178	C₂×#100	—	67	2	no	—
179	C₂×C₂²⋊C₈	—	67	2	no	—
180	C₄×M₁₆	—	67	1	no	—
181	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
182	C₂×C₄×C₈	—	67	1	no	—
183	#343	—	67	2	no	—
184	#323	—	67	4	no	—
185	#322	—	134	1	no	—
186	#322	—	134	1	no	—
187	#322	—	134	1	no	—
188	C₈²	—	134	1	no	—
189	C₂²×D₄	—	67	1	no	—
190	C₂³×C₄	—	67	2	no	—
191	C₂×M₁₆	—	268	4	no	—
192	C₂×M₁₆	—	67	2	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
196	C₂²×C₈	—	67	1	no	—
197	C₂²×C₈	—	67	1	no	—
198	C₂²×C₈	C₄×D₆₇	1	2	yes	—
199	C₄×D₄	—	134	1	no	—
200	C₄×D₄	—	134	1	no	—
201	C₄×D₄	—	134	1	no	—
202	C₄×D₄	—	134	1	no	—
203	C₂×C₄⋊C₄	—	67	1	no	—
204	C₂×C₂²⋊C₄	—	67	2	no	—
205	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	134	1	no	—
210	#100	—	134	4	no	—
211	C₂²⋊C₈	—	67	2	no	—
212	C₂²⋊C₈	—	134	2	no	—
213	C₂²⋊C₈	—	134	2	no	—
214	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	134	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	67	1	no	—
220	C₄×C₈	—	67	1	no	—
221	C₄×C₈	—	67	1	no	—
222	C₄×C₈	—	67	1	no	—
223	C₄×C₈	—	134	1	no	—
224	C₄×C₈	—	134	1	no	—
225	C₂⁴	—	67	2	no	—
226	C₂×D₄	—	134	1	no	—
227	C₂×D₄	—	134	1	no	—
228	C₂×D₄	—	134	2	no	—
229	C₂×D₄	—	134	2	no	—
230	C₂²×C₄	—	268	2	no	—
231	C₂²×C₄	—	268	2	no	—
232	C₂²×C₄	—	67	2	no	—
233	C₂²×C₄	—	134	2	no	—
234	C₂²×C₄	—	134	2	no	—
235	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	268	4	no	—
240	M₁₆	—	268	4	no	—
241	M₁₆	—	134	2	no	—
242	M₁₆	—	134	2	no	—
243	M₁₆	#11761611	1	4	yes	complemented
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	268	1	no	—
249	C₂×C₈	—	67	1	no	—
250	C₂×C₈	—	67	1	no	—
251	C₂×C₈	—	67	1	no	—
252	C₂×C₈	—	67	1	no	—
253	C₂×C₈	—	134	2	no	—
254	C₂×C₈	—	134	2	no	—
255	C₂×C₈	—	134	2	no	—
256	C₄⋊C₄	—	134	1	no	—
257	C₄⋊C₄	—	134	1	no	—
258	C₂²⋊C₄	—	134	2	no	—
259	C₂²⋊C₄	—	134	2	no	—
260	C₄²	D₄×D₆₇	1	1	yes	characteristic
261	C₄²	D₄×D₆₇	1	1	yes	characteristic
262	C₄²	#11761627	1	1	yes	characteristic
263	C₄²	#11761629	1	1	yes	characteristic
264	C₂³	—	268	2	no	—
265	C₂³	—	268	2	no	—
266	C₂³	—	134	2	no	—
267	C₂³	—	134	2	no	—
268	C₂³	—	134	2	no	—
269	C₂³	C₄²×D₆₇	1	1	yes	characteristic
270	D₄	—	268	2	no	—
271	D₄	—	268	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	268	4	no	—
276	C₂×C₄	—	268	2	no	—
277	C₂×C₄	—	268	2	no	—
278	C₂×C₄	—	268	2	no	—
279	C₂×C₄	—	134	2	no	—
280	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
281	C₂×C₄	#14524001	1	1	yes	characteristic
282	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#14524005	1	1	yes	characteristic
285	C₂×C₄	#14524005	1	1	yes	characteristic
286	C₂×C₄	C₂×#11761611	1	2	yes	—
287	C₂×C₄	#14524051	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	268	2	no	—
292	C₈	—	268	2	no	—
293	C₈	—	67	2	no	—
294	C₈	—	134	1	no	—
295	C₈	—	134	1	no	—
296	C₈	—	134	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	268	2	no	—
300	C₂²	—	268	2	no	—
301	C₂²	—	268	2	no	—
302	C₂²	—	536	2	no	—
303	C₂²	—	536	2	no	—
304	C₂²	#16863243	1	1	yes	characteristic
305	C₂²	C₄×#11761611	1	1	yes	characteristic
306	C₂²	#16863353	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	268	2	no	—
311	C₄	#100×D₆₇	1	1	yes	characteristic
312	C₄	#16863379	1	1	yes	characteristic
313	C₄	#16863464	1	2	yes	—
314	C₄	#16863473	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	536	2	no	—
318	C₂	#22559989	1	1	yes	characteristic
319	C₂	#22560000	1	1	yes	characteristic
320	C₂	#22560008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₆₇
commutator	C₂³×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1283	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1283	D₆₇
hypocenter	C₆₇	#8166
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
4. #22558499	C₂	semidirect
5. M₁₆×C₆₇⋊C₈	C₂	semidirect
6. #22558059	C₂	semidirect
7. C₆₇×#1283	C₂	semidirect
94. M₁₆×C₆₇	M₁₆	semidirect
135. C₆₇⋊C₈	#100	semidirect
167. #1283	D₆₇	semidirect
168. C₆₇	#8166	semidirect
243. M₁₆	#11761611	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863473	2
C₄×#14523959	2
#22558529	2
#22558499	2
M₁₆×C₆₇⋊C₈	2
#22558059	2
C₆₇×#1283	2
#8166	67