Subgroups — #30748992

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761628	C₂	1	1	yes	maximal, characteristic
2	#22560226	C₂	1	2	yes	maximal
3	C₄×#14523975	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558110	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11761628	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic, complemented
9	C₈×D₂₆₈	C₄	1	1	yes	characteristic, complemented
10	C₈×D₂₆₈	C₄	1	1	yes	characteristic, complemented
11	C₈×C₂²⋊D₆₇	—	2	1	no	—
12	C₈×C₂²⋊D₆₇	—	2	1	no	—
13	#16863449	C₂²	1	1	yes	characteristic
14	#16863447	—	2	1	no	—
15	C₂×#14523975	—	2	1	no	—
16	C₂×#14523969	—	2	2	no	—
17	C₄×C₈×D₆₇	—	2	1	no	—
18	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
19	#16863159	—	2	1	no	—
20	#16863157	—	2	2	no	—
21	#423×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#334×C₆₇	—	2	1	no	—
24	#334×C₆₇	—	2	1	no	—
25	C₂×#11761628	D₄	1	1	yes	characteristic
26	C₂×C₈×D₆₇	—	2	1	no	—
27	C₂×C₈×D₆₇	—	2	1	no	—
28	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
30	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
31	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
32	C₄×C₂²⋊D₆₇	—	2	1	no	—
33	C₄×C₂²⋊D₆₇	—	2	1	no	—
34	#14524002	D₄	1	1	yes	characteristic
35	#14524000	—	2	1	no	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄×Dic₆₇	—	2	2	no	—
38	C₄²×D₆₇	—	2	1	no	—
39	#14523975	—	2	4	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	Dic₆₇⋊C₈	—	2	1	no	—
43	Dic₆₇⋊C₈	—	2	1	no	—
44	#14523969	—	4	4	no	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
46	(C₂×D₆₇)⋊C₈	—	2	1	no	—
47	(C₂×D₆₇)⋊C₈	—	2	1	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₈×Dic₆₇	—	2	1	no	—
50	C₈×Dic₆₇	—	2	1	no	—
51	C₆₇⋊#95	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	—	2	2	no	—
57	C₄⋊C₈×C₆₇	—	2	2	no	—
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	#11761628	—	2	2	no	—
63	#11761628	SD₁₆	1	4	yes	—
64	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₆₇	—	2	1	no	—
67	C₂²×Dic₆₇	—	4	2	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	1	no	—
71	C₂×C₆₇⋊C₈	—	2	1	no	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₈×D₆₇	—	4	2	no	—
74	C₈×D₆₇	—	4	2	no	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	—	4	2	no	—
77	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
78	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
79	Dic₆₇⋊C₄	—	2	1	no	—
80	Dic₆₇⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₆₇	—	2	1	no	—
82	(C₂×D₆₇)⋊C₄	—	2	1	no	—
83	(C₂×D₆₇)⋊C₄	—	2	1	no	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₄×Dic₆₇	—	2	1	no	—
87	C₄×Dic₆₇	—	4	2	no	—
88	C₄×Dic₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
92	M₁₆×C₆₇	—	4	2	no	—
93	M₁₆×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	4	1	no	—
96	C₂×C₈×C₆₇	—	4	2	no	—
97	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₆₇	#41	1	1	yes	characteristic
105	C₂³×C₆₇	#121	1	1	yes	characteristic
106	C₂²×D₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	4	1	no	—
110	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₆₇	#135	1	2	yes	—
118	C₂²⋊D₆₇	—	2	4	no	—
119	C₂×Dic₆₇	—	2	1	no	—
120	C₂×Dic₆₇	—	2	1	no	—
121	C₂×Dic₆₇	—	2	1	no	—
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	—	8	2	no	—
124	C₂×Dic₆₇	—	8	2	no	—
125	D₂₆₈	—	2	1	no	—
126	D₂₆₈	#106	1	2	yes	complemented
127	C₄×D₆₇	—	2	4	no	—
128	C₄×D₆₇	—	4	2	no	—
129	C₄×D₆₇	—	4	1	no	—
130	C₄×D₆₇	—	4	1	no	—
131	Dic₁₃₄	—	2	1	no	—
132	Dic₁₃₄	#106	1	2	yes	complemented
133	C₈×C₆₇	—	2	1	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	8	2	no	—
137	C₈×C₆₇	#104	1	2	yes	complemented
138	C₆₇⋊C₈	—	4	1	no	—
139	C₆₇⋊C₈	—	4	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	#380	1	1	yes	characteristic
143	C₂²×C₆₇	#465	1	2	yes	—
144	C₂×D₆₇	—	2	2	no	—
145	C₂×D₆₇	—	4	1	no	—
146	C₂×D₆₇	—	4	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
151	C₄×C₆₇	#421	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
153	C₄×C₆₇	#427	1	1	yes	characteristic
154	C₄×C₆₇	#465	1	2	yes	—
155	Dic₆₇	—	2	2	no	—
156	Dic₆₇	—	4	1	no	—
157	Dic₆₇	—	4	1	no	—
158	Dic₆₇	—	8	4	no	—
159	#8666	—	67	1	no	maximal, Sylow
160	C₂×C₆₇	—	2	2	no	—
161	C₂×C₆₇	#1806	1	1	yes	characteristic
162	C₂×C₆₇	#1807	1	1	yes	characteristic
163	C₂×C₆₇	#1814	1	1	yes	characteristic
164	D₆₇	—	4	2	no	—
165	C₈×#41	—	67	1	no	—
166	#1806	—	67	2	no	—
167	C₄×#106	—	67	1	no	—
168	#1566	D₆₇	1	1	yes	characteristic, complemented
169	#1303	—	67	2	no	—
170	C₆₇	#8666	1	1	yes	characteristic, complemented, Sylow
171	C₄×#41	—	67	1	no	—
172	C₈×Q₈	—	67	1	no	—
173	C₈×Q₈	—	67	1	no	—
174	C₈×D₄	—	67	1	no	—
175	C₈×D₄	—	67	1	no	—
176	C₈×D₄	—	134	1	no	—
177	C₈×D₄	—	134	1	no	—
178	#431	—	67	1	no	—
179	#431	—	134	1	no	—
180	C₂×#106	—	134	1	no	—
181	#423	C₂×D₆₇	1	2	yes	—
182	C₂×#104	—	134	2	no	—
183	C₂×C₄×C₈	—	134	1	no	—
184	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
185	C₄×C₄⋊C₄	—	67	2	no	—
186	#340	—	134	1	no	—
187	#339	—	134	2	no	—
188	#334	—	2	1	no	—
189	#334	—	2	1	no	—
190	C₂×#41	—	67	1	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂²×C₈	—	134	1	no	—
193	C₂²×C₈	—	134	1	no	—
194	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
195	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
196	C₄×Q₈	—	67	1	no	—
197	C₄×Q₈	—	67	1	no	—
198	C₄×D₄	—	67	1	no	—
199	C₄×D₄	—	67	1	no	—
200	C₄×D₄	—	134	1	no	—
201	C₄×D₄	—	134	1	no	—
202	#117	—	67	1	no	—
203	#117	—	134	1	no	—
204	C₂×C₄⋊C₄	—	134	2	no	—
205	C₂×C₄²	—	134	2	no	—
206	C₂×C₄²	—	134	1	no	—
207	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
208	#106	—	134	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	67	1	no	—
212	C₄⋊C₈	—	67	1	no	—
213	C₄⋊C₈	—	134	1	no	—
214	C₄⋊C₈	—	134	1	no	—
215	#104	—	268	4	no	—
216	C₂²⋊C₈	—	134	1	no	—
217	C₂²⋊C₈	—	134	1	no	—
218	C₂²⋊C₈	—	134	1	no	—
219	C₄×C₈	—	134	1	no	—
220	C₄×C₈	—	134	1	no	—
221	C₄×C₈	—	134	1	no	—
222	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
223	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
226	#95	—	134	2	no	—
227	#41	—	67	4	no	—
228	#41	—	134	2	no	—
229	C₂×Q₈	—	67	1	no	—
230	C₂×D₄	—	67	1	no	—
231	C₂×D₄	—	134	1	no	—
232	C₂²×C₄	—	268	2	no	—
233	C₂²×C₄	—	134	1	no	—
234	C₂²×C₄	—	134	1	no	—
235	C₂²×C₄	—	134	1	no	—
236	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
238	C₂²×C₄	#11761629	1	1	yes	characteristic
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	268	2	no	—
245	C₂×C₈	—	268	2	no	—
246	C₂×C₈	—	134	1	no	—
247	C₂×C₈	—	134	1	no	—
248	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
249	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
250	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
251	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
252	C₄⋊C₄	—	268	2	no	—
253	C₄⋊C₄	—	67	1	no	—
254	C₄⋊C₄	—	67	1	no	—
255	C₄⋊C₄	—	134	4	no	—
256	C₄⋊C₄	—	134	1	no	—
257	C₄⋊C₄	—	134	1	no	—
258	C₂²⋊C₄	—	134	1	no	—
259	C₂²⋊C₄	—	134	1	no	—
260	C₂²⋊C₄	—	134	1	no	—
261	C₄²	—	268	2	no	—
262	C₄²	—	268	2	no	—
263	C₄²	—	134	1	no	—
264	C₄²	—	134	1	no	—
265	C₄²	—	134	1	no	—
266	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
268	C₄²	Q₈×D₆₇	1	1	yes	characteristic
269	C₄²	#11761627	1	1	yes	characteristic
270	C₂³	—	134	1	no	—
271	C₂³	#14524018	1	1	yes	characteristic
272	Q₈	—	67	2	no	—
273	Q₈	—	134	1	no	—
274	D₄	—	67	2	no	—
275	D₄	—	134	4	no	—
276	D₄	—	134	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	268	2	no	—
281	C₂×C₄	—	268	1	no	—
282	C₂×C₄	—	268	1	no	—
283	C₂×C₄	—	268	4	no	—
284	C₂×C₄	—	536	2	no	—
285	C₂×C₄	—	536	2	no	—
286	C₂×C₄	—	134	1	no	—
287	C₂×C₄	—	134	1	no	—
288	C₂×C₄	—	134	1	no	—
289	C₂×C₄	—	134	4	no	—
290	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₄	#14524001	1	1	yes	characteristic
293	C₂×C₄	#14524014	1	1	yes	characteristic
294	C₂×C₄	#14524019	1	1	yes	characteristic
295	C₂×C₄	#14524023	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
297	C₂×C₄	#14524064	1	2	yes	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	8	2	no	—
302	C₈	—	268	1	no	—
303	C₈	—	268	1	no	—
304	C₈	#14523969	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	268	1	no	—
308	C₂²	—	268	1	no	—
309	C₂²	—	134	2	no	—
310	C₂²	#16863253	1	1	yes	characteristic
311	C₂²	#16863626	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	268	1	no	—
316	C₄	—	268	1	no	—
317	C₄	—	536	4	no	—
318	C₄	—	134	2	no	—
319	C₄	C₂×#14523969	1	1	yes	characteristic
320	C₄	#16863412	1	1	yes	characteristic
321	C₄	#106×D₆₇	1	1	yes	characteristic
322	C₄	#16863431	1	1	yes	characteristic
323	C₄	#16863622	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	268	2	no	—
326	C₂	#22560226	1	1	yes	characteristic
327	C₂	#22560231	1	1	yes	characteristic
328	C₂	#22560264	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#11761628	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#8666
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. C₄×#14523975	C₂	semidirect
4. #22558110	C₂	semidirect
5. C₆₇×#1566	C₂	semidirect
7. C₈×Dic₁₃₄	C₄	semidirect
8. C₈×Dic₁₃₄	C₄	semidirect
9. C₈×D₂₆₈	C₄	semidirect
10. C₈×D₂₆₈	C₄	semidirect
126. D₂₆₈	#106	semidirect
132. Dic₁₃₄	#106	semidirect
137. C₈×C₆₇	#104	semidirect
168. #1566	D₆₇	semidirect
170. C₆₇	#8666	semidirect
304. C₈	#14523969	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11761628	2
#22560226	2
C₄×#14523975	2
#22558110	2
C₆₇×#1566	2
#8666	67