Subgroups — #30748974

Order 17152.

450 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#22560380	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558635	C₂	1	1	yes	maximal, characteristic
5	#22558633	C₂	1	1	yes	maximal, characteristic
6	#22558106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761611	—	2	1	no	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	#16863473	—	2	1	no	—
11	#16863473	—	2	1	no	—
12	#16863473	C₄	1	2	yes	—
13	#16863473	C₄	1	2	yes	—
14	#16863449	C₂²	1	1	yes	characteristic
15	C₂×#14523978	—	2	1	no	—
16	C₂×#14523978	—	2	1	no	—
17	#16863420	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	#16863148	—	2	1	no	—
21	#16863146	—	2	1	no	—
22	#16863125	—	2	2	no	—
23	#16863123	—	2	2	no	—
24	#431×C₆₇	C₂²	1	1	yes	characteristic
25	#423×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×D₂₆₈	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₆₇	—	2	1	no	—
29	C₂×#11761612	—	2	1	no	—
30	C₂×#11761611	—	2	2	no	—
31	C₂×#11761611	—	2	2	no	—
32	C₂×#11761611	—	4	2	no	—
33	C₂×#11761611	—	4	2	no	—
34	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
36	C₄×D₂₆₈	—	2	1	no	—
37	C₄×D₂₆₈	—	2	1	no	—
38	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
39	C₄×D₂₆₈	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
41	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
42	#14523978	—	4	2	no	—
43	#14523978	—	4	2	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
48	(C₂×D₆₇)⋊C₈	—	2	2	no	—
49	(C₂×D₆₇)⋊C₈	—	2	2	no	—
50	C₄×C₆₇⋊C₈	—	2	1	no	—
51	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
54	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₆₇	—	2	1	no	—
64	C₄×C₈×C₆₇	—	2	1	no	—
65	C₂³×D₆₇	—	2	1	no	—
66	C₂×D₂₆₈	—	2	1	no	—
67	C₂×D₂₆₈	—	4	2	no	—
68	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
69	C₂²×Dic₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	2	no	—
71	C₂×C₄×D₆₇	—	2	2	no	—
72	C₂×C₄×D₆₇	—	4	1	no	—
73	C₂×C₄×D₆₇	—	4	1	no	—
74	C₂×C₄×D₆₇	—	4	1	no	—
75	C₂×C₄×D₆₇	—	4	1	no	—
76	#11761612	—	4	1	no	—
77	#11761612	—	4	1	no	—
78	#11761611	—	4	4	no	—
79	#11761611	—	4	4	no	—
80	C₂×C₆₇⋊C₈	—	2	1	no	—
81	C₂×C₆₇⋊C₈	—	2	1	no	—
82	C₂×C₆₇⋊C₈	—	4	1	no	—
83	C₂×C₆₇⋊C₈	—	4	1	no	—
84	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
89	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
90	C₄⋊Dic₆₇	—	2	1	no	—
91	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
92	(C₂×D₆₇)⋊C₄	—	2	2	no	—
93	(C₂×D₆₇)⋊C₄	—	2	2	no	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
97	M₁₆×C₆₇	—	4	1	no	—
98	M₁₆×C₆₇	—	4	1	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	—	4	1	no	—
103	C₂×C₈×C₆₇	—	4	1	no	—
104	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₇	#41	1	1	yes	characteristic
108	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×D₆₇	—	2	2	no	—
110	C₂²×D₆₇	—	4	2	no	—
111	C₂²×D₆₇	—	4	2	no	—
112	C₂²×D₆₇	—	4	1	no	—
113	C₂²×D₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	2	no	—
127	C₂×Dic₆₇	—	4	1	no	—
128	C₂×Dic₆₇	—	4	1	no	—
129	D₂₆₈	—	2	4	no	—
130	D₂₆₈	—	4	2	no	—
131	C₄×D₆₇	—	4	2	no	—
132	C₄×D₆₇	—	4	2	no	—
133	C₄×D₆₇	—	4	2	no	—
134	C₄×D₆₇	—	4	2	no	—
135	C₈×C₆₇	—	4	2	no	—
136	C₈×C₆₇	—	8	1	no	—
137	C₈×C₆₇	—	8	1	no	—
138	C₆₇⋊C₈	—	2	1	no	—
139	C₆₇⋊C₈	—	2	1	no	—
140	C₆₇⋊C₈	—	8	1	no	—
141	C₆₇⋊C₈	—	8	1	no	—
142	C₆₇⋊C₈	#102	1	2	yes	—
143	C₆₇⋊C₈	#102	1	2	yes	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	#380	1	1	yes	characteristic
147	C₂²×C₆₇	#474	1	1	yes	characteristic
148	C₂²×C₆₇	#476	1	1	yes	characteristic
149	C₂×D₆₇	—	4	2	no	—
150	C₂×D₆₇	—	4	2	no	—
151	C₂×D₆₇	—	4	2	no	—
152	C₂×D₆₇	—	8	2	no	—
153	C₂×D₆₇	—	8	2	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
158	C₄×C₆₇	#417	1	1	yes	characteristic
159	C₄×C₆₇	#430	1	2	yes	—
160	C₄×C₆₇	#459	1	1	yes	characteristic
161	C₄×C₆₇	#461	1	1	yes	characteristic
162	Dic₆₇	—	4	2	no	—
163	#8662	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	#1802	1	1	yes	characteristic
167	C₂×C₆₇	#1811	1	1	yes	characteristic
168	C₂×C₆₇	#1814	1	1	yes	characteristic
169	D₆₇	—	8	2	no	—
170	C₂×#436	—	67	1	no	—
171	#1848	—	67	1	no	—
172	#1564	—	67	1	no	—
173	#1486	—	67	1	no	—
174	#1486	—	67	1	no	—
175	#1301	—	67	1	no	—
176	#1301	D₆₇	1	1	yes	characteristic, complemented
177	C₆₇	#8662	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	134	1	no	—
179	C₂×C₄×D₄	—	67	1	no	—
180	#436	—	67	2	no	—
181	#436	—	67	2	no	—
182	#436	—	134	1	no	—
183	#436	—	134	1	no	—
184	#431	—	67	1	no	—
185	#431	C₂×D₆₇	1	1	yes	characteristic
186	C₂×#108	—	134	1	no	—
187	C₂×#108	—	134	1	no	—
188	#423	—	67	1	no	—
189	#423	C₂×D₆₇	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₈	—	134	1	no	—
192	C₂×C₄×C₈	—	67	1	no	—
193	#335	—	134	1	no	—
194	#334	—	134	1	no	—
195	D₄⋊C₈	—	134	2	no	—
196	D₄⋊C₈	—	134	2	no	—
197	C₂²×D₄	—	67	1	no	—
198	C₂³×C₄	—	134	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	268	2	no	—
201	C₂×M₁₆	—	268	2	no	—
202	C₂×M₁₆	—	134	2	no	—
203	C₂×M₁₆	—	134	2	no	—
204	C₂×M₁₆	—	134	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	67	1	no	—
207	C₂²×C₈	—	67	1	no	—
208	C₄×D₄	—	67	2	no	—
209	C₄×D₄	—	67	2	no	—
210	C₄×D₄	—	134	1	no	—
211	C₄×D₄	—	134	1	no	—
212	C₂×C₄⋊C₄	—	67	1	no	—
213	C₂×C₂²⋊C₄	—	134	1	no	—
214	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
215	#108	—	268	2	no	—
216	#108	—	268	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	134	1	no	—
222	C₄⋊C₈	—	134	1	no	—
223	C₄⋊C₈	—	134	1	no	—
224	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	134	2	no	—
227	C₂²⋊C₈	—	134	2	no	—
228	C₂²⋊C₈	—	134	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	67	1	no	—
231	C₄×C₈	—	67	1	no	—
232	C₄×C₈	—	67	1	no	—
233	C₄×C₈	—	67	1	no	—
234	C₄×C₈	—	134	1	no	—
235	C₂⁴	—	134	1	no	—
236	C₂×D₄	—	268	2	no	—
237	C₂×D₄	—	67	2	no	—
238	C₂×D₄	—	134	1	no	—
239	C₂²×C₄	—	268	1	no	—
240	C₂²×C₄	—	268	1	no	—
241	C₂²×C₄	—	268	1	no	—
242	C₂²×C₄	—	268	1	no	—
243	C₂²×C₄	—	134	1	no	—
244	C₂²×C₄	—	134	2	no	—
245	C₂²×C₄	—	134	2	no	—
246	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
248	C₂²×C₄	#11761628	1	1	yes	characteristic
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	268	4	no	—
252	M₁₆	—	268	4	no	—
253	M₁₆	—	268	1	no	—
254	M₁₆	—	268	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	268	1	no	—
261	C₂×C₈	—	268	1	no	—
262	C₂×C₈	—	67	1	no	—
263	C₂×C₈	—	67	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	67	1	no	—
267	C₂×C₈	—	67	1	no	—
268	C₂×C₈	—	134	1	no	—
269	C₂×C₈	—	134	1	no	—
270	C₄⋊C₄	—	67	2	no	—
271	C₄⋊C₄	—	134	1	no	—
272	C₂²⋊C₄	—	134	2	no	—
273	C₂²⋊C₄	—	134	2	no	—
274	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
275	C₄²	D₄×D₆₇	1	1	yes	characteristic
276	C₄²	D₄×D₆₇	1	1	yes	characteristic
277	C₄²	#11761628	1	1	yes	characteristic
278	C₂³	—	268	2	no	—
279	C₂³	—	268	2	no	—
280	C₂³	—	268	1	no	—
281	C₂³	—	268	1	no	—
282	C₂³	—	134	2	no	—
283	C₂³	#14524021	1	1	yes	characteristic
284	D₄	—	268	2	no	—
285	D₄	—	134	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	268	1	no	—
289	C₂×C₄	—	268	1	no	—
290	C₂×C₄	—	268	2	no	—
291	C₂×C₄	—	268	2	no	—
292	C₂×C₄	—	268	2	no	—
293	C₂×C₄	—	268	2	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
296	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
297	C₂×C₄	#14524000	1	1	yes	characteristic
298	C₂×C₄	#14524016	1	1	yes	characteristic
299	C₂×C₄	#14524016	1	1	yes	characteristic
300	C₂×C₄	#14524021	1	1	yes	characteristic
301	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
303	C₂×C₄	#14524070	1	1	yes	characteristic
304	C₂×C₄	#14524075	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	—	8	1	no	—
308	C₈	—	536	1	no	—
309	C₈	—	536	1	no	—
310	C₈	—	67	2	no	—
311	C₈	—	67	2	no	—
312	C₈	—	134	1	no	—
313	C₈	—	134	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	268	2	no	—
317	C₂²	—	268	2	no	—
318	C₂²	—	268	2	no	—
319	C₂²	—	536	2	no	—
320	C₂²	—	536	2	no	—
321	C₂²	#16863255	1	1	yes	characteristic
322	C₂²	#16863675	1	1	yes	characteristic
323	C₂²	#16863685	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	268	2	no	—
328	C₄	#102×D₆₇	1	1	yes	characteristic
329	C₄	#16863396	1	1	yes	characteristic
330	C₄	#16863443	1	2	yes	—
331	C₄	#16863590	1	1	yes	characteristic
332	C₄	#16863600	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	536	2	no	—
336	C₂	#22560208	1	1	yes	characteristic
337	C₂	#22560250	1	1	yes	characteristic
338	C₂	#22560262	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524016
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1301	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×D₂₆₈	Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1301	D₆₇
hypocenter	C₆₇	#8662
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #22560380	C₂	semidirect
3. #22558955	C₂	semidirect
6. #22558106	C₂	semidirect
7. C₆₇×#1301	C₂	semidirect
176. #1301	D₆₇	semidirect
177. C₆₇	#8662	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
#22560380	2
#22558955	2
#22558635	2
#22558633	2
#22558106	2
C₆₇×#1301	2
#8662	67