Subgroups — #30748540

Order 17152.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562686	C₂	1	1	yes	maximal, characteristic
2	#22560361	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863128	C₂	1	2	yes	maximal
4	#22558049	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863302	C₄	1	1	yes	characteristic
11	#16863302	C₄	1	1	yes	characteristic
12	#16863294	C₄	1	1	yes	characteristic
13	#16863294	C₄	1	1	yes	characteristic
14	#16863280	—	2	1	no	—
15	#16863280	—	2	1	no	—
16	C₄×Dic₆₇⋊C₄	—	2	1	no	—
17	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863242	—	2	1	no	—
19	#16863128	—	2	4	no	—
20	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	#336×C₆₇	—	2	1	no	—
24	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₄×Dic₁₃₄	—	2	1	no	—
27	C₄×Dic₁₃₄	—	2	1	no	—
28	C₄×Dic₁₃₄	D₄	1	2	yes	—
29	C₄×Dic₁₃₄	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₆₇⋊C₄	—	2	1	no	—
34	C₂×Dic₆₇⋊C₄	—	2	1	no	—
35	C₂×Dic₆₇⋊C₄	—	2	1	no	—
36	C₂×C₄×Dic₆₇	—	2	1	no	—
37	C₂×C₄×Dic₆₇	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
41	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
42	C₆₇⋊#95	—	2	1	no	—
43	C₆₇⋊#95	—	2	1	no	—
44	C₆₇⋊#95	—	2	1	no	—
45	C₆₇⋊#95	—	2	1	no	—
46	C₆₇⋊#95	—	2	1	no	—
47	C₆₇⋊#95	—	2	1	no	—
48	C₆₇⋊#95	—	2	1	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	C₄×C₈×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	—	2	2	no	—
60	C₂×Dic₁₃₄	—	2	2	no	—
61	C₂×Dic₁₃₄	—	2	2	no	—
62	C₂×Dic₁₃₄	—	2	1	no	—
63	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₆₇	—	2	1	no	—
65	C₂²×Dic₆₇	—	2	1	no	—
66	C₂²×Dic₆₇	—	2	1	no	—
67	C₂²×Dic₆₇	—	2	1	no	—
68	C₂×C₆₇⋊C₈	—	4	2	no	—
69	C₂×C₆₇⋊C₈	—	4	2	no	—
70	C₂×C₆₇⋊C₈	—	4	2	no	—
71	C₄⋊Dic₆₇	—	2	1	no	—
72	C₄⋊Dic₆₇	—	2	2	no	—
73	C₄⋊Dic₆₇	—	2	2	no	—
74	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
75	Dic₆₇⋊C₄	—	2	2	no	—
76	Dic₆₇⋊C₄	—	2	2	no	—
77	Dic₆₇⋊C₄	—	4	2	no	—
78	Dic₆₇⋊C₄	—	4	2	no	—
79	C₄×Dic₆₇	—	2	2	no	—
80	C₄×Dic₆₇	—	2	2	no	—
81	C₄×Dic₆₇	—	4	1	no	—
82	C₄×Dic₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	—	4	2	no	—
92	C₂×C₈×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	4	1	no	—
99	C₄²×C₆₇	SD₁₆	1	2	yes	—
100	C₄²×C₆₇	Q₁₆	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₄²×C₆₇	#41	1	1	yes	characteristic
104	C₄²×C₆₇	#41	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	#103	1	2	yes	—
118	C₂×C₄×C₆₇	#103	1	2	yes	—
119	C₂×C₄×C₆₇	#104	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#131	1	2	yes	—
127	C₂×C₄×C₆₇	#131	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
132	C₂×Dic₆₇	—	2	2	no	—
133	C₂×Dic₆₇	—	2	2	no	—
134	C₂×Dic₆₇	—	4	1	no	—
135	C₂×Dic₆₇	—	4	1	no	—
136	C₂×Dic₆₇	—	4	1	no	—
137	C₂×Dic₆₇	—	4	1	no	—
138	C₂×Dic₆₇	—	4	1	no	—
139	C₂×Dic₆₇	—	4	1	no	—
140	C₂×Dic₆₇	—	4	1	no	—
141	C₂×Dic₆₇	—	4	1	no	—
142	C₂×Dic₆₇	—	4	1	no	—
143	C₂×Dic₆₇	—	4	1	no	—
144	Dic₁₃₄	—	2	4	no	—
145	Dic₁₃₄	—	4	2	no	—
146	C₈×C₆₇	—	4	4	no	—
147	C₆₇⋊C₈	—	8	2	no	—
148	C₂²×C₆₇	#380	1	1	yes	characteristic
149	C₂²×C₆₇	#408	1	1	yes	characteristic
150	C₂²×C₆₇	#408	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
152	C₂²×C₆₇	#417	1	1	yes	characteristic
153	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
154	C₂²×C₆₇	#421	1	1	yes	characteristic
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	—	4	1	no	—
161	C₄×C₆₇	#424	1	2	yes	—
162	C₄×C₆₇	#433	1	2	yes	—
163	C₄×C₆₇	#460	1	1	yes	characteristic
164	C₄×C₆₇	#462	1	1	yes	characteristic
165	C₄×C₆₇	#475	1	1	yes	characteristic
166	C₄×C₆₇	#477	1	1	yes	characteristic
167	Dic₆₇	—	4	2	no	—
168	Dic₆₇	—	4	2	no	—
169	Dic₆₇	—	8	1	no	—
170	Dic₆₇	—	8	1	no	—
171	#8568	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	#1481	1	1	yes	characteristic
173	C₂×C₆₇	#1483	1	1	yes	characteristic
174	C₂×C₆₇	#1493	1	1	yes	characteristic
175	C₂×C₆₇	#1495	1	1	yes	characteristic
176	C₂×C₆₇	#1798	1	1	yes	characteristic
177	C₂×C₆₇	#1803	1	1	yes	characteristic
178	C₂×C₆₇	#1806	1	1	yes	characteristic
179	#2295	—	67	1	no	—
180	#1842	—	67	1	no	—
181	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
182	C₂×Q₈⋊C₈	—	67	2	no	—
183	#1279	—	67	2	no	—
184	C₆₇	#8568	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	—	67	1	no	—
186	C₂×C₄⋊C₈	—	2	1	no	—
187	C₂×C₄⋊C₈	—	67	2	no	—
188	C₂×C₄⋊C₈	—	134	1	no	—
189	C₂×C₄⋊C₈	—	134	1	no	—
190	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
191	#391	—	67	1	no	—
192	#391	—	67	1	no	—
193	#389	—	67	1	no	—
194	#389	—	67	1	no	—
195	#387	—	134	1	no	—
196	#387	—	134	1	no	—
197	C₄×C₄⋊C₄	—	67	1	no	—
198	C₄×C₄⋊C₄	—	134	1	no	—
199	#376	—	134	1	no	—
200	C₄³	C₂×D₆₇	1	1	yes	characteristic
201	#336	—	2	1	no	—
202	Q₈⋊C₈	—	134	4	no	—
203	C₂²×Q₈	—	67	1	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	134	2	no	—
206	C₄×Q₈	—	67	2	no	—
207	C₄×Q₈	—	67	2	no	—
208	C₄×Q₈	—	134	1	no	—
209	C₄×Q₈	—	134	1	no	—
210	C₂×C₄⋊C₄	—	67	1	no	—
211	C₂×C₄⋊C₄	—	67	1	no	—
212	C₂×C₄⋊C₄	—	67	1	no	—
213	C₂×C₄⋊C₄	—	134	1	no	—
214	C₂×C₄⋊C₄	—	134	1	no	—
215	C₂×C₄⋊C₄	—	134	1	no	—
216	C₂×C₄²	—	134	1	no	—
217	C₂×C₄²	—	134	1	no	—
218	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
220	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
221	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	268	2	no	—
227	C₄⋊C₈	—	268	2	no	—
228	C₄⋊C₈	—	134	2	no	—
229	C₄⋊C₈	—	134	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	#95	—	134	1	no	—
233	#95	—	134	1	no	—
234	#95	—	134	1	no	—
235	#95	—	134	1	no	—
236	#95	—	134	1	no	—
237	#95	—	134	1	no	—
238	#95	—	134	1	no	—
239	C₂×Q₈	—	67	2	no	—
240	C₂×Q₈	—	134	2	no	—
241	C₂×Q₈	—	134	2	no	—
242	C₂×Q₈	—	134	1	no	—
243	C₂²×C₄	—	134	1	no	—
244	C₂²×C₄	—	134	1	no	—
245	C₂²×C₄	—	134	1	no	—
246	C₂²×C₄	—	134	1	no	—
247	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	268	2	no	—
258	C₂×C₈	—	268	2	no	—
259	C₂×C₈	—	268	2	no	—
260	C₄⋊C₄	—	268	2	no	—
261	C₄⋊C₄	—	268	2	no	—
262	C₄⋊C₄	—	67	2	no	—
263	C₄⋊C₄	—	134	2	no	—
264	C₄⋊C₄	—	134	2	no	—
265	C₄⋊C₄	—	134	1	no	—
266	C₄⋊C₄	—	134	2	no	—
267	C₄⋊C₄	—	134	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	268	1	no	—
275	C₄²	—	268	1	no	—
276	C₄²	—	134	2	no	—
277	C₄²	—	134	2	no	—
278	C₄²	#11761617	1	2	yes	—
279	C₄²	Dic₂₆₈	1	2	yes	—
280	C₄²	D₄×D₆₇	1	1	yes	characteristic
281	C₄²	Q₈×D₆₇	1	1	yes	characteristic
282	C₄²	#11761627	1	1	yes	characteristic
283	C₄²	#11761629	1	1	yes	characteristic
284	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
285	Q₈	—	268	2	no	—
286	Q₈	—	134	4	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	268	1	no	—
299	C₂×C₄	—	268	1	no	—
300	C₂×C₄	—	268	1	no	—
301	C₂×C₄	—	268	1	no	—
302	C₂×C₄	—	268	1	no	—
303	C₂×C₄	—	268	1	no	—
304	C₂×C₄	—	268	1	no	—
305	C₂×C₄	—	268	1	no	—
306	C₂×C₄	—	268	1	no	—
307	C₂×C₄	—	268	1	no	—
308	C₂×C₄	—	134	2	no	—
309	C₂×C₄	—	134	2	no	—
310	C₂×C₄	#14523968	1	2	yes	—
311	C₂×C₄	#14523968	1	2	yes	—
312	C₂×C₄	#14523971	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₄	#14524001	1	1	yes	characteristic
315	C₂×C₄	#14524014	1	1	yes	characteristic
316	C₂×C₄	#14524018	1	1	yes	characteristic
317	C₂×C₄	#14524019	1	1	yes	characteristic
318	C₂×C₄	#14524023	1	1	yes	characteristic
319	C₂×C₄	#14524049	1	2	yes	—
320	C₂×C₄	#14524051	1	2	yes	—
321	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
322	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
323	C₂×C₄	#14524071	1	1	yes	characteristic
324	C₂×C₄	#14524076	1	1	yes	characteristic
325	C₈	—	4	4	no	—
326	C₈	—	536	2	no	—
327	C₂²	#16863253	1	1	yes	characteristic
328	C₂²	#16863362	1	1	yes	characteristic
329	C₂²	#16863364	1	1	yes	characteristic
330	C₂²	C₂×#14523968	1	1	yes	characteristic
331	C₂²	#16863398	1	1	yes	characteristic
332	C₂²	C₂×#14523971	1	1	yes	characteristic
333	C₂²	#16863414	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	268	2	no	—
341	C₄	—	268	2	no	—
342	C₄	—	536	1	no	—
343	C₄	—	536	1	no	—
344	C₄	#16863421	1	2	yes	—
345	C₄	#16863456	1	2	yes	—
346	C₄	#16863597	1	1	yes	characteristic
347	C₄	#16863607	1	1	yes	characteristic
348	C₄	#16863682	1	1	yes	characteristic
349	C₄	#16863692	1	1	yes	characteristic
350	C₂	#22558613	1	1	yes	characteristic
351	C₂	#22558623	1	1	yes	characteristic
352	C₂	#22558665	1	1	yes	characteristic
353	C₂	#22558675	1	1	yes	characteristic
354	C₂	#22560190	1	1	yes	characteristic
355	C₂	#22560215	1	1	yes	characteristic
356	C₂	#22560230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#8568
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
184. C₆₇	#8568	semidirect

Maximal subgroups

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

subgroup	index
#22562686	2
#22560361	2
C₂×#16863128	2
#22558049	2
C₄×C₄⋊C₈×C₆₇	2
#8568	67