Subgroups — #541×D₆₇ #22573571

Order 8576.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864009	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864008	C₂	1	1	yes	maximal, characteristic, complemented
3	#16864007	C₂	1	1	yes	maximal, characteristic, complemented
4	#16864006	C₂	1	2	yes	maximal, complemented
5	#16864005	C₂	1	4	yes	maximal, complemented
6	#16864004	C₂	1	2	yes	maximal, complemented
7	#16864003	C₂	1	4	yes	maximal, complemented
8	C₄⋊Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#126×D₆₇	C₂	1	4	yes	maximal
10	#125×D₆₇	C₂	1	4	yes	maximal, complemented
11	#125×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²⋊Q₈×D₆₇	C₂	1	2	yes	maximal
13	C₄×Q₈×D₆₇	C₂	1	2	yes	maximal, complemented
14	#117×D₆₇	C₂	1	1	yes	maximal, characteristic
15	#541×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₆₇	C₂²	1	2	yes	—
17	C₂²×C₄×D₆₇	C₂²	1	1	yes	characteristic
18	#14524043	C₂²	1	1	yes	characteristic, complemented
19	#14524042	C₂²	1	4	yes	complemented
20	#14524041	C₂²	1	2	yes	complemented
21	#14524038	C₂²	1	4	yes	—
22	#14524038	C₂²	1	4	yes	—
23	#14524038	C₂²	1	4	yes	—
24	#14524037	C₂²	1	4	yes	—
25	#14524036	C₂²	1	4	yes	—
26	#14524036	C₂²	1	4	yes	—
27	#14524036	C₂²	1	4	yes	—
28	#14524035	C₂²	1	4	yes	complemented
29	#14524035	C₂²	1	1	yes	characteristic, complemented
30	#14524034	C₂²	1	4	yes	complemented
31	#14524034	C₂²	1	2	yes	complemented
32	#14524034	C₂²	1	4	yes	complemented
33	#14524033	C₂²	1	4	yes	complemented
34	#14524033	C₂²	1	4	yes	complemented
35	#14524033	C₂²	1	4	yes	complemented
36	#14524033	C₂²	1	4	yes	complemented
37	#14524033	C₂²	1	4	yes	complemented
38	#14524023	C₂²	1	2	yes	—
39	#14524022	C₂²	1	2	yes	—
40	#14524021	C₂²	1	4	yes	—
41	#14524020	C₂²	1	4	yes	—
42	#14524019	C₂²	1	2	yes	—
43	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
44	C₄×Dic₁₃₄	C₂²	1	4	yes	complemented
45	C₄×Dic₁₃₄	C₂²	1	2	yes	complemented
46	#14524008	C₂²	1	4	yes	complemented
47	#14524008	C₂²	1	2	yes	complemented
48	#14524002	C₂²	1	1	yes	characteristic
49	#14524001	C₂²	1	2	yes	—
50	#14524000	C₂²	1	1	yes	characteristic
51	#14524000	C₂²	1	1	yes	characteristic
52	#14523999	C₂²	1	2	yes	—
53	C₄⋊C₄×D₆₇	C₂²	1	2	yes	—
54	C₄⋊C₄×D₆₇	C₂²	1	2	yes	—
55	C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
56	C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
57	C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
58	C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
59	C₂²⋊C₄×D₆₇	C₂²	1	2	yes	—
60	C₂²⋊C₄×D₆₇	C₂²	1	4	yes	—
61	C₄²×D₆₇	C₂²	1	1	yes	characteristic
62	C₄²×D₆₇	C₂²	1	1	yes	characteristic
63	C₄²×D₆₇	C₂²	1	4	yes	—
64	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
65	#126×C₆₇	C₂²	1	4	yes	—
66	#125×C₆₇	C₂²	1	4	yes	complemented
67	#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
68	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
69	C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
70	#117×C₆₇	C₂²	1	1	yes	characteristic
71	C₂³×D₆₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₃₄	C₂³	1	4	yes	—
73	C₂×Dic₁₃₄	C₂³	1	2	yes	—
74	Q₈×D₆₇	—	2	8	no	—
75	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₆₇	—	2	4	no	—
77	C₂×C₄×D₆₇	C₂³	1	2	yes	—
78	C₂×C₄×D₆₇	C₂³	1	2	yes	—
79	C₂×C₄×D₆₇	C₂³	1	4	yes	—
80	C₂×C₄×D₆₇	C₂³	1	4	yes	—
81	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₆₇	C₂³	1	2	yes	—
86	C₄⋊Dic₆₇	C₂³	1	2	yes	—
87	C₄⋊Dic₆₇	C₂³	1	4	yes	—
88	C₄⋊Dic₆₇	C₂³	1	4	yes	—
89	C₄⋊Dic₆₇	C₂³	1	4	yes	—
90	C₄⋊Dic₆₇	C₂³	1	4	yes	—
91	Dic₆₇⋊C₄	C₂³	1	2	yes	—
92	Dic₆₇⋊C₄	C₂³	1	2	yes	—
93	Dic₆₇⋊C₄	C₂³	1	4	yes	—
94	Dic₆₇⋊C₄	C₂³	1	4	yes	—
95	Dic₆₇⋊C₄	C₂³	1	4	yes	—
96	Dic₆₇⋊C₄	C₂³	1	4	yes	—
97	Dic₆₇⋊C₄	C₂³	1	4	yes	—
98	Dic₆₇⋊C₄	C₂³	1	4	yes	—
99	Dic₆₇⋊C₄	C₂³	1	4	yes	—
100	Dic₆₇⋊C₄	C₂³	1	4	yes	—
101	Dic₆₇⋊C₄	C₂³	1	4	yes	—
102	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
103	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
104	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
105	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
106	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
107	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
108	C₄×Dic₆₇	C₂³	1	2	yes	—
109	C₄×Dic₆₇	C₂³	1	2	yes	—
110	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
111	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
112	C₄×Dic₆₇	C₂³	1	4	yes	—
113	C₄×Dic₆₇	C₂³	1	4	yes	—
114	C₄×Dic₆₇	C₂³	1	4	yes	—
115	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₄²×C₆₇	C₂³	1	4	yes	—
128	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂²×D₆₇	—	2	2	no	—
130	C₂²×D₆₇	—	2	4	no	—
131	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
132	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
133	Q₈×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
136	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
137	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
138	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₂×Dic₆₇	—	2	2	no	—
142	C₂×Dic₆₇	C₂⁴	1	2	yes	—
143	C₂×Dic₆₇	C₂⁴	1	2	yes	—
144	C₂×Dic₆₇	C₂⁴	1	4	yes	—
145	C₂×Dic₆₇	C₂⁴	1	4	yes	—
146	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
147	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
148	C₄×D₆₇	—	2	2	no	—
149	C₄×D₆₇	—	2	8	no	—
150	C₄×D₆₇	—	2	4	no	—
151	C₄×D₆₇	—	2	4	no	—
152	C₄×D₆₇	—	2	8	no	—
153	C₄×D₆₇	—	2	2	no	—
154	C₄×D₆₇	—	2	2	no	—
155	C₄×D₆₇	#41	1	4	yes	—
156	Dic₁₃₄	—	2	4	no	—
157	Dic₁₃₄	—	2	8	no	—
158	C₂²×C₆₇	—	2	1	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
161	C₂×D₆₇	—	2	1	no	—
162	C₂×D₆₇	—	2	2	no	—
163	C₂×D₆₇	—	4	4	no	—
164	C₂×D₆₇	C₂×#41	1	2	yes	—
165	C₂×D₆₇	2₋¹⁺⁴	1	4	yes	—
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	2	4	no	—
171	C₄×C₆₇	C₂×#41	1	2	yes	—
172	Dic₆₇	—	2	4	no	—
173	Dic₆₇	—	2	1	no	—
174	Dic₆₇	—	2	2	no	—
175	Dic₆₇	—	2	2	no	—
176	Dic₆₇	—	2	4	no	—
177	Dic₆₇	C₂×#41	1	2	yes	—
178	C₂×C₆₇	—	4	1	no	—
179	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
180	C₂×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
181	D₆₇	—	4	1	no	—
182	D₆₇	#541	1	4	yes	direct product factor
183	C₂×#541	—	67	1	no	maximal, Sylow
184	C₆₇	C₂×#541	1	1	yes	characteristic, complemented, Sylow
185	#541	—	67	1	no	—
186	#541	—	67	4	no	—
187	#541	—	67	4	no	—
188	#541	—	67	1	no	—
189	#541	—	67	2	no	—
190	#541	—	67	2	no	—
191	#541	—	67	1	no	—
192	#541	D₆₇	1	1	yes	characteristic, direct product factor
193	C₂×C₄⋊Q₈	—	67	1	no	—
194	C₂×#126	—	67	4	no	—
195	C₂×#125	—	67	1	no	—
196	C₂×#125	—	67	4	no	—
197	C₂×C₂²⋊Q₈	—	67	2	no	—
198	C₂×C₄×Q₈	—	67	2	no	—
199	C₂×#117	—	67	1	no	—
200	C₂²×Q₈	—	67	2	no	—
201	C₂³×C₄	—	67	1	no	—
202	C₄⋊Q₈	—	67	2	no	—
203	C₄⋊Q₈	—	67	4	no	—
204	C₄⋊Q₈	—	67	1	no	—
205	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
206	#126	—	67	4	no	—
207	#126	—	67	4	no	—
208	#126	—	67	4	no	—
209	#126	—	67	4	no	—
210	#126	—	67	4	no	—
211	#126	—	67	4	no	—
212	#126	—	67	4	no	—
213	#126	C₂×D₆₇	1	4	yes	—
214	#125	—	67	4	no	—
215	#125	—	67	4	no	—
216	#125	—	67	4	no	—
217	#125	—	67	4	no	—
218	#125	—	67	4	no	—
219	#125	—	67	2	no	—
220	#125	—	67	4	no	—
221	#125	—	67	4	no	—
222	#125	—	67	1	no	—
223	#125	—	67	4	no	—
224	#125	C₂×D₆₇	1	1	yes	characteristic, complemented
225	#125	C₂×D₆₇	1	4	yes	complemented
226	C₂²⋊Q₈	—	67	2	no	—
227	C₂²⋊Q₈	—	67	2	no	—
228	C₂²⋊Q₈	—	67	4	no	—
229	C₂²⋊Q₈	—	67	2	no	—
230	C₂²⋊Q₈	—	67	4	no	—
231	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
232	C₄×Q₈	—	67	2	no	—
233	C₄×Q₈	—	67	4	no	—
234	C₄×Q₈	—	67	2	no	—
235	C₄×Q₈	—	67	4	no	—
236	C₄×Q₈	—	67	2	no	—
237	C₄×Q₈	C₂×D₆₇	1	2	yes	complemented
238	#117	—	67	1	no	—
239	#117	—	67	2	no	—
240	#117	—	67	2	no	—
241	#117	—	67	1	no	—
242	#117	—	67	1	no	—
243	#117	C₂×D₆₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	67	2	no	—
245	C₂×C₄⋊C₄	—	67	2	no	—
246	C₂×C₄⋊C₄	—	67	4	no	—
247	C₂×C₄⋊C₄	—	67	4	no	—
248	C₂×C₄⋊C₄	—	67	4	no	—
249	C₂×C₄⋊C₄	—	67	4	no	—
250	C₂×C₂²⋊C₄	—	67	2	no	—
251	C₂×C₂²⋊C₄	—	67	4	no	—
252	C₂×C₄²	—	67	1	no	—
253	C₂×C₄²	—	67	1	no	—
254	C₂×C₄²	—	67	4	no	—
255	C₂⁴	—	67	1	no	—
256	C₂×Q₈	—	67	4	no	—
257	C₂×Q₈	—	67	2	no	—
258	C₂×Q₈	—	134	8	no	—
259	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
260	C₂²×C₄	—	67	1	no	—
261	C₂²×C₄	—	67	1	no	—
262	C₂²×C₄	—	67	2	no	—
263	C₂²×C₄	—	67	2	no	—
264	C₂²×C₄	—	67	4	no	—
265	C₂²×C₄	—	67	4	no	—
266	C₂²×C₄	—	67	1	no	—
267	C₂²×C₄	—	67	1	no	—
268	C₂²×C₄	—	67	1	no	—
269	C₂²×C₄	—	134	4	no	—
270	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
271	C₄⋊C₄	—	67	2	no	—
272	C₄⋊C₄	—	67	2	no	—
273	C₄⋊C₄	—	67	2	no	—
274	C₄⋊C₄	—	67	4	no	—
275	C₄⋊C₄	—	67	2	no	—
276	C₄⋊C₄	—	67	4	no	—
277	C₄⋊C₄	—	67	4	no	—
278	C₄⋊C₄	—	67	4	no	—
279	C₄⋊C₄	—	67	4	no	—
280	C₄⋊C₄	—	67	4	no	—
281	C₄⋊C₄	—	67	4	no	—
282	C₄⋊C₄	—	67	4	no	—
283	C₄⋊C₄	—	67	4	no	—
284	C₄⋊C₄	—	67	4	no	—
285	C₄⋊C₄	—	67	4	no	—
286	C₄⋊C₄	—	67	4	no	—
287	C₄⋊C₄	—	67	4	no	—
288	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
289	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
290	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
291	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
292	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
293	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
294	C₂²⋊C₄	—	67	2	no	—
295	C₂²⋊C₄	—	67	4	no	—
296	C₂²⋊C₄	—	67	2	no	—
297	C₂²⋊C₄	—	67	2	no	—
298	C₂²⋊C₄	—	67	4	no	—
299	C₂²⋊C₄	—	67	4	no	—
300	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
301	C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
302	C₄²	—	67	2	no	—
303	C₄²	—	67	1	no	—
304	C₄²	—	67	2	no	—
305	C₄²	—	67	1	no	—
306	C₄²	—	67	4	no	—
307	C₄²	—	67	4	no	—
308	C₄²	—	67	4	no	—
309	C₄²	C₂²×D₆₇	1	1	yes	characteristic
310	C₄²	C₂²×D₆₇	1	1	yes	characteristic
311	C₄²	C₂²×D₆₇	1	4	yes	—
312	C₂³	—	67	1	no	—
313	C₂³	—	67	1	no	—
314	C₂³	—	134	2	no	—
315	C₂³	—	134	4	no	—
316	C₂³	C₂³×D₆₇	1	1	yes	characteristic
317	Q₈	—	2	4	no	—
318	Q₈	—	134	8	no	—
319	Q₈	—	134	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	67	1	no	—
322	C₂×C₄	—	67	4	no	—
323	C₂×C₄	—	67	2	no	—
324	C₂×C₄	—	67	2	no	—
325	C₂×C₄	—	67	4	no	—
326	C₂×C₄	—	67	4	no	—
327	C₂×C₄	—	67	1	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	—	134	8	no	—
331	C₂×C₄	—	134	4	no	—
332	C₂×C₄	—	134	4	no	—
333	C₂×C₄	—	134	8	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	—	134	2	no	—
336	C₂×C₄	C₂³×D₆₇	1	2	yes	—
337	C₂×C₄	C₂³×D₆₇	1	2	yes	—
338	C₂×C₄	C₂³×D₆₇	1	4	yes	—
339	C₂×C₄	C₂³×D₆₇	1	4	yes	—
340	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	268	4	no	—
345	C₂²	—	67	2	no	—
346	C₂²	—	67	4	no	—
347	C₂²	—	134	1	no	—
348	C₂²	—	134	2	no	—
349	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	4	no	—
355	C₄	—	67	2	no	—
356	C₄	—	134	1	no	—
357	C₄	—	134	4	no	—
358	C₄	—	134	2	no	—
359	C₄	—	134	2	no	—
360	C₄	—	134	4	no	—
361	C₄	#41×D₆₇	1	2	yes	—
362	C₂	—	4	1	no	—
363	C₂	—	268	1	no	—
364	C₂	—	67	4	no	—
365	C₂	C₂×#41×D₆₇	1	1	yes	characteristic
366	C₂	2₋¹⁺⁴×D₆₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#541×C₆₇	C₂
hypercenter	#541	D₆₇
hypocenter	C₆₇	C₂×#541
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #541

Complements

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

N	H ≅ G/N	type
1. #16864009	C₂	semidirect
2. #16864008	C₂	semidirect
3. #16864007	C₂	semidirect
4. #16864006	C₂	semidirect
5. #16864005	C₂	semidirect
6. #16864004	C₂	semidirect
7. #16864003	C₂	semidirect
8. C₄⋊Q₈×D₆₇	C₂	semidirect
10. #125×D₆₇	C₂	semidirect
11. #125×D₆₇	C₂	semidirect
13. C₄×Q₈×D₆₇	C₂	semidirect
15. #541×C₆₇	C₂	semidirect
18. #14524043	C₂²	semidirect
19. #14524042	C₂²	semidirect
20. #14524041	C₂²	semidirect
28. #14524035	C₂²	semidirect
29. #14524035	C₂²	semidirect
30. #14524034	C₂²	semidirect
31. #14524034	C₂²	semidirect
32. #14524034	C₂²	semidirect
33. #14524033	C₂²	semidirect
34. #14524033	C₂²	semidirect
35. #14524033	C₂²	semidirect
36. #14524033	C₂²	semidirect
37. #14524033	C₂²	semidirect
43. Q₈×Dic₆₇	C₂²	semidirect
44. C₄×Dic₁₃₄	C₂²	semidirect
45. C₄×Dic₁₃₄	C₂²	semidirect
46. #14524008	C₂²	semidirect
47. #14524008	C₂²	semidirect
64. C₄⋊Q₈×C₆₇	C₂²	semidirect
66. #125×C₆₇	C₂²	semidirect
67. #125×C₆₇	C₂²	semidirect
69. C₄×Q₈×C₆₇	C₂²	semidirect
182. D₆₇	#541	direct (N × H)
184. C₆₇	C₂×#541	semidirect
192. #541	D₆₇	direct (N × H)
205. C₄⋊Q₈	C₂×D₆₇	semidirect
224. #125	C₂×D₆₇	semidirect
225. #125	C₂×D₆₇	semidirect
237. C₄×Q₈	C₂×D₆₇	semidirect

Subgroups — #541×D67 #22573571

Conjugacy classes of subgroups

Complements

Subgroups — #541×D₆₇ #22573571