Subgroups — #22568872

Order 8576.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863864	C₂	1	1	yes	maximal, characteristic
2	#16863457	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863456	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863456	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863451	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863450	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863447	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863446	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#16863361	C₂	1	1	yes	maximal, characteristic
12	#16863361	C₂	1	1	yes	maximal, characteristic
13	#16863353	C₂	1	1	yes	maximal, characteristic, complemented
14	#16863350	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11761628	C₂²	1	1	yes	characteristic
17	C₂×#11761611	C₂²	1	1	yes	characteristic
18	C₂×#11761611	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
21	#14524008	C₄	1	1	yes	characteristic
22	#14524008	C₄	1	1	yes	characteristic
23	#14524005	C₄	1	1	yes	characteristic
24	#14524005	C₄	1	1	yes	characteristic
25	(C₂²⋊D₆₇)⋊C₄	C₄	1	1	yes	characteristic
26	(C₂²⋊D₆₇)⋊C₄	C₄	1	1	yes	characteristic
27	(C₂²⋊D₆₇)⋊C₄	C₄	1	1	yes	characteristic
28	(C₂²⋊D₆₇)⋊C₄	C₄	1	1	yes	characteristic
29	#14524000	C₂²	1	1	yes	characteristic
30	#14524000	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₄²×D₆₇	C₂²	1	1	yes	characteristic
33	C₄²×D₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
36	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
42	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
43	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
46	#14523954	C₂²	1	1	yes	characteristic
47	#14523953	C₂²	1	1	yes	characteristic
48	#14523953	C₂²	1	1	yes	characteristic
49	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
52	#117×C₆₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
57	#97×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
59	#11761628	—	2	2	no	—
60	#11761628	—	2	2	no	—
61	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
62	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
73	#11761611	—	2	2	no	—
74	#11761611	—	2	2	no	—
75	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×D₆₇	—	2	2	no	—
80	C₈×D₆₇	—	2	2	no	—
81	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
82	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
83	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
84	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
96	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₆₇	—	2	2	no	—
120	C₂²⋊D₆₇	—	2	2	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	C₂×Dic₆₇	—	4	1	no	—
123	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
129	D₂₆₈	—	2	2	no	—
130	C₄×D₆₇	—	2	1	no	—
131	C₄×D₆₇	—	2	1	no	—
132	C₄×D₆₇	—	2	2	no	—
133	C₄×D₆₇	—	2	2	no	—
134	C₄×D₆₇	—	4	1	no	—
135	C₄×D₆₇	—	4	1	no	—
136	C₄×D₆₇	#41	1	2	yes	—
137	C₄×D₆₇	#41	1	2	yes	—
138	Dic₁₃₄	—	2	2	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₆₇⋊C₈	—	2	1	no	—
144	C₆₇⋊C₈	—	2	1	no	—
145	C₆₇⋊C₈	—	2	1	no	—
146	C₆₇⋊C₈	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₆₇	—	2	1	no	—
152	C₂×D₆₇	—	2	1	no	—
153	C₂×D₆₇	—	2	1	no	—
154	C₂×D₆₇	—	2	1	no	—
155	C₂×D₆₇	—	2	1	no	—
156	C₂×D₆₇	#117	1	2	yes	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	2	1	no	—
162	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
164	Dic₆₇	—	2	1	no	—
165	Dic₆₇	—	2	1	no	—
166	Dic₆₇	—	2	1	no	—
167	Dic₆₇	—	2	1	no	—
168	Dic₆₇	#117	1	2	yes	—
169	Dic₆₇	#131	1	2	yes	—
170	C₂×C₆₇	—	4	1	no	—
171	C₂×C₆₇	C₂×#117	1	1	yes	characteristic
172	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
173	C₂×C₆₇	#568	1	1	yes	characteristic
174	D₆₇	—	2	2	no	—
175	D₆₇	—	4	1	no	—
176	#2925	—	67	1	no	maximal, Sylow
177	C₆₇	#2925	1	1	yes	characteristic, complemented, Sylow
178	C₄×#41	—	67	1	no	—
179	#433	—	67	1	no	—
180	#433	—	67	1	no	—
181	#433	—	67	1	no	—
182	#433	D₆₇	1	1	yes	characteristic, complemented
183	#432	—	67	1	no	—
184	#432	—	67	1	no	—
185	#431	—	67	1	no	—
186	#431	—	67	1	no	—
187	#423	—	67	1	no	—
188	C₂×C₄⋊C₈	—	67	1	no	—
189	#408	—	67	1	no	—
190	#408	—	67	1	no	—
191	#405	—	67	1	no	—
192	#405	—	67	1	no	—
193	C₂×#41	—	67	1	no	—
194	C₂×M₁₆	—	67	1	no	—
195	C₂×M₁₆	—	67	1	no	—
196	C₂²×C₈	—	67	1	no	—
197	C₂²×C₈	—	67	1	no	—
198	C₄×Q₈	—	67	1	no	—
199	C₄×Q₈	—	67	1	no	—
200	C₄×D₄	—	67	1	no	—
201	C₄×D₄	—	67	1	no	—
202	C₄×D₄	—	67	1	no	—
203	C₄×D₄	—	67	1	no	—
204	C₄×D₄	—	67	1	no	—
205	C₄×D₄	—	67	1	no	—
206	#117	—	67	1	no	—
207	#117	—	67	1	no	—
208	#117	C₂×D₆₇	1	1	yes	characteristic
209	C₂×C₄²	—	67	1	no	—
210	C₂×C₄²	—	67	1	no	—
211	C₂×C₄²	—	67	1	no	—
212	C₄⋊C₈	—	67	1	no	—
213	C₄⋊C₈	—	67	1	no	—
214	C₄⋊C₈	—	67	1	no	—
215	C₄⋊C₈	—	67	1	no	—
216	C₄⋊C₈	—	67	1	no	—
217	C₄⋊C₈	—	67	1	no	—
218	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
220	C₂²⋊C₈	—	67	1	no	—
221	C₂²⋊C₈	—	67	1	no	—
222	C₂²⋊C₈	—	67	1	no	—
223	C₂²⋊C₈	—	67	1	no	—
224	C₂²⋊C₈	—	67	1	no	—
225	C₂²⋊C₈	—	67	1	no	—
226	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
227	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
228	#97	—	67	1	no	—
229	#97	—	67	1	no	—
230	#97	—	67	1	no	—
231	#97	C₂×D₆₇	1	1	yes	characteristic
232	C₄×C₈	—	67	1	no	—
233	C₄×C₈	—	67	1	no	—
234	C₄×C₈	—	67	1	no	—
235	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
236	#41	—	134	2	no	—
237	#41	—	134	2	no	—
238	C₂×Q₈	—	67	1	no	—
239	C₂×D₄	—	67	1	no	—
240	C₂×D₄	—	67	1	no	—
241	C₂×D₄	—	67	1	no	—
242	C₂²×C₄	—	67	1	no	—
243	C₂²×C₄	—	67	1	no	—
244	C₂²×C₄	—	67	1	no	—
245	C₂²×C₄	—	67	1	no	—
246	C₂²×C₄	—	67	1	no	—
247	C₂²×C₄	—	67	1	no	—
248	C₂²×C₄	—	67	1	no	—
249	C₂²×C₄	—	67	1	no	—
250	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
251	M₁₆	—	134	2	no	—
252	M₁₆	—	134	2	no	—
253	C₂×C₈	—	67	1	no	—
254	C₂×C₈	—	67	1	no	—
255	C₂×C₈	—	67	1	no	—
256	C₂×C₈	—	67	1	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
263	C₄⋊C₄	—	67	1	no	—
264	C₄⋊C₄	—	67	1	no	—
265	C₄⋊C₄	—	67	1	no	—
266	C₄⋊C₄	—	67	1	no	—
267	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
268	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
269	C₂²⋊C₄	—	67	1	no	—
270	C₂²⋊C₄	—	67	1	no	—
271	C₂²⋊C₄	—	67	1	no	—
272	C₂²⋊C₄	—	67	1	no	—
273	C₂²⋊C₄	C₄×D₆₇	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₆₇	1	1	yes	characteristic
275	C₄²	—	67	1	no	—
276	C₄²	—	67	1	no	—
277	C₄²	—	67	1	no	—
278	C₄²	—	67	1	no	—
279	C₄²	—	67	1	no	—
280	C₄²	—	67	1	no	—
281	C₄²	—	67	1	no	—
282	C₄²	—	67	1	no	—
283	C₄²	C₂²×D₆₇	1	1	yes	characteristic
284	C₄²	C₂²×D₆₇	1	1	yes	characteristic
285	C₂³	—	67	1	no	—
286	C₂³	—	67	1	no	—
287	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
288	Q₈	—	134	2	no	—
289	D₄	—	134	2	no	—
290	D₄	—	134	2	no	—
291	D₄	—	134	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	268	1	no	—
295	C₂×C₄	—	268	1	no	—
296	C₂×C₄	—	268	1	no	—
297	C₂×C₄	—	67	1	no	—
298	C₂×C₄	—	67	1	no	—
299	C₂×C₄	—	67	2	no	—
300	C₂×C₄	—	67	2	no	—
301	C₂×C₄	—	67	1	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	1	no	—
304	C₂×C₄	—	67	1	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	—	134	2	no	—
307	C₂×C₄	—	134	2	no	—
308	C₂×C₄	—	134	1	no	—
309	C₂×C₄	—	134	1	no	—
310	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	134	1	no	—
321	C₈	—	134	1	no	—
322	C₈	—	134	1	no	—
323	C₈	—	134	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	67	2	no	—
328	C₂²	—	134	1	no	—
329	C₂²	—	134	1	no	—
330	C₂²	—	134	1	no	—
331	C₂²	—	134	1	no	—
332	C₂²	—	134	1	no	—
333	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	67	2	no	—
340	C₄	—	67	2	no	—
341	C₄	—	134	1	no	—
342	C₄	—	134	1	no	—
343	C₄	—	134	1	no	—
344	C₄	—	134	1	no	—
345	C₄	#41×D₆₇	1	1	yes	characteristic
346	C₄	#41×D₆₇	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	268	1	no	—
349	C₂	—	134	2	no	—
350	C₂	#117×D₆₇	1	1	yes	characteristic
351	C₂	#131×D₆₇	1	1	yes	characteristic
352	C₂	#16864161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#433×C₆₇	C₂
hypercenter	#433	D₆₇
hypocenter	C₆₇	#2925
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #16863457	C₂	semidirect
3. #16863456	C₂	semidirect
4. #16863456	C₂	semidirect
5. #16863451	C₂	semidirect
6. #16863450	C₂	semidirect
7. #16863447	C₂	semidirect
8. #16863446	C₂	semidirect
9. #16863418	C₂	semidirect
10. C₄⋊C₈×D₆₇	C₂	semidirect
13. #16863353	C₂	semidirect
14. #16863350	C₂	semidirect
15. #433×C₆₇	C₂	semidirect
177. C₆₇	#2925	semidirect
182. #433	D₆₇	semidirect