Subgroups — #30825172

Order 17152.

594 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569336	C₂	1	2	yes	maximal
2	#22569175	C₂	1	1	yes	maximal, characteristic
3	#22558995	C₂	1	2	yes	maximal, complemented
4	#22558988	C₂	1	2	yes	maximal, complemented
5	#22558982	C₂	1	2	yes	maximal, complemented
6	#22558973	C₂	1	2	yes	maximal, complemented
7	#22558958	C₂	1	2	yes	maximal, complemented
8	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#11761628	C₂²	1	1	yes	characteristic
11	#16863526	C₄	1	1	yes	characteristic
12	#16863526	C₄	1	1	yes	characteristic
13	#16863524	—	2	2	no	—
14	#16863475	C₄	1	1	yes	characteristic
15	#16863475	C₄	1	1	yes	characteristic
16	#16863469	—	2	2	no	—
17	#16863467	C₄	1	2	yes	—
18	#16863467	C₄	1	2	yes	—
19	#16863464	—	2	2	no	—
20	(C₂²⋊D₆₇)⋊C₈	—	2	2	no	—
21	#16863452	C₂²	1	2	yes	—
22	#16863449	C₂²	1	1	yes	characteristic
23	#16863448	C₂²	1	2	yes	—
24	#16863447	C₂²	1	2	yes	—
25	#16863418	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
27	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
28	C₄×#11761611	C₂²	1	2	yes	—
29	#16863149	C₂²	1	2	yes	—
30	#16863149	C₂²	1	1	yes	characteristic
31	#16863149	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	2	yes	—
33	#16863148	C₂²	1	2	yes	—
34	#16863146	C₂²	1	2	yes	—
35	#16863146	C₂²	1	2	yes	—
36	#432×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#335×C₆₇	C₂²	1	1	yes	characteristic
40	#334×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	Q₈	1	1	yes	characteristic
42	C₂×#11761611	—	2	2	no	—
43	C₂×#11761611	C₂×C₄	1	2	yes	—
44	C₂×#11761611	C₂×C₄	1	2	yes	—
45	C₂²×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₈×D₆₇	—	2	2	no	—
47	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
49	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₆₈	D₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₆₇	D₄	1	2	yes	—
53	#14524002	Q₈	1	1	yes	characteristic
54	#14524000	Q₈	1	2	yes	—
55	C₄²×D₆₇	C₂³	1	2	yes	—
56	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
57	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
58	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
59	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₆₇⋊C₈	—	2	2	no	—
62	Dic₆₇⋊C₈	—	2	2	no	—
63	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
65	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
66	(C₂×D₆₇)⋊C₈	—	2	2	no	—
67	(C₂×D₆₇)⋊C₈	—	2	2	no	—
68	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
69	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
70	#14523953	—	2	2	no	—
71	#14523953	C₂×C₄	1	2	yes	—
72	#14523953	C₂×C₄	1	2	yes	—
73	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
74	C₈×Dic₆₇	—	2	2	no	—
75	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₆₇	—	2	2	no	—
81	#97×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	#11761628	—	4	2	no	—
87	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	2	yes	—
90	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
93	#11761611	—	2	4	no	—
94	#11761611	—	4	2	no	—
95	#11761611	—	8	2	no	—
96	C₂×C₆₇⋊C₈	—	2	2	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	8	2	no	—
99	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
101	C₈×D₆₇	—	8	2	no	—
102	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
104	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
105	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₂²⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
107	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
108	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
109	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
110	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
111	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	4	1	no	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂²⋊D₆₇	—	2	4	no	—
141	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
144	D₂₆₈	—	4	1	no	—
145	C₄×D₆₇	—	2	2	no	—
146	C₄×D₆₇	—	2	2	no	—
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	2	2	no	—
149	C₄×D₆₇	—	4	2	no	—
150	Dic₁₃₄	—	4	1	no	—
151	C₈×C₆₇	—	2	1	no	—
152	C₈×C₆₇	—	2	1	no	—
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	—	4	2	no	—
155	C₈×C₆₇	—	4	2	no	—
156	C₈×C₆₇	#131	1	2	yes	—
157	C₆₇⋊C₈	—	2	2	no	—
158	C₆₇⋊C₈	—	2	2	no	—
159	C₆₇⋊C₈	—	4	2	no	—
160	C₆₇⋊C₈	—	4	2	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	#430	1	2	yes	—
164	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂×D₆₇	—	2	2	no	—
166	C₂×D₆₇	—	2	2	no	—
167	C₂×D₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
173	C₄×C₆₇	#568	1	1	yes	characteristic
174	C₄×C₆₇	#576	1	1	yes	characteristic
175	C₄×C₆₇	#578	1	1	yes	characteristic
176	Dic₆₇	—	2	2	no	—
177	Dic₆₇	—	2	2	no	—
178	Dic₆₇	—	2	2	no	—
179	#17840	—	67	1	no	maximal, Sylow
180	C₂×C₆₇	—	2	2	no	—
181	C₂×C₆₇	#2906	1	1	yes	characteristic
182	C₂×C₆₇	#2913	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
184	D₆₇	—	4	2	no	—
185	#2990	—	67	2	no	—
186	#2965	—	67	1	no	—
187	#1573	—	67	2	no	—
188	#1572	—	67	2	no	—
189	#1571	—	67	2	no	—
190	#1569	—	67	2	no	—
191	#1568	D₆₇	1	1	yes	characteristic, complemented
192	#1565	—	67	2	no	—
193	C₂×#335	—	67	1	no	—
194	C₆₇	#17840	1	1	yes	characteristic, complemented, Sylow
195	C₄×#41	—	67	1	no	—
196	#446	—	67	1	no	—
197	#446	—	67	1	no	—
198	#446	—	134	2	no	—
199	#436	—	67	1	no	—
200	#436	—	67	1	no	—
201	#435	—	67	2	no	—
202	#435	—	67	2	no	—
203	#435	—	134	2	no	—
204	#435	—	134	2	no	—
205	C₈×D₄	—	134	2	no	—
206	#432	—	67	2	no	—
207	#432	C₂×D₆₇	1	2	yes	—
208	#431	—	67	1	no	—
209	#431	—	67	2	no	—
210	#431	—	67	2	no	—
211	#423	—	67	2	no	—
212	C₂×C₄⋊C₈	—	67	2	no	—
213	C₂×C₄⋊C₈	—	67	2	no	—
214	C₄×M₁₆	—	67	2	no	—
215	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
216	#335	—	67	2	no	—
217	#335	—	67	2	no	—
218	#335	—	67	1	no	—
219	#335	—	67	1	no	—
220	#335	—	67	2	no	—
221	#335	C₂×D₆₇	1	1	yes	characteristic
222	#335	C₂×D₆₇	1	1	yes	characteristic
223	#334	—	67	2	no	—
224	#334	—	67	2	no	—
225	#334	C₂×D₆₇	1	2	yes	—
226	C₂×#41	—	67	1	no	—
227	C₂×M₁₆	—	67	2	no	—
228	C₂×M₁₆	—	67	2	no	—
229	C₂×M₁₆	—	134	2	no	—
230	C₂²×C₈	—	134	2	no	—
231	C₂²×C₈	—	134	2	no	—
232	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
234	C₄×Q₈	—	67	1	no	—
235	C₄×Q₈	—	67	1	no	—
236	C₄×D₄	—	67	2	no	—
237	C₄×D₄	—	67	2	no	—
238	C₄×D₄	—	67	1	no	—
239	C₄×D₄	—	67	1	no	—
240	#117	—	67	1	no	—
241	#117	—	67	2	no	—
242	C₂×C₄²	—	67	2	no	—
243	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
244	C₄⋊C₈	—	67	2	no	—
245	C₄⋊C₈	—	67	2	no	—
246	C₄⋊C₈	—	67	1	no	—
247	C₄⋊C₈	—	67	1	no	—
248	C₄⋊C₈	—	67	2	no	—
249	C₄⋊C₈	—	67	2	no	—
250	C₄⋊C₈	—	134	2	no	—
251	C₄⋊C₈	—	134	2	no	—
252	C₄⋊C₈	—	134	2	no	—
253	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
254	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	67	2	no	—
257	C₂²⋊C₈	—	67	2	no	—
258	C₂²⋊C₈	—	67	2	no	—
259	C₂²⋊C₈	—	134	2	no	—
260	C₂²⋊C₈	—	134	2	no	—
261	#97	—	2	2	no	—
262	#97	—	67	2	no	—
263	#97	—	67	2	no	—
264	#97	—	134	2	no	—
265	C₄×C₈	—	67	2	no	—
266	C₄×C₈	—	134	2	no	—
267	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
268	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	#41	—	268	2	no	—
272	C₂×Q₈	—	67	1	no	—
273	C₂×D₄	—	67	2	no	—
274	C₂×D₄	—	67	1	no	—
275	C₂²×C₄	—	67	2	no	—
276	C₂²×C₄	—	67	2	no	—
277	C₂²×C₄	—	67	2	no	—
278	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
280	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
281	M₁₆	—	268	2	no	—
282	M₁₆	—	536	2	no	—
283	M₁₆	—	134	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	536	2	no	—
289	C₂×C₈	—	536	2	no	—
290	C₂×C₈	—	67	2	no	—
291	C₂×C₈	—	67	2	no	—
292	C₂×C₈	—	134	2	no	—
293	C₂×C₈	—	134	2	no	—
294	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
298	C₄⋊C₄	—	67	2	no	—
299	C₄⋊C₄	—	67	2	no	—
300	C₄⋊C₄	—	67	1	no	—
301	C₄⋊C₄	—	67	1	no	—
302	C₂²⋊C₄	—	67	2	no	—
303	C₂²⋊C₄	—	67	2	no	—
304	C₂²⋊C₄	—	67	2	no	—
305	C₄²	—	67	2	no	—
306	C₄²	—	67	2	no	—
307	C₄²	—	67	2	no	—
308	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
309	C₄²	D₄×D₆₇	1	1	yes	characteristic
310	C₄²	Q₈×D₆₇	1	1	yes	characteristic
311	C₄²	C₂³×D₆₇	1	1	yes	characteristic
312	C₂³	—	67	2	no	—
313	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
314	Q₈	—	268	1	no	—
315	D₄	—	268	1	no	—
316	D₄	—	134	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	268	2	no	—
323	C₂×C₄	—	67	2	no	—
324	C₂×C₄	—	67	2	no	—
325	C₂×C₄	—	67	2	no	—
326	C₂×C₄	—	134	2	no	—
327	C₂×C₄	—	134	2	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	268	2	no	—
342	C₈	—	268	2	no	—
343	C₈	—	134	2	no	—
344	C₈	—	134	2	no	—
345	C₈	#14524051	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	134	2	no	—
349	C₂²	—	134	2	no	—
350	C₂²	—	134	2	no	—
351	C₂²	#16863443	1	2	yes	—
352	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	134	2	no	—
358	C₄	—	134	2	no	—
359	C₄	—	134	2	no	—
360	C₄	C₂×#14524051	1	1	yes	characteristic
361	C₄	#16864162	1	1	yes	characteristic
362	C₄	#16864198	1	1	yes	characteristic
363	C₄	#16864210	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	268	2	no	—
366	C₂	#22568778	1	1	yes	characteristic
367	C₂	#22568806	1	1	yes	characteristic
368	C₂	C₂×#16863443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1568	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	Q₈
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#17840
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #22558995	C₂	semidirect
4. #22558988	C₂	semidirect
5. #22558982	C₂	semidirect
6. #22558973	C₂	semidirect
7. #22558958	C₂	semidirect
8. C₂×#16863149	C₂	semidirect
9. C₆₇×#1568	C₂	semidirect
191. #1568	D₆₇	semidirect
194. C₆₇	#17840	semidirect

Maximal subgroups

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

subgroup	index
#22569336	2
#22569175	2
#22558995	2
#22558988	2
#22558982	2
#22558973	2
#22558958	2
C₂×#16863149	2
C₆₇×#1568	2
#17840	67