Subgroups — #30825671

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569291	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22559116	C₂	1	2	yes	maximal, complemented
4	#22559115	C₂	1	2	yes	maximal, complemented
5	#22559107	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559106	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559105	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558956	C₂	1	1	yes	maximal, characteristic
9	#22558602	C₂	1	2	yes	maximal
10	#22558491	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₆₇	C₂²	1	2	yes	—
14	#16863527	C₂²	1	2	yes	—
15	#16863526	C₂²	1	2	yes	—
16	#16863525	—	2	2	no	—
17	#16863524	—	2	2	no	—
18	#16863523	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic
21	#16863490	C₄	1	1	yes	characteristic
22	C₄×#11761618	C₂²	1	2	yes	—
23	#16863488	C₄	1	2	yes	—
24	#16863452	—	2	2	no	—
25	#16863451	—	2	2	no	—
26	#16863448	C₂²	1	2	yes	—
27	#16863447	C₂²	1	1	yes	characteristic
28	#16863447	C₂²	1	1	yes	characteristic
29	#107×D₆₇	C₄	1	1	yes	characteristic
30	#103×D₆₇	C₄	1	2	yes	—
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863149	C₂²	1	1	yes	characteristic
33	#16863146	C₂²	1	2	yes	—
34	#16863128	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863118	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#335×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
47	C₂×#11761618	C₂×C₄	1	2	yes	—
48	Q₁₆×D₆₇	—	2	4	no	—
49	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₃₄	—	2	2	no	—
53	C₄×Dic₁₃₄	C₂³	1	2	yes	—
54	#14524008	—	2	2	no	—
55	#14524008	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×D₆₇	—	2	2	no	—
57	C₄⋊C₄×D₆₇	C₂×C₄	1	2	yes	—
58	C₄²×D₆₇	—	2	2	no	—
59	C₄²×D₆₇	C₂³	1	1	yes	characteristic
60	#14523977	C₂×C₄	1	1	yes	characteristic
61	#14523976	C₂×C₄	1	2	yes	—
62	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
63	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
64	Dic₆₇⋊C₈	—	2	2	no	—
65	Dic₆₇⋊C₈	—	2	1	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
68	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
69	#14523968	C₂×C₄	1	2	yes	—
70	#14523967	C₂×C₄	1	2	yes	—
71	#14523966	C₂×C₄	1	2	yes	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	1	no	—
74	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
75	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
76	#14523954	—	2	2	no	—
77	#14523953	—	2	2	no	—
78	#14523953	—	2	2	no	—
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₆₇	—	2	2	no	—
83	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
87	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	—	2	2	no	—
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	C₂×C₄	1	2	yes	—
91	#97×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₃₄	—	2	2	no	—
96	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
97	Q₈×D₆₇	—	2	4	no	—
98	Q₈×D₆₇	—	2	4	no	—
99	C₂×C₄×D₆₇	—	2	2	no	—
100	C₂×C₄×D₆₇	—	2	2	no	—
101	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
104	Dic₂₆₈	—	2	2	no	—
105	#11761618	—	2	4	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	1	no	—
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₈×D₆₇	—	2	2	no	—
113	C₈×D₆₇	M₁₆	1	4	yes	—
114	C₄⋊Dic₆₇	—	2	2	no	—
115	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
116	Dic₆₇⋊C₄	—	2	2	no	—
117	Dic₆₇⋊C₄	—	2	2	no	—
118	Dic₆₇⋊C₄	—	2	2	no	—
119	C₄×Dic₆₇	—	2	2	no	—
120	C₄×Dic₆₇	—	2	2	no	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₆₇	#41	1	1	yes	characteristic
125	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
126	Q₁₆×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	2	no	—
139	Q₈×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
150	C₄×D₆₇	—	2	1	no	—
151	C₄×D₆₇	—	2	1	no	—
152	C₄×D₆₇	—	4	1	no	—
153	C₄×D₆₇	—	4	4	no	—
154	C₄×D₆₇	—	8	2	no	—
155	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
156	C₄×D₆₇	#131	1	2	yes	—
157	Dic₁₃₄	—	2	2	no	—
158	Dic₁₃₄	—	2	2	no	—
159	Dic₁₃₄	—	4	4	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	4	1	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
166	C₆₇⋊C₈	—	2	1	no	—
167	C₆₇⋊C₈	—	2	1	no	—
168	C₆₇⋊C₈	—	2	1	no	—
169	C₆₇⋊C₈	—	4	1	no	—
170	C₆₇⋊C₈	—	4	2	no	—
171	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
172	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₆₇	—	2	1	no	—
174	C₂×D₆₇	—	2	1	no	—
175	C₂×D₆₇	#436	1	2	yes	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	4	2	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
180	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
181	C₄×C₆₇	#578	1	1	yes	characteristic
182	C₄×C₆₇	#578	1	1	yes	characteristic
183	Dic₆₇	—	2	1	no	—
184	Dic₆₇	—	2	1	no	—
185	Dic₆₇	—	4	2	no	—
186	Dic₆₇	—	4	2	no	—
187	Dic₆₇	#436	1	2	yes	—
188	#17897	—	67	1	no	maximal, Sylow
189	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
190	C₂×C₆₇	#2950	1	1	yes	characteristic
191	C₂×C₆₇	#2957	1	1	yes	characteristic
192	D₆₇	—	2	2	no	—
193	#2983	—	67	2	no	—
194	C₂×C₄×Q₁₆	—	67	1	no	—
195	#1593	—	67	2	no	—
196	#1593	—	67	2	no	—
197	#1591	—	67	1	no	—
198	#1591	—	67	1	no	—
199	#1591	—	67	1	no	—
200	#1591	D₆₇	1	1	yes	characteristic, complemented
201	#1564	—	67	1	no	—
202	#1479	—	67	2	no	—
203	#1450	—	67	1	no	—
204	C₆₇	#17897	1	1	yes	characteristic, complemented, Sylow
205	C₂²×Q₁₆	—	67	1	no	—
206	C₂×C₄×Q₈	—	67	2	no	—
207	#446	—	67	2	no	—
208	#446	—	67	2	no	—
209	#446	—	67	2	no	—
210	#446	—	134	2	no	—
211	#446	—	134	2	no	—
212	#446	C₂×D₆₇	1	2	yes	—
213	C₄×Q₁₆	—	67	1	no	—
214	C₄×Q₁₆	—	67	2	no	—
215	C₄×Q₁₆	—	67	1	no	—
216	C₄×Q₁₆	—	67	2	no	—
217	C₄×Q₁₆	—	67	1	no	—
218	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
219	#432	—	134	2	no	—
220	#432	—	134	2	no	—
221	#431	—	67	1	no	—
222	#431	—	67	1	no	—
223	#431	—	67	2	no	—
224	C₂×#107	—	67	1	no	—
225	C₂×#103	—	67	2	no	—
226	C₂×C₄×C₈	—	67	1	no	—
227	#335	—	67	1	no	—
228	#335	C₂×D₆₇	1	1	yes	characteristic
229	#334	—	67	2	no	—
230	Q₈⋊C₈	—	67	2	no	—
231	Q₈⋊C₈	—	67	2	no	—
232	Q₈⋊C₈	—	67	2	no	—
233	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
234	#322	—	67	1	no	—
235	#322	—	67	1	no	—
236	C₈²	—	67	1	no	—
237	C₈²	C₂×D₆₇	1	1	yes	characteristic
238	C₂²×Q₈	—	67	2	no	—
239	C₂×Q₁₆	—	67	2	no	—
240	C₂×Q₁₆	—	67	1	no	—
241	C₂×Q₁₆	—	134	4	no	—
242	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
243	C₂²×C₈	—	67	1	no	—
244	C₂²×C₈	—	67	1	no	—
245	C₄×Q₈	—	67	2	no	—
246	C₄×Q₈	—	67	2	no	—
247	C₄×Q₈	—	67	2	no	—
248	C₄×Q₈	—	134	2	no	—
249	C₄×Q₈	—	134	2	no	—
250	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
251	C₂×C₄⋊C₄	—	67	2	no	—
252	C₂×C₄⋊C₄	—	134	2	no	—
253	C₂×C₄²	—	67	1	no	—
254	C₂×C₄²	—	134	2	no	—
255	#107	—	67	1	no	—
256	#107	—	67	2	no	—
257	#107	C₄×D₆₇	1	1	yes	characteristic
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	67	1	no	—
260	C₄⋊C₈	—	67	1	no	—
261	C₄⋊C₈	—	67	2	no	—
262	C₄⋊C₈	—	134	2	no	—
263	C₄⋊C₈	—	134	1	no	—
264	C₄⋊C₈	—	134	2	no	—
265	C₄⋊C₈	—	134	2	no	—
266	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
267	#103	—	67	2	no	—
268	#103	—	67	2	no	—
269	#103	—	67	2	no	—
270	#103	C₄×D₆₇	1	2	yes	—
271	C₂²⋊C₈	—	67	1	no	—
272	C₂²⋊C₈	—	67	1	no	—
273	C₂²⋊C₈	—	134	1	no	—
274	C₂²⋊C₈	—	134	2	no	—
275	#97	—	2	2	no	—
276	#97	—	134	2	no	—
277	#97	—	134	2	no	—
278	#97	—	134	2	no	—
279	C₄×C₈	—	67	1	no	—
280	C₄×C₈	—	67	1	no	—
281	C₄×C₈	—	67	1	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	—	134	2	no	—
285	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
288	C₂×Q₈	—	67	2	no	—
289	C₂×Q₈	—	134	2	no	—
290	C₂×Q₈	—	134	4	no	—
291	C₂×Q₈	—	134	4	no	—
292	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
293	C₂²×C₄	—	67	1	no	—
294	C₂²×C₄	—	67	1	no	—
295	C₂²×C₄	—	67	1	no	—
296	C₂²×C₄	—	134	2	no	—
297	C₂²×C₄	—	134	2	no	—
298	Q₁₆	—	2	2	no	—
299	Q₁₆	—	134	4	no	—
300	Q₁₆	—	134	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	67	1	no	—
304	C₂×C₈	—	67	4	no	—
305	C₂×C₈	—	67	1	no	—
306	C₂×C₈	—	67	1	no	—
307	C₂×C₈	—	67	1	no	—
308	C₂×C₈	—	134	2	no	—
309	C₂×C₈	—	134	2	no	—
310	C₂×C₈	—	134	1	no	—
311	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
313	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
314	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	67	2	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	—	134	2	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	—	134	2	no	—
321	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
322	C₄²	—	2	2	no	—
323	C₄²	—	67	1	no	—
324	C₄²	—	67	1	no	—
325	C₄²	—	67	1	no	—
326	C₄²	—	134	2	no	—
327	C₄²	—	134	2	no	—
328	C₄²	—	134	2	no	—
329	C₄²	C₂³×D₆₇	1	1	yes	characteristic
330	C₂³	—	67	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	268	4	no	—
334	Q₈	—	134	2	no	—
335	Q₈	—	134	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	268	1	no	—
339	C₂×C₄	—	268	4	no	—
340	C₂×C₄	—	536	2	no	—
341	C₂×C₄	—	67	1	no	—
342	C₂×C₄	—	67	2	no	—
343	C₂×C₄	—	67	2	no	—
344	C₂×C₄	—	67	1	no	—
345	C₂×C₄	—	67	1	no	—
346	C₂×C₄	—	134	2	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	1	no	—
349	C₂×C₄	—	134	1	no	—
350	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
352	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	268	1	no	—
359	C₈	—	268	2	no	—
360	C₈	—	67	2	no	—
361	C₈	—	134	1	no	—
362	C₈	—	134	1	no	—
363	C₈	—	134	1	no	—
364	C₈	M₁₆×D₆₇	1	2	yes	—
365	C₂²	—	67	2	no	—
366	C₂²	—	134	1	no	—
367	C₂²	—	134	1	no	—
368	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	268	2	no	—
373	C₄	—	268	2	no	—
374	C₄	—	67	2	no	—
375	C₄	—	134	1	no	—
376	C₄	—	134	1	no	—
377	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
378	C₄	#131×D₆₇	1	1	yes	characteristic
379	C₄	#16864209	1	1	yes	characteristic
380	C₄	#16864210	1	1	yes	characteristic
381	C₂	—	134	2	no	—
382	C₂	#436×D₆₇	1	1	yes	characteristic
383	C₂	#22569072	1	1	yes	characteristic
384	C₂	#22569121	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×#1591	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1591	D₆₇
hypocenter	C₆₇	#17897
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #22559116	C₂	semidirect
4. #22559115	C₂	semidirect
5. #22559107	C₂	semidirect
6. #22559106	C₂	semidirect
7. #22559105	C₂	semidirect
11. C₆₇×#1591	C₂	semidirect
200. #1591	D₆₇	semidirect
204. C₆₇	#17897	semidirect

Maximal subgroups

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

subgroup	index
#22569291	2
C₄×Q₁₆×D₆₇	2
#22559116	2
#22559115	2
#22559107	2
#22559106	2
#22559105	2
#22558956	2
#22558602	2
#22558491	2
C₆₇×#1591	2
#17897	67