Subgroups — #30863997

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570531	C₂	1	1	yes	maximal, characteristic
2	#22569149	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561189	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561042	C₂	1	1	yes	maximal, characteristic
5	#22560923	C₂	1	2	yes	maximal, complemented
6	#22560916	C₂	1	2	yes	maximal, complemented
7	#22560346	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560051	C₂	1	2	yes	maximal, complemented
9	#22560044	C₂	1	2	yes	maximal, complemented
10	#22559897	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2932	C₂	1	1	yes	maximal, characteristic
12	#16863953	C₂²	1	1	yes	characteristic
13	#16863750	C₄	1	4	yes	complemented
14	#16863746	C₄	1	2	yes	complemented
15	#16863742	C₄	1	2	yes	complemented
16	#16863524	—	2	4	no	—
17	#16863422	C₂²	1	2	yes	—
18	#16863420	C₂²	1	1	yes	characteristic
19	C₂×#14523970	C₂²	1	2	yes	—
20	C₂×#14523970	C₂²	1	2	yes	—
21	#16863392	C₂²	1	2	yes	—
22	#16863392	C₂²	1	2	yes	—
23	#16863351	C₂²	1	2	yes	—
24	C₂×#14523953	C₂²	1	1	yes	characteristic
25	C₂×#14523953	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
29	#16863242	C₂²	1	1	yes	characteristic
30	#16863160	C₂²	1	2	yes	—
31	#16863160	C₂²	1	2	yes	—
32	#16863157	C₂²	1	2	yes	—
33	#16863157	C₂²	1	2	yes	—
34	C₂×#131×C₆₇	C₂²	1	2	yes	—
35	#532×C₆₇	C₂²	1	1	yes	characteristic
36	#435×C₆₇	—	2	4	no	—
37	#423×C₆₇	C₂²	1	1	yes	characteristic
38	#408×C₆₇	C₂²	1	2	yes	—
39	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#11761612	C₂³	1	2	yes	—
41	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	#14524043	C₂×C₄	1	2	yes	—
44	#14524035	C₂×C₄	1	2	yes	—
45	#14524022	—	2	2	no	—
46	C₄×Dic₁₃₄	—	2	2	no	—
47	#14524002	C₂³	1	2	yes	—
48	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	Dic₆₇⋊C₈	—	2	4	no	—
52	#14523970	—	2	4	no	—
53	#14523970	—	2	4	no	—
54	#14523967	C₂×C₄	1	4	yes	—
55	#14523967	C₂×C₄	1	4	yes	—
56	#14523964	C₂×C₄	1	4	yes	—
57	#14523964	C₂×C₄	1	4	yes	—
58	#14523953	—	2	2	no	—
59	#14523953	—	2	2	no	—
60	#14523953	—	2	2	no	—
61	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
63	C₈×Dic₆₇	—	2	2	no	—
64	C₈×Dic₆₇	—	2	2	no	—
65	C₈×Dic₆₇	—	2	2	no	—
66	C₆₇⋊#95	—	2	2	no	—
67	C₂×#41×C₆₇	C₂³	1	2	yes	—
68	#131×C₆₇	—	2	4	no	—
69	#131×C₆₇	—	2	4	no	—
70	C₂×M₁₆×C₆₇	—	2	2	no	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	—	2	2	no	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
77	#124×C₆₇	C₂×C₄	1	2	yes	—
78	#121×C₆₇	—	2	2	no	—
79	C₄×D₄×C₆₇	—	2	2	no	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	C₂²⋊C₈×C₆₇	—	2	4	no	—
83	#97×C₆₇	D₄	1	4	yes	—
84	C₂×Dic₁₃₄	—	2	2	no	—
85	C₂²×Dic₆₇	—	2	1	no	—
86	C₂²×Dic₆₇	—	2	1	no	—
87	#11761612	—	2	2	no	—
88	#11761612	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	1	no	—
93	C₂×C₆₇⋊C₈	—	2	2	no	—
94	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
96	C₄⋊Dic₆₇	—	2	2	no	—
97	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
98	Dic₆₇⋊C₄	—	2	4	no	—
99	C₂²⋊Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	1	no	—
102	C₄×Dic₆₇	—	2	1	no	—
103	C₄×Dic₆₇	#41	1	2	yes	—
104	C₄×Dic₆₇	#41	1	2	yes	—
105	#41×C₆₇	—	2	2	no	—
106	#41×C₆₇	—	2	2	no	—
107	#41×C₆₇	—	2	2	no	—
108	#41×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	#41	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₆₇	—	2	2	no	—
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	4	4	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	4	4	no	—
123	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	#41	1	2	yes	—
126	C₂×C₈×C₆₇	#41	1	2	yes	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₂²⋊C₄×C₆₇	—	2	4	no	—
129	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₆₇	C₂×D₄	1	2	yes	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	2	no	—
135	Q₈×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₆₇	—	2	4	no	—
154	C₂×Dic₆₇	—	2	2	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	4	2	no	—
157	C₂×Dic₆₇	—	4	2	no	—
158	Dic₁₃₄	—	8	2	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₈×C₆₇	#124	1	4	yes	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₆₇⋊C₈	—	2	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	—	4	4	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₆₇	#529	1	1	yes	characteristic
170	C₂²×C₆₇	#529	1	1	yes	characteristic
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	4	2	no	—
174	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
175	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
176	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
177	C₄×C₆₇	#532	1	1	yes	characteristic
178	Dic₆₇	—	4	2	no	—
179	Dic₆₇	—	4	2	no	—
180	Dic₆₇	—	4	4	no	—
181	#21659	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	2	1	no	—
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	—	8	2	no	—
185	C₂×C₆₇	C₄×#124	1	1	yes	characteristic
186	C₂×C₆₇	#2957	1	2	yes	—
187	#3131	—	67	1	no	—
188	#2961	—	67	1	no	—
189	#2932	D₆₇	1	1	yes	characteristic
190	#1995	—	67	1	no	—
191	#1963	—	67	1	no	—
192	#1938	—	67	2	no	—
193	#1937	—	67	2	no	—
194	#1837	—	67	1	no	—
195	#1765	—	67	2	no	—
196	#1764	—	67	2	no	—
197	#1726	—	67	1	no	—
198	C₆₇	#21659	1	1	yes	characteristic, complemented, Sylow
199	C₂×#131	C₂×D₆₇	1	2	yes	—
200	#533	—	67	1	no	—
201	#532	C₂×D₆₇	1	1	yes	characteristic
202	#488	—	67	4	no	—
203	#487	—	67	2	no	—
204	#486	—	67	2	no	—
205	#446	—	134	4	no	—
206	#435	—	2	4	no	—
207	#424	—	67	2	no	—
208	#423	—	67	1	no	—
209	#423	C₂×D₆₇	1	1	yes	characteristic
210	C₂×#104	—	67	2	no	—
211	C₂×#104	—	67	2	no	—
212	#416	—	67	2	no	—
213	#416	—	67	2	no	—
214	#408	C₂×D₆₇	1	2	yes	—
215	#405	—	67	2	no	—
216	C₂×#97	—	67	1	no	—
217	C₂×#97	—	67	1	no	—
218	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
219	C₂×C₄×C₈	—	67	1	no	—
220	C₂×C₄×C₈	—	67	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	#376	—	67	1	no	—
223	#341	—	67	2	no	—
224	#341	—	67	2	no	—
225	#339	—	67	2	no	—
226	#339	—	67	2	no	—
227	C₂×#41	C₂²×D₆₇	1	2	yes	—
228	#131	—	2	4	no	—
229	#131	—	2	4	no	—
230	C₂×M₁₆	—	2	2	no	—
231	C₂×M₁₆	—	67	2	no	—
232	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	—	67	1	no	—
235	C₂²×C₈	—	67	1	no	—
236	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₄⋊Q₈	—	67	2	no	—
239	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
240	#127	C₄×D₆₇	1	1	yes	characteristic
241	#125	—	67	2	no	—
242	#124	C₄×D₆₇	1	2	yes	—
243	C₂²⋊Q₈	—	134	2	no	—
244	#121	—	2	2	no	—
245	C₄×Q₈	—	134	2	no	—
246	C₄×D₄	—	2	2	no	—
247	#117	—	67	2	no	—
248	C₂×C₄²	—	67	1	no	—
249	C₂×C₄²	—	67	1	no	—
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	134	4	no	—
253	C₄⋊C₈	—	134	2	no	—
254	#104	—	134	4	no	—
255	#104	—	134	4	no	—
256	#103	—	67	4	no	—
257	#103	—	67	4	no	—
258	#102	—	67	4	no	—
259	#102	—	67	4	no	—
260	C₂²⋊C₈	—	2	4	no	—
261	#97	—	134	2	no	—
262	#97	—	134	2	no	—
263	#97	—	134	2	no	—
264	#97	C₂²⋊D₆₇	1	4	yes	—
265	C₄×C₈	—	67	2	no	—
266	C₄×C₈	—	67	2	no	—
267	C₄×C₈	—	134	2	no	—
268	C₄×C₈	—	134	2	no	—
269	C₄×C₈	—	134	2	no	—
270	#95	—	134	2	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	C₂×Q₈	—	134	2	no	—
276	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	134	1	no	—
282	C₂²×C₄	—	134	1	no	—
283	C₂²×C₄	#11761628	1	2	yes	—
284	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	4	4	no	—
288	M₁₆	—	134	2	no	—
289	M₁₆	—	134	2	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	4	no	—
294	C₂×C₈	—	67	2	no	—
295	C₂×C₈	—	67	2	no	—
296	C₂×C₈	—	134	2	no	—
297	C₂×C₈	—	134	1	no	—
298	C₂×C₈	—	134	1	no	—
299	C₂×C₈	—	134	1	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
302	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
303	C₂×C₈	#11761627	1	2	yes	—
304	C₂×C₈	#11761627	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	67	4	no	—
307	C₄⋊C₄	—	134	4	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₂²⋊C₄	—	2	4	no	—
310	C₂²⋊C₄	—	134	2	no	—
311	C₄²	—	67	2	no	—
312	C₄²	—	67	2	no	—
313	C₄²	—	134	2	no	—
314	C₄²	—	134	1	no	—
315	C₄²	—	134	1	no	—
316	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
318	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
319	C₂³	—	2	2	no	—
320	C₂³	C₂×#11761628	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	536	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	268	2	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	134	4	no	—
336	C₂×C₄	—	134	2	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
339	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
340	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
342	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
343	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
345	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	4	2	no	—
348	C₈	—	268	2	no	—
349	C₈	—	134	2	no	—
350	C₈	—	134	2	no	—
351	C₈	#14524030	1	4	yes	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	4	no	—
354	C₂²	—	4	2	no	—
355	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
356	C₂²	#16863928	1	1	yes	characteristic
357	C₂²	#16863928	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	268	2	no	—
362	C₄	—	268	2	no	—
363	C₄	—	268	4	no	—
364	C₄	C₄×#11761627	1	1	yes	characteristic
365	C₄	C₄×#11761627	1	1	yes	characteristic
366	C₄	C₂×#14524030	1	1	yes	characteristic
367	C₄	#16863942	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	C₄×#14524030	1	1	yes	characteristic
372	C₂	#22569124	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2932	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2932	D₆₇
hypocenter	C₆₇	#21659
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #2932

Complements

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

N	H ≅ G/N	type
2. #22569149	C₂	semidirect
3. #22561189	C₂	semidirect
5. #22560923	C₂	semidirect
6. #22560916	C₂	semidirect
7. #22560346	C₂	semidirect
8. #22560051	C₂	semidirect
9. #22560044	C₂	semidirect
10. #22559897	C₂	semidirect
13. #16863750	C₄	semidirect
14. #16863746	C₄	semidirect
15. #16863742	C₄	semidirect
198. C₆₇	#21659	semidirect

Maximal subgroups

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

subgroup	index
#22570531	2
#22569149	2
#22561189	2
#22561042	2
#22560923	2
#22560916	2
#22560346	2
#22560051	2
#22560044	2
#22559897	2
C₆₇×#2932	2
#21659	67