Subgroups — #30864461

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570637	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863519	C₂	1	2	yes	maximal, complemented
3	C₂×#16863514	C₂	1	2	yes	maximal, complemented
4	#22568928	C₂	1	2	yes	maximal, complemented
5	#22561102	C₂	1	2	yes	maximal, complemented
6	#22561097	C₂	1	2	yes	maximal, complemented
7	#22561041	C₂	1	2	yes	maximal, complemented
8	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524065	C₂²	1	2	yes	—
11	C₂×#14524065	C₂²	1	2	yes	—
12	C₂×#14524049	C₂²	1	4	yes	—
13	#16863946	C₂²	1	2	yes	—
14	#16863780	C₄	1	2	yes	complemented
15	#16863780	C₄	1	2	yes	complemented
16	#16863776	C₄	1	1	yes	characteristic, complemented
17	#16863773	C₄	1	1	yes	characteristic, complemented
18	#16863769	C₄	1	2	yes	complemented
19	#16863519	—	2	8	no	—
20	#16863514	—	2	8	no	—
21	#16863469	—	2	8	no	—
22	C₂×#14523979	C₂²	1	2	yes	—
23	C₂×#14523979	C₂²	1	2	yes	—
24	#16863420	C₂²	1	2	yes	—
25	C₂×#14523971	C₂²	1	4	yes	—
26	C₂×#14523971	C₂²	1	4	yes	—
27	#16863362	C₂²	1	4	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#376×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#11761628	C₂³	1	4	yes	—
36	#14524065	—	2	4	no	—
37	#14524065	—	2	4	no	—
38	#14524065	—	2	4	no	—
39	#14524065	—	2	4	no	—
40	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
41	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
42	C₂×#11761617	C₂×C₄	1	4	yes	—
43	C₂×#11761617	C₂×C₄	1	4	yes	—
44	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
45	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
46	#14524049	—	2	8	no	—
47	#14524049	—	2	8	no	—
48	C₂×#11761612	C₂³	1	4	yes	—
49	C₂×#11761611	—	2	4	no	—
50	C₂×C₈×D₆₇	—	2	4	no	—
51	#14524043	C₂×C₄	1	2	yes	—
52	#14524040	C₂×C₄	1	2	yes	—
53	#14524031	C₂×C₄	1	4	yes	—
54	#14524017	—	2	4	no	—
55	C₄×D₂₆₈	—	2	4	no	—
56	#14523979	—	2	4	no	—
57	#14523979	—	2	4	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
59	#14523971	—	2	8	no	—
60	#14523971	—	2	8	no	—
61	(C₂×D₆₇)⋊C₈	—	2	8	no	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	#97×C₆₇	—	2	2	no	—
70	#97×C₆₇	D₄	1	4	yes	—
71	#97×C₆₇	D₄	1	4	yes	complemented
72	C₄×C₈×C₆₇	—	2	2	no	—
73	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
78	#95×C₆₇	—	2	2	no	—
79	#11761628	—	2	4	no	—
80	#11761628	—	2	4	no	—
81	#11761628	—	2	4	no	—
82	#11761628	—	2	4	no	—
83	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
84	C₂×D₂₆₈	—	2	4	no	—
85	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
86	C₂×C₂²⋊D₆₇	—	2	4	no	—
87	C₂×C₄×D₆₇	—	2	4	no	—
88	Dic₂₆₈	—	2	4	no	—
89	Dic₂₆₈	—	2	4	no	—
90	#11761617	—	2	8	no	—
91	#11761617	—	2	8	no	—
92	D₅₃₆	—	2	4	no	—
93	D₅₃₆	—	2	4	no	—
94	#11761612	—	2	4	no	—
95	#11761612	—	2	4	no	—
96	#11761611	—	4	8	no	—
97	C₂×C₆₇⋊C₈	—	2	4	no	—
98	C₈×D₆₇	—	4	8	no	—
99	C₄⋊Dic₆₇	—	2	4	no	—
100	(C₂×D₆₇)⋊C₄	—	2	8	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	#41	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₆₇	#41	1	2	yes	—
118	C₂×C₈×C₆₇	#41	1	2	yes	—
119	C₄²×C₆₇	—	2	1	no	—
120	C₄²×C₆₇	—	2	1	no	—
121	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂×D₄	1	2	yes	—
124	C₄²×C₆₇	C₂×D₄	1	2	yes	—
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂²×D₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₆₇	—	4	4	no	—
142	C₂²⋊D₆₇	—	4	4	no	—
143	C₂×Dic₆₇	—	2	4	no	—
144	D₂₆₈	—	2	4	no	—
145	D₂₆₈	—	2	4	no	—
146	D₂₆₈	—	8	4	no	—
147	C₄×D₆₇	—	4	4	no	—
148	C₄×D₆₇	—	4	4	no	—
149	Dic₁₃₄	—	2	4	no	—
150	Dic₁₃₄	—	2	4	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	C₄×D₄	1	2	yes	—
153	C₈×C₆₇	C₄×D₄	1	2	yes	—
154	C₈×C₆₇	C₄×D₄	1	2	yes	—
155	C₈×C₆₇	C₄×D₄	1	2	yes	—
156	C₈×C₆₇	#127	1	4	yes	—
157	C₆₇⋊C₈	—	4	4	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₆₇	#532	1	1	yes	characteristic
161	C₂²×C₆₇	#532	1	1	yes	characteristic
162	C₂×D₆₇	—	4	4	no	—
163	C₂×D₆₇	—	4	8	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
171	C₄×C₆₇	#532	1	1	yes	characteristic
172	Dic₆₇	—	4	4	no	—
173	#21707	—	67	1	no	maximal, Sylow
174	C₂×C₆₇	—	2	1	no	—
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
177	C₂×C₆₇	#2957	1	2	yes	—
178	D₆₇	—	8	4	no	—
179	#3147	—	67	1	no	—
180	C₂×#444	—	67	2	no	—
181	C₂×#443	—	67	2	no	—
182	#2932	—	67	2	no	—
183	#1975	—	67	2	no	—
184	#1974	—	67	2	no	—
185	#1963	—	67	2	no	—
186	#1848	—	67	1	no	—
187	#1726	D₆₇	1	1	yes	characteristic, complemented
188	C₆₇	#21707	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	—	67	2	no	—
190	C₂×#135	—	67	2	no	—
191	C₂×#131	—	67	4	no	—
192	#532	—	67	2	no	—
193	#495	—	67	2	no	—
194	#495	—	67	2	no	—
195	#494	—	67	1	no	—
196	#493	—	67	1	no	—
197	#492	—	67	2	no	—
198	#444	—	134	8	no	—
199	#443	—	134	8	no	—
200	#435	—	134	8	no	—
201	C₂×#108	—	67	2	no	—
202	C₂×#108	—	67	2	no	—
203	#423	—	67	2	no	—
204	C₂×#104	—	67	4	no	—
205	C₂×#104	—	67	4	no	—
206	#408	—	67	4	no	—
207	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
208	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
209	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
213	#376	C₂×D₆₇	1	1	yes	characteristic
214	C₂×#41	—	67	4	no	—
215	#135	—	134	4	no	—
216	#135	—	134	4	no	—
217	#135	—	134	4	no	—
218	#135	—	134	4	no	—
219	C₂×Q₁₆	—	67	2	no	—
220	C₂×Q₁₆	—	67	2	no	—
221	C₂×SD₁₆	—	67	4	no	—
222	C₂×SD₁₆	—	67	4	no	—
223	C₂×D₈	—	67	2	no	—
224	C₂×D₈	—	67	2	no	—
225	#131	—	134	8	no	—
226	#131	—	134	8	no	—
227	C₂×M₁₆	—	67	4	no	—
228	C₂×M₁₆	—	134	4	no	—
229	C₂²×C₈	—	134	4	no	—
230	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	C₄⋊Q₈	—	67	2	no	—
235	#127	—	67	2	no	—
236	#124	—	67	4	no	—
237	#121	—	134	4	no	—
238	C₄×D₄	—	134	4	no	—
239	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
242	#108	—	134	4	no	—
243	#108	—	134	4	no	—
244	C₄⋊C₈	—	134	4	no	—
245	#104	—	134	8	no	—
246	#104	—	134	8	no	—
247	C₂²⋊C₈	—	134	8	no	—
248	#97	—	2	2	no	—
249	#97	D₂₆₈	1	4	yes	—
250	#97	D₂₆₈	1	4	yes	complemented
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
253	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
254	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
255	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
256	C₄×C₈	D₂₆₈	1	4	yes	complemented
257	#95	—	2	2	no	—
258	#41	—	134	4	no	—
259	#41	—	134	4	no	—
260	#41	—	134	4	no	—
261	#41	—	134	4	no	—
262	C₂×Q₈	—	67	4	no	—
263	C₂×D₄	—	67	4	no	—
264	C₂×D₄	—	134	4	no	—
265	C₂×D₄	—	134	4	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	134	4	no	—
269	C₂²×C₄	#11761628	1	2	yes	—
270	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
271	Q₁₆	—	134	4	no	—
272	Q₁₆	—	134	4	no	—
273	SD₁₆	—	134	8	no	—
274	SD₁₆	—	134	8	no	—
275	D₈	—	134	4	no	—
276	D₈	—	134	4	no	—
277	M₁₆	—	268	8	no	—
278	M₁₆	—	134	4	no	—
279	M₁₆	—	134	4	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	268	8	no	—
283	C₂×C₈	—	134	4	no	—
284	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
288	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
289	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
290	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
291	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
292	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
293	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
294	C₂×C₈	#11761628	1	2	yes	—
295	C₂×C₈	#11761628	1	2	yes	—
296	C₄⋊C₄	—	134	4	no	—
297	C₂²⋊C₄	—	134	8	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₄²	C₂×D₂₆₈	1	2	yes	—
303	C₄²	C₂×D₂₆₈	1	2	yes	—
304	C₄²	C₂×D₂₆₈	1	2	yes	—
305	C₂³	—	134	4	no	—
306	C₂³	C₂×#11761628	1	1	yes	characteristic
307	Q₈	—	134	4	no	—
308	Q₈	—	134	4	no	—
309	D₄	—	268	4	no	—
310	D₄	—	268	4	no	—
311	D₄	—	536	4	no	—
312	D₄	—	134	4	no	—
313	D₄	—	134	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	268	4	no	—
320	C₂×C₄	—	268	4	no	—
321	C₂×C₄	—	134	4	no	—
322	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
323	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
324	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
328	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
329	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	268	4	no	—
332	C₈	C₄×D₂₆₈	1	2	yes	—
333	C₈	C₄×D₂₆₈	1	2	yes	—
334	C₈	C₄×D₂₆₈	1	2	yes	—
335	C₈	C₄×D₂₆₈	1	2	yes	—
336	C₈	#14524040	1	4	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	268	8	no	—
339	C₂²	—	268	4	no	—
340	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
341	C₂²	#16863946	1	1	yes	characteristic
342	C₂²	#16863946	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	268	4	no	—
348	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
349	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
350	C₄	C₂×#14524040	1	1	yes	characteristic
351	C₄	#16863946	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	536	4	no	—
355	C₂	C₄×#14524040	1	1	yes	characteristic
356	C₂	#22569127	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570637	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21707
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#16863519	C₂	semidirect
3. C₂×#16863514	C₂	semidirect
4. #22568928	C₂	semidirect
5. #22561102	C₂	semidirect
6. #22561097	C₂	semidirect
7. #22561041	C₂	semidirect
8. #22560381	C₂	semidirect
9. C₆₇×#1726	C₂	semidirect
14. #16863780	C₄	semidirect
15. #16863780	C₄	semidirect
16. #16863776	C₄	semidirect
17. #16863773	C₄	semidirect
18. #16863769	C₄	semidirect
71. #97×C₆₇	D₄	semidirect
77. C₄×C₈×C₆₇	D₄	semidirect
187. #1726	D₆₇	semidirect
188. C₆₇	#21707	semidirect
250. #97	D₂₆₈	semidirect
256. C₄×C₈	D₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22570637	2
C₂×#16863519	2
C₂×#16863514	2
#22568928	2
#22561102	2
#22561097	2
#22561041	2
#22560381	2
C₆₇×#1726	2
#21707	67