Subgroups — #30864682

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570719	C₂	1	1	yes	maximal, characteristic
2	#22569150	C₂	1	1	yes	maximal, characteristic
3	#22569146	C₂	1	1	yes	maximal, characteristic
4	#22561190	C₂	1	1	yes	maximal, characteristic
5	#22560961	C₂	1	2	yes	maximal
6	#22560956	C₂	1	2	yes	maximal
7	#22560380	C₂	1	1	yes	maximal, characteristic
8	#22560092	C₂	1	2	yes	maximal
9	#22560087	C₂	1	2	yes	maximal
10	#22559896	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1995	C₂	1	1	yes	maximal, characteristic
12	#16863953	C₂²	1	1	yes	characteristic
13	#16863799	C₄	1	2	yes	complemented
14	#16863795	C₄	1	2	yes	complemented
15	#16863795	C₄	1	2	yes	complemented
16	#16863792	C₄	1	2	yes	complemented
17	#16863527	—	2	4	no	—
18	#16863525	—	2	4	no	—
19	C₂×#14523978	C₂²	1	2	yes	—
20	C₂×#14523978	C₂²	1	2	yes	—
21	#16863432	C₂²	1	2	yes	—
22	#16863432	C₂²	1	2	yes	—
23	#16863424	C₂²	1	2	yes	—
24	#16863423	C₂²	1	2	yes	—
25	#16863420	C₂²	1	1	yes	characteristic
26	#16863354	C₂²	1	2	yes	—
27	#16863352	C₂²	1	2	yes	—
28	C₂×#14523954	C₂²	1	1	yes	characteristic
29	C₂×#14523954	C₂²	1	1	yes	characteristic
30	C₂×#14523954	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
34	#16863156	C₂²	1	2	yes	—
35	#16863156	C₂²	1	2	yes	—
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	#423×C₆₇	C₂²	1	1	yes	characteristic
38	#376×C₆₇	C₂²	1	1	yes	characteristic
39	#339×C₆₇	C₂²	1	2	yes	—
40	#339×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761612	C₂³	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	#14524043	C₂×C₄	1	2	yes	—
47	#14524035	C₂×C₄	1	2	yes	—
48	#14524022	—	2	2	no	—
49	C₄×Dic₁₃₄	—	2	2	no	—
50	#14524002	C₂³	1	2	yes	—
51	#14523978	—	2	4	no	—
52	#14523978	—	2	4	no	—
53	#14523976	C₂×C₄	1	4	yes	—
54	#14523976	C₂×C₄	1	4	yes	—
55	#14523974	C₂×C₄	1	4	yes	—
56	#14523974	C₂×C₄	1	4	yes	—
57	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
59	Dic₆₇⋊C₈	—	2	4	no	—
60	#14523954	—	2	2	no	—
61	#14523954	—	2	2	no	—
62	#14523954	D₄	1	4	yes	—
63	#14523954	Q₈	1	4	yes	complemented
64	#14523953	—	2	2	no	—
65	C₄×C₆₇⋊C₈	—	2	2	no	—
66	C₄×C₆₇⋊C₈	—	2	2	no	—
67	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₆₇⋊C₈	Q₈	1	4	yes	complemented
72	C₈×Dic₆₇	—	2	2	no	—
73	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
75	#125×C₆₇	C₂×C₄	1	2	yes	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₄×Q₈×C₆₇	—	2	2	no	—
78	#117×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	C₂×Dic₁₃₄	—	2	2	no	—
85	#11761612	—	2	2	no	—
86	#11761612	—	2	2	no	—
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₆₇⋊C₈	#41	1	2	yes	—
101	C₂×C₆₇⋊C₈	#41	1	2	yes	—
102	C₄⋊Dic₆₇	—	2	2	no	—
103	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
104	Dic₆₇⋊C₄	—	2	4	no	—
105	C₂²⋊Dic₆₇	—	2	2	no	—
106	C₄×Dic₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	#41	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₆₇	—	2	2	no	—
113	M₁₆×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
118	C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	1	no	—
121	C₄²×C₆₇	—	2	1	no	—
122	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	2	yes	—
125	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
126	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
127	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
128	Q₈×C₆₇	—	8	2	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
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	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₆₇	—	2	4	no	—
144	Dic₁₃₄	—	8	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₆₇⋊C₈	—	2	2	no	—
147	C₆₇⋊C₈	—	4	2	no	—
148	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
149	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
150	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
151	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
152	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂²×C₆₇	#533	1	1	yes	characteristic
156	C₂²×C₆₇	#533	1	1	yes	characteristic
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	—	4	4	no	—
162	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₄×C₆₇	#532	1	1	yes	characteristic
166	Dic₆₇	—	4	4	no	—
167	#21733	—	67	1	no	maximal, Sylow
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	2	1	no	—
170	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₆₇	#2957	1	2	yes	—
172	#3160	—	67	1	no	—
173	#2961	—	67	1	no	—
174	#2961	—	67	1	no	—
175	#1995	—	67	1	no	—
176	#1995	D₆₇	1	1	yes	characteristic
177	#1946	—	67	2	no	—
178	#1945	—	67	2	no	—
179	#1848	—	67	1	no	—
180	#1773	—	67	2	no	—
181	#1772	—	67	2	no	—
182	#1726	—	67	1	no	—
183	C₆₇	#21733	1	1	yes	characteristic, complemented, Sylow
184	#533	—	67	1	no	—
185	#533	C₂×D₆₇	1	1	yes	characteristic
186	#500	—	67	2	no	—
187	#499	—	67	2	no	—
188	#499	—	67	2	no	—
189	#498	—	67	2	no	—
190	#446	—	134	4	no	—
191	#446	—	134	4	no	—
192	C₂×#108	—	67	2	no	—
193	C₂×#108	—	67	2	no	—
194	#427	—	67	2	no	—
195	#427	—	67	2	no	—
196	#424	—	67	2	no	—
197	#424	—	67	2	no	—
198	#423	—	67	1	no	—
199	#423	C₂×D₆₇	1	1	yes	characteristic
200	#405	—	67	2	no	—
201	#405	—	67	2	no	—
202	C₂×#97	—	67	1	no	—
203	C₂×#97	—	67	1	no	—
204	C₂×#97	—	67	1	no	—
205	C₂×C₄×C₈	—	67	1	no	—
206	C₂×C₄×C₈	—	67	1	no	—
207	C₂×C₄×C₈	—	67	1	no	—
208	#376	C₂×D₆₇	1	1	yes	characteristic
209	#339	—	67	2	no	—
210	#339	—	67	2	no	—
211	#339	C₂×D₆₇	1	2	yes	—
212	#339	C₂×D₆₇	1	2	yes	—
213	C₂×M₁₆	—	67	2	no	—
214	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
215	C₂²×C₈	—	67	1	no	—
216	C₂²×C₈	—	67	1	no	—
217	C₂²×C₈	—	67	1	no	—
218	C₂²×C₈	—	67	1	no	—
219	C₄⋊Q₈	—	67	2	no	—
220	C₄⋊Q₈	C₄×D₆₇	1	2	yes	—
221	#125	—	67	2	no	—
222	#125	C₄×D₆₇	1	2	yes	—
223	C₂²⋊Q₈	—	2	2	no	—
224	C₂²⋊Q₈	—	134	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	134	2	no	—
227	#117	—	67	2	no	—
228	#117	C₂²×D₆₇	1	2	yes	—
229	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
232	#108	—	134	4	no	—
233	#108	—	134	4	no	—
234	#107	—	67	4	no	—
235	#107	—	67	4	no	—
236	#106	—	67	4	no	—
237	#106	—	67	4	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	134	4	no	—
240	C₄⋊C₈	—	134	4	no	—
241	C₄⋊C₈	—	134	2	no	—
242	#97	—	67	4	no	—
243	#97	—	67	4	no	—
244	#97	—	134	2	no	—
245	#97	—	134	2	no	—
246	#97	—	134	2	no	—
247	C₄×C₈	—	67	1	no	—
248	C₄×C₈	—	67	1	no	—
249	C₄×C₈	—	67	1	no	—
250	C₄×C₈	—	67	1	no	—
251	C₄×C₈	—	67	4	no	—
252	C₄×C₈	—	134	2	no	—
253	C₄×C₈	—	134	2	no	—
254	C₄×C₈	—	134	2	no	—
255	#95	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	134	2	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	#11761627	1	2	yes	—
261	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	134	2	no	—
265	M₁₆	—	134	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	67	1	no	—
268	C₂×C₈	—	67	1	no	—
269	C₂×C₈	—	67	1	no	—
270	C₂×C₈	—	67	1	no	—
271	C₂×C₈	—	67	2	no	—
272	C₂×C₈	—	67	2	no	—
273	C₂×C₈	—	67	2	no	—
274	C₂×C₈	—	67	2	no	—
275	C₂×C₈	—	67	2	no	—
276	C₂×C₈	—	67	2	no	—
277	C₂×C₈	—	67	2	no	—
278	C₂×C₈	—	67	2	no	—
279	C₂×C₈	—	134	2	no	—
280	C₂×C₈	—	134	1	no	—
281	C₂×C₈	—	134	1	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	67	4	no	—
285	C₄⋊C₄	—	134	4	no	—
286	C₄⋊C₄	—	134	2	no	—
287	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	134	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	134	2	no	—
294	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
295	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
296	C₄²	D₄×D₆₇	1	2	yes	—
297	C₄²	Q₈×D₆₇	1	2	yes	—
298	C₄²	Q₈×D₆₇	1	2	yes	—
299	C₂³	C₂×#11761627	1	1	yes	characteristic
300	Q₈	—	8	2	no	—
301	Q₈	—	536	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	134	4	no	—
309	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
310	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
315	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
316	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
317	C₈	—	4	2	no	—
318	C₈	—	268	2	no	—
319	C₈	—	67	2	no	—
320	C₈	—	67	2	no	—
321	C₈	—	67	2	no	—
322	C₈	—	67	2	no	—
323	C₈	—	67	4	no	—
324	C₈	—	134	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
327	C₂²	#16863948	1	1	yes	characteristic
328	C₂²	#16863948	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	2	no	—
333	C₄	—	4	4	no	—
334	C₄	—	268	4	no	—
335	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
336	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
337	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
338	C₄	#16863945	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	#22562778	1	1	yes	characteristic
342	C₂	#22569122	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1995	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1995	D₆₇
hypocenter	C₆₇	#21733
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

Complements

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

N	H ≅ G/N	type
13. #16863799	C₄	semidirect
14. #16863795	C₄	semidirect
15. #16863795	C₄	semidirect
16. #16863792	C₄	semidirect
63. #14523954	Q₈	semidirect
71. C₄×C₆₇⋊C₈	Q₈	semidirect
183. C₆₇	#21733	semidirect

Maximal subgroups

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

subgroup	index
#22570719	2
#22569150	2
#22569146	2
#22561190	2
#22560961	2
#22560956	2
#22560380	2
#22560092	2
#22560087	2
#22559896	2
C₆₇×#1995	2
#21733	67