Subgroups — #31010571

Order 17152.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570721	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863513	C₂	1	1	yes	maximal, characteristic
3	#22562383	C₂	1	4	yes	maximal, complemented
4	#22562378	C₂	1	4	yes	maximal, complemented
5	#22561982	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561981	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863216	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561945	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2145	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	—	2	1	no	—
11	#16863949	—	2	1	no	—
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	#16863800	—	2	1	no	—
14	#16863797	—	2	1	no	—
15	#16863796	C₂²	1	2	yes	complemented
16	#16863793	C₂²	1	2	yes	complemented
17	#16863513	—	2	1	no	—
18	#16863513	—	2	1	no	—
19	#16863513	C₂²	1	2	yes	—
20	#16863513	C₂²	1	2	yes	—
21	C₂×#14523979	C₂²	1	1	yes	characteristic
22	#16863434	C₂²	1	1	yes	characteristic
23	#16863432	—	2	1	no	—
24	C₂×#14523970	—	2	1	no	—
25	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
26	#16863219	C₂²	1	4	yes	complemented
27	#16863218	C₂²	1	4	yes	complemented
28	#16863216	C₂²	1	2	yes	complemented
29	#16863216	C₂²	1	2	yes	complemented
30	#16863206	C₂²	1	4	yes	—
31	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
33	#427×C₆₇	C₂²	1	1	yes	characteristic
34	#360×C₆₇	C₂²	1	4	yes	—
35	#14524050	—	2	2	no	—
36	#14524050	—	2	2	no	—
37	#14524050	—	4	1	no	—
38	#14524050	—	4	1	no	—
39	C₂×#11761612	—	2	1	no	—
40	C₂×#11761612	—	4	1	no	—
41	C₂²×C₆₇⋊C₈	—	4	1	no	—
42	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
43	#14524042	—	2	2	no	—
44	#14524034	—	2	2	no	—
45	#14524020	—	4	2	no	—
46	C₄×Dic₁₃₄	—	4	2	no	—
47	#14524002	—	2	1	no	—
48	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
49	#14523983	—	2	1	no	—
50	#14523983	C₂³	1	2	yes	—
51	C₂×C₆₇⋊C₁₆	—	2	1	no	—
52	#14523979	—	2	1	no	—
53	#14523979	C₂³	1	2	yes	—
54	#14523977	—	2	1	no	—
55	#14523976	—	2	2	no	—
56	#14523975	C₂³	1	2	yes	—
57	#14523974	—	2	2	no	—
58	#14523970	—	2	2	no	—
59	#14523970	—	2	2	no	—
60	C₈×Dic₆₇	D₄	1	1	yes	characteristic
61	C₈×Dic₆₇	D₄	1	1	yes	characteristic
62	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
63	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
64	C₂×#41×C₆₇	—	2	1	no	—
65	#135×C₆₇	—	2	1	no	—
66	#135×C₆₇	—	2	1	no	—
67	#135×C₆₇	Q₈	1	2	yes	complemented
68	#135×C₆₇	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
70	C₂×SD₁₆×C₆₇	—	2	1	no	—
71	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic, complemented
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	#117×C₆₇	—	2	1	no	—
74	M₃₂×C₆₇	—	2	1	no	—
75	M₃₂×C₆₇	C₂³	1	2	yes	—
76	C₂×C₁₆×C₆₇	—	2	1	no	—
77	#107×C₆₇	—	2	1	no	—
78	#106×C₆₇	C₂³	1	2	yes	—
79	C₂×Dic₁₃₄	—	4	2	no	—
80	C₂²×Dic₆₇	—	2	1	no	—
81	#11761612	—	2	2	no	—
82	#11761612	—	4	1	no	—
83	#11761612	—	4	2	no	—
84	#11761612	—	4	2	no	—
85	C₂×C₆₇⋊C₈	—	4	1	no	—
86	C₂×C₆₇⋊C₈	—	4	2	no	—
87	C₂×C₆₇⋊C₈	—	4	2	no	—
88	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
90	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	4	2	no	—
96	Dic₆₇⋊C₄	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	C₂²⋊Dic₆₇	—	4	1	no	—
99	C₄×Dic₆₇	—	2	1	no	—
100	C₄×Dic₆₇	—	4	1	no	—
101	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
103	C₆₇⋊C₁₆	—	2	2	no	—
104	#41×C₆₇	—	2	2	no	—
105	#41×C₆₇	—	2	2	no	—
106	#41×C₆₇	—	4	1	no	—
107	#41×C₆₇	—	4	1	no	—
108	C₂×Q₈×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	4	1	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₆₇	—	2	1	no	—
114	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	—
115	SD₁₆×C₆₇	—	2	2	no	—
116	SD₁₆×C₆₇	—	2	2	no	—
117	D₈×C₆₇	—	2	1	no	—
118	D₈×C₆₇	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
122	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
123	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	1	no	—
127	C₄²×C₆₇	—	4	1	no	—
128	C₁₆×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	4	1	no	—
130	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₆₇	—	2	2	no	—
132	Q₈×C₆₇	—	4	1	no	—
133	D₄×C₆₇	—	2	2	no	—
134	D₄×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₆₇	—	2	1	no	—
148	C₂×Dic₆₇	—	2	1	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	C₂×Dic₆₇	—	4	2	no	—
151	Dic₁₃₄	—	8	4	no	—
152	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₆₇⋊C₈	—	4	2	no	—
157	C₆₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
158	C₆₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
159	C₆₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
160	C₆₇⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₆₇	—	4	1	no	—
162	C₂²×C₆₇	—	4	2	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	#476	1	1	yes	characteristic
165	C₂²×C₆₇	#476	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	#476	1	1	yes	characteristic
170	C₄×C₆₇	#476	1	1	yes	characteristic
171	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
173	Dic₆₇	—	2	2	no	—
174	Dic₆₇	—	4	1	no	—
175	Dic₆₇	—	8	2	no	—
176	#33637	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	—	2	1	no	—
179	C₂×C₆₇	—	2	1	no	—
180	C₂×C₆₇	—	8	2	no	—
181	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
182	#3160	—	67	1	no	—
183	C₂×#443	—	67	1	no	—
184	#2231	—	67	4	no	—
185	#2230	—	67	4	no	—
186	#2155	—	67	1	no	—
187	#2155	—	67	1	no	—
188	C₂×#364	—	67	1	no	—
189	#2145	—	67	1	no	—
190	#2145	D₆₇	1	1	yes	characteristic
191	C₆₇	#33637	1	1	yes	characteristic, complemented, Sylow
192	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
193	C₂×#131	—	134	1	no	—
194	#533	—	134	1	no	—
195	C₂×M₃₂	—	67	1	no	—
196	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
197	#500	—	134	1	no	—
198	#499	—	67	2	no	—
199	#499	—	134	1	no	—
200	#498	—	67	2	no	—
201	#443	—	67	2	no	—
202	#443	—	67	2	no	—
203	#443	—	134	1	no	—
204	#443	—	134	1	no	—
205	C₂×#108	—	67	1	no	—
206	#427	—	67	1	no	—
207	#427	—	134	1	no	—
208	#427	C₂×D₆₇	1	1	yes	characteristic
209	C₂×#104	—	134	1	no	—
210	C₂×C₄×C₈	—	67	1	no	—
211	#365	—	67	4	no	—
212	#365	—	67	4	no	—
213	#364	—	67	2	no	—
214	#364	—	67	2	no	—
215	#360	—	67	4	no	—
216	#360	C₂×D₆₇	1	4	yes	—
217	C₂×#41	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	#135	Dic₁₃₄	1	2	yes	complemented
221	#135	C₂²×D₆₇	1	2	yes	—
222	C₂×Q₁₆	Dic₁₃₄	1	1	yes	characteristic, complemented
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×D₈	Dic₁₃₄	1	1	yes	characteristic, complemented
225	#131	—	268	1	no	—
226	#131	—	268	1	no	—
227	#131	—	134	2	no	—
228	#131	—	134	2	no	—
229	C₂×M₁₆	—	268	1	no	—
230	C₂×M₁₆	—	134	1	no	—
231	C₂²×C₈	—	268	1	no	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	C₄⋊Q₈	—	134	2	no	—
235	#125	—	134	2	no	—
236	C₂²⋊Q₈	—	268	2	no	—
237	C₄×Q₈	—	268	2	no	—
238	#117	—	2	1	no	—
239	#117	—	134	1	no	—
240	C₂×C₄²	—	67	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	—	67	2	no	—
243	M₃₂	—	134	1	no	—
244	M₃₂	C₂²×D₆₇	1	2	yes	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	134	1	no	—
247	#108	—	67	2	no	—
248	#108	—	134	1	no	—
249	#107	—	2	1	no	—
250	#107	—	134	2	no	—
251	#107	—	134	1	no	—
252	#106	—	67	2	no	—
253	#106	—	134	2	no	—
254	#106	C₂²×D₆₇	1	2	yes	—
255	#104	—	134	2	no	—
256	#104	—	134	2	no	—
257	C₄×C₈	—	67	1	no	—
258	C₄×C₈	—	67	1	no	—
259	C₄×C₈	—	67	1	no	—
260	C₄×C₈	—	67	1	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	4	1	no	—
264	#41	—	4	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	268	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	134	1	no	—
271	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂×Dic₁₃₄	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	C₂×Dic₁₃₄	1	2	yes	—
278	M₁₆	—	268	1	no	—
279	M₁₆	—	268	2	no	—
280	M₁₆	—	268	2	no	—
281	M₁₆	—	134	2	no	—
282	C₂×C₈	—	268	1	no	—
283	C₂×C₈	—	268	2	no	—
284	C₂×C₈	—	268	2	no	—
285	C₂×C₈	—	67	1	no	—
286	C₂×C₈	—	67	1	no	—
287	C₂×C₈	—	67	1	no	—
288	C₂×C₈	—	67	1	no	—
289	C₂×C₈	—	67	1	no	—
290	C₂×C₈	—	67	1	no	—
291	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
293	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
294	C₂×C₈	#11761627	1	1	yes	characteristic
295	C₂×C₈	#11761627	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	268	2	no	—
299	C₄⋊C₄	—	268	2	no	—
300	C₄⋊C₄	—	268	2	no	—
301	C₄⋊C₄	—	134	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	268	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	268	1	no	—
306	C₄²	—	67	1	no	—
307	C₄²	—	67	1	no	—
308	C₄²	—	134	1	no	—
309	C₁₆	—	2	2	no	—
310	C₁₆	—	134	2	no	—
311	C₂³	—	4	1	no	—
312	C₂³	#14524067	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	536	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	268	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	134	1	no	—
327	C₂×C₄	—	134	1	no	—
328	C₂×C₄	#14524020	1	1	yes	characteristic
329	C₂×C₄	#14524020	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
332	C₂×C₄	#14524067	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
334	C₈	—	268	2	no	—
335	C₈	—	67	1	no	—
336	C₈	—	67	1	no	—
337	C₈	—	67	1	no	—
338	C₈	—	67	1	no	—
339	C₈	#14524020	1	1	yes	characteristic
340	C₈	#14524020	1	1	yes	characteristic
341	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
342	C₈	C₂×#11761627	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#16863684	1	1	yes	characteristic
347	C₂²	#16863684	1	1	yes	characteristic
348	C₂²	C₂×#14524067	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	268	1	no	—
352	C₄	—	536	2	no	—
353	C₄	—	134	2	no	—
354	C₄	#16863684	1	1	yes	characteristic
355	C₄	#16863684	1	1	yes	characteristic
356	C₄	C₂×#14524020	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#16863684	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863684
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2145	C₂
socle	C₂×C₆₇	C₂×#476
prime radical	C₂×#135×C₆₇	C₂²
hypercenter	#2145	D₆₇
hypocenter	C₆₇	#33637
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
1. #22570721	C₂	semidirect
3. #22562383	C₂	semidirect
4. #22562378	C₂	semidirect
5. #22561982	C₂	semidirect
6. #22561981	C₂	semidirect
7. C₂×#16863216	C₂	semidirect
15. #16863796	C₂²	semidirect
16. #16863793	C₂²	semidirect
26. #16863219	C₂²	semidirect
27. #16863218	C₂²	semidirect
28. #16863216	C₂²	semidirect
29. #16863216	C₂²	semidirect
67. #135×C₆₇	Q₈	semidirect
69. C₂×Q₁₆×C₆₇	Q₈	semidirect
71. C₂×D₈×C₆₇	Q₈	semidirect
191. C₆₇	#33637	semidirect
220. #135	Dic₁₃₄	semidirect
222. C₂×Q₁₆	Dic₁₃₄	semidirect
224. C₂×D₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570721	2
C₂×#16863513	2
#22562383	2
#22562378	2
#22561982	2
#22561981	2
C₂×#16863216	2
#22561945	2
C₆₇×#2145	2
#33637	67