Subgroups — #31009188

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#22569056	C₂	1	2	yes	maximal, complemented
3	#22562083	C₂	1	2	yes	maximal, complemented
4	#22562082	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562055	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561990	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561923	C₂	1	2	yes	maximal, complemented
10	#22561804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₆₇	C₂²	1	2	yes	—
13	#16863868	—	2	2	no	—
14	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
15	C₂×#14523984	C₂²	1	2	yes	—
16	D₁₆×D₆₇	C₄	1	8	yes	complemented
17	C₂×#14523982	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	#16863506	—	2	2	no	—
20	#16863505	—	2	2	no	—
21	C₄×D₅₃₆	C₂²	1	2	yes	complemented
22	D₈×Dic₆₇	C₂²	1	2	yes	complemented
23	#16863478	C₂²	1	2	yes	complemented
24	#16863433	C₂²	1	2	yes	—
25	#16863399	—	2	2	no	—
26	#16863350	C₂²	1	1	yes	characteristic
27	#16863223	C₂²	1	2	yes	complemented
28	#16863222	C₂²	1	1	yes	characteristic, complemented
29	#16863198	C₂²	1	2	yes	complemented
30	#16863197	C₂²	1	2	yes	complemented
31	#16863196	C₂²	1	2	yes	complemented
32	#16863172	C₂²	1	1	yes	characteristic, complemented
33	#16863171	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
37	#366×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#357×C₆₇	C₂²	1	2	yes	complemented
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₆₇	—	2	2	no	—
41	C₂×D₅₃₆	C₂³	1	2	yes	—
42	C₂×#11761613	—	2	2	no	—
43	D₈×D₆₇	—	2	4	no	—
44	D₈×D₆₇	C₂×C₄	1	8	yes	complemented
45	C₂×#11761611	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	C₄×D₂₆₈	—	2	2	no	—
48	D₄×Dic₆₇	—	2	2	no	—
49	#14524005	—	2	2	no	—
50	C₄×C₂²⋊D₆₇	—	4	2	no	—
51	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
52	#14524001	—	2	2	no	—
53	#14524000	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₆₇)⋊C₄	—	4	2	no	—
55	C₂²⋊C₄×D₆₇	—	4	2	no	—
56	D₁₀₇₂	C₂×C₄	1	4	yes	complemented
57	#14523984	C₂×C₄	1	8	yes	complemented
58	#14523982	—	2	2	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	complemented
62	#14523977	C₂³	1	2	yes	—
63	#14523974	—	2	2	no	—
64	#14523965	—	2	2	no	—
65	#14523964	—	2	2	no	—
66	#14523963	—	2	2	no	—
67	#14523954	D₄	1	1	yes	characteristic
68	#14523953	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₆₇	C₂³	1	2	yes	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	D₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#102×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×D₆₇	—	4	2	no	—
79	C₂×D₂₆₈	—	2	2	no	—
80	C₂×C₂²⋊D₆₇	—	4	2	no	—
81	D₄×D₆₇	—	2	8	no	—
82	D₄×D₆₇	—	4	4	no	—
83	C₂²×Dic₆₇	—	4	2	no	—
84	C₂×C₄×D₆₇	—	4	2	no	—
85	C₂×C₄×D₆₇	—	4	2	no	—
86	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
87	D₅₃₆	—	2	2	no	—
88	D₅₃₆	C₂²×C₄	1	4	yes	—
89	#11761613	—	2	8	no	—
90	#11761611	—	2	2	no	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₆₇	C₂²×C₄	1	2	yes	—
94	C₈×D₆₇	#41	1	2	yes	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
97	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
98	C₂²⋊Dic₆₇	—	4	2	no	—
99	(C₂×D₆₇)⋊C₄	—	4	2	no	—
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	2	no	—
102	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
103	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
104	C₄×Dic₆₇	—	4	2	no	—
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	D₈×C₆₇	—	2	2	no	—
111	D₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂³×C₆₇	—	4	2	no	—
120	C₂²×D₆₇	—	4	2	no	—
121	C₂²×D₆₇	—	4	8	no	—
122	C₂²×D₆₇	—	16	2	no	—
123	C₂²×D₆₇	—	16	2	no	—
124	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
125	D₄×C₆₇	—	2	4	no	—
126	D₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	16	2	no	—
129	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₆₇	—	4	8	no	—
133	C₂×Dic₆₇	—	4	2	no	—
134	C₂×Dic₆₇	—	16	2	no	—
135	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
138	D₂₆₈	—	2	4	no	—
139	D₂₆₈	—	4	2	no	—
140	C₄×D₆₇	—	16	2	no	—
141	C₄×D₆₇	—	16	2	no	—
142	C₄×D₆₇	C₄×D₄	1	2	yes	—
143	C₄×D₆₇	#135	1	2	yes	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₆₇⋊C₈	—	2	1	no	—
148	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
149	C₂²×C₆₇	—	4	4	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₆₇	—	2	1	no	—
154	C₂×D₆₇	—	2	1	no	—
155	C₂×D₆₇	—	4	4	no	—
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	8	2	no	—
158	C₂×D₆₇	—	16	8	no	—
159	C₂×D₆₇	C₄×D₈	1	2	yes	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	2	1	no	—
162	C₄×C₆₇	—	8	2	no	—
163	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
164	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
165	Dic₆₇	—	2	1	no	—
166	Dic₆₇	—	2	1	no	—
167	Dic₆₇	—	8	2	no	—
168	Dic₆₇	C₄×D₈	1	2	yes	—
169	#33456	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	—	8	4	no	—
171	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
172	C₂×C₆₇	#3415	1	1	yes	characteristic
173	C₂×C₆₇	#3415	1	1	yes	characteristic
174	D₆₇	—	2	2	no	—
175	D₆₇	—	8	4	no	—
176	C₂²×D₁₆	—	67	1	no	—
177	#2948	—	67	2	no	—
178	#2177	—	67	2	no	—
179	#2177	—	67	2	no	—
180	C₄×D₁₆	—	67	1	no	—
181	C₄×D₁₆	—	67	1	no	—
182	C₄×D₁₆	—	67	1	no	—
183	C₄×D₁₆	D₆₇	1	1	yes	characteristic, complemented
184	#2158	—	67	1	no	—
185	#2140	—	67	2	no	—
186	#2108	—	67	1	no	—
187	C₆₇	#33456	1	1	yes	characteristic, complemented, Sylow
188	C₂²×D₈	—	67	2	no	—
189	#518	—	134	2	no	—
190	C₂×D₁₆	—	67	2	no	—
191	C₂×D₁₆	—	67	8	no	—
192	C₂×D₁₆	—	67	1	no	—
193	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
194	C₂×M₃₂	—	67	1	no	—
195	C₂²×C₁₆	—	67	1	no	—
196	#442	—	134	2	no	—
197	#442	—	134	2	no	—
198	C₄×D₈	—	67	2	no	—
199	C₄×D₈	—	67	2	no	—
200	C₄×D₈	—	67	2	no	—
201	C₄×D₈	C₂×D₆₇	1	2	yes	complemented
202	#427	—	67	2	no	—
203	#418	—	134	2	no	—
204	#405	—	67	1	no	—
205	#367	—	67	2	no	—
206	#366	—	67	1	no	—
207	#366	C₂×D₆₇	1	1	yes	characteristic, complemented
208	#357	—	67	2	no	—
209	#357	—	67	2	no	—
210	#357	—	67	2	no	—
211	#357	C₂×D₆₇	1	2	yes	complemented
212	#346	—	67	1	no	—
213	#346	—	67	1	no	—
214	C₄×C₁₆	—	67	1	no	—
215	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
216	C₂²×D₄	—	134	2	no	—
217	C₂×D₈	—	67	8	no	—
218	C₂×D₈	—	67	2	no	—
219	C₂×D₈	—	134	4	no	—
220	C₂×D₈	—	134	2	no	—
221	C₂×D₈	C₂²×D₆₇	1	2	yes	—
222	C₂×M₁₆	—	67	1	no	—
223	C₂²×C₈	—	67	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	268	2	no	—
226	C₄×D₄	—	268	2	no	—
227	C₄×D₄	—	134	2	no	—
228	C₄×D₄	—	134	2	no	—
229	C₄×D₄	—	134	2	no	—
230	#117	—	67	1	no	—
231	#117	—	134	2	no	—
232	C₂×C₂²⋊C₄	—	268	2	no	—
233	C₂×C₂²⋊C₄	—	268	2	no	—
234	D₁₆	—	67	8	no	—
235	D₁₆	—	67	4	no	—
236	D₁₆	C₄×D₆₇	1	4	yes	complemented
237	M₃₂	—	134	2	no	—
238	C₂×C₁₆	—	67	1	no	—
239	C₂×C₁₆	—	67	4	no	—
240	C₂×C₁₆	—	67	1	no	—
241	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
242	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
243	#107	—	67	2	no	—
244	#107	C₂²×D₆₇	1	2	yes	—
245	#106	—	134	2	no	—
246	#102	—	2	2	no	—
247	#102	—	134	2	no	—
248	#102	—	134	2	no	—
249	#102	—	134	2	no	—
250	#97	—	67	1	no	—
251	#97	—	67	1	no	—
252	C₄×C₈	—	67	1	no	—
253	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
254	C₂⁴	—	268	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	268	4	no	—
257	C₂×D₄	—	268	2	no	—
258	C₂×D₄	—	134	8	no	—
259	C₂×D₄	—	134	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	268	2	no	—
262	C₂²×C₄	—	268	2	no	—
263	C₂²×C₄	—	268	2	no	—
264	C₂²×C₄	—	67	1	no	—
265	D₈	—	2	2	no	—
266	D₈	—	67	4	no	—
267	D₈	—	134	8	no	—
268	D₈	—	134	2	no	—
269	D₈	C₂×C₄×D₆₇	1	4	yes	—
270	M₁₆	—	134	2	no	—
271	C₂×C₈	—	67	1	no	—
272	C₂×C₈	—	67	2	no	—
273	C₂×C₈	—	67	2	no	—
274	C₂×C₈	—	67	1	no	—
275	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	67	1	no	—
279	C₄⋊C₄	—	67	1	no	—
280	C₄⋊C₄	—	134	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	268	2	no	—
283	C₂²⋊C₄	—	268	2	no	—
284	C₂²⋊C₄	—	268	2	no	—
285	C₂²⋊C₄	—	268	2	no	—
286	C₂²⋊C₄	—	67	1	no	—
287	C₂²⋊C₄	—	67	1	no	—
288	C₄²	—	268	2	no	—
289	C₄²	—	67	1	no	—
290	C₄²	D₄×D₆₇	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	67	2	no	—
293	C₁₆	—	134	1	no	—
294	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
295	C₂³	—	4	2	no	—
296	C₂³	—	268	8	no	—
297	C₂³	—	268	2	no	—
298	C₂³	—	1072	2	no	—
299	C₂³	—	1072	2	no	—
300	C₂³	—	67	1	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	2	no	—
303	D₄	—	268	8	no	—
304	D₄	—	268	2	no	—
305	D₄	—	134	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	268	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	—	1072	2	no	—
310	C₂×C₄	—	1072	2	no	—
311	C₂×C₄	—	1072	2	no	—
312	C₂×C₄	—	67	1	no	—
313	C₂×C₄	—	67	2	no	—
314	C₂×C₄	—	67	2	no	—
315	C₂×C₄	—	67	1	no	—
316	C₂×C₄	—	67	1	no	—
317	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
318	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	67	2	no	—
322	C₈	—	134	1	no	—
323	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
324	C₈	#41×D₆₇	1	1	yes	characteristic
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	268	4	no	—
329	C₂²	—	536	2	no	—
330	C₂²	—	536	2	no	—
331	C₂²	—	1072	8	no	—
332	C₂²	—	67	2	no	—
333	C₂²	—	134	1	no	—
334	C₂²	—	134	1	no	—
335	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	536	2	no	—
340	C₄	—	67	2	no	—
341	C₄	—	134	1	no	—
342	C₄	—	134	1	no	—
343	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
344	C₄	#135×D₆₇	1	1	yes	characteristic
345	C₂	—	8	4	no	—
346	C₂	—	536	4	no	—
347	C₂	—	134	2	no	—
348	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
349	C₂	#22573346	1	1	yes	characteristic
350	C₂	#22573347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×D₁₆×D₆₇	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×D₁₆	D₆₇
hypocenter	C₆₇	#33456
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. #22569056	C₂	semidirect
3. #22562083	C₂	semidirect
4. #22562082	C₂	semidirect
5. C₄×D₁₀₇₂	C₂	semidirect
6. D₁₆×Dic₆₇	C₂	semidirect
7. #22562055	C₂	semidirect
8. #22561990	C₂	semidirect
9. #22561923	C₂	semidirect
10. #22561804	C₂	semidirect
11. C₄×D₁₆×C₆₇	C₂	semidirect
16. D₁₆×D₆₇	C₄	semidirect
21. C₄×D₅₃₆	C₂²	semidirect
22. D₈×Dic₆₇	C₂²	semidirect
23. #16863478	C₂²	semidirect
27. #16863223	C₂²	semidirect
28. #16863222	C₂²	semidirect
29. #16863198	C₂²	semidirect
30. #16863197	C₂²	semidirect
31. #16863196	C₂²	semidirect
32. #16863172	C₂²	semidirect
33. #16863171	C₂²	semidirect
34. C₁₆×Dic₆₇	C₂²	semidirect
36. C₄×D₈×C₆₇	C₂²	semidirect
37. #366×C₆₇	C₂²	semidirect
38. #357×C₆₇	C₂²	semidirect
39. C₄×C₁₆×C₆₇	C₂²	semidirect
44. D₈×D₆₇	C₂×C₄	semidirect
56. D₁₀₇₂	C₂×C₄	semidirect
57. #14523984	C₂×C₄	semidirect
61. C₁₆×D₆₇	C₂×C₄	semidirect
72. D₁₆×C₆₇	C₂×C₄	semidirect
183. C₄×D₁₆	D₆₇	semidirect
187. C₆₇	#33456	semidirect
201. C₄×D₈	C₂×D₆₇	semidirect
207. #366	C₂×D₆₇	semidirect
211. #357	C₂×D₆₇	semidirect
215. C₄×C₁₆	C₂×D₆₇	semidirect
236. D₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₆₇	2
#22569056	2
#22562083	2
#22562082	2
C₄×D₁₀₇₂	2
D₁₆×Dic₆₇	2
#22562055	2
#22561990	2
#22561923	2
#22561804	2
C₄×D₁₆×C₆₇	2
#33456	67