Subgroups — #30825752

Order 17152.

698 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570218	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
4	#22559068	C₂	1	4	yes	maximal, complemented
5	#22559023	C₂	1	4	yes	maximal, complemented
6	#22558984	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558635	C₂	1	1	yes	maximal, characteristic
8	#22558613	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₆₇	—	2	1	no	—
11	#16863953	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
13	#16863727	C₄	1	2	yes	—
14	#16863687	C₄	1	2	yes	—
15	#16863602	C₄	1	2	yes	—
16	#16863549	C₄	1	2	yes	—
17	#16863526	—	2	1	no	—
18	#16863526	C₂²	1	2	yes	complemented
19	C₈×Dic₁₃₄	—	2	1	no	—
20	C₈×Dic₁₃₄	C₂²	1	2	yes	complemented
21	C₄×#11761617	C₂²	1	4	yes	—
22	C₈×D₂₆₈	—	2	1	no	—
23	C₈×D₂₆₈	—	2	1	no	—
24	C₈×D₂₆₈	C₂²	1	2	yes	—
25	C₈×D₂₆₈	C₂²	1	2	yes	—
26	#16863422	—	2	1	no	—
27	#16863420	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
30	#16863351	—	2	1	no	—
31	#16863149	C₂²	1	4	yes	complemented
32	#16863128	C₂²	1	4	yes	complemented
33	#16863125	C₂²	1	4	yes	—
34	#431×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#322×C₆₇	C₂²	1	2	yes	complemented
38	#322×C₆₇	C₂²	1	2	yes	complemented
39	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₆₇	—	2	1	no	—
41	C₂×#11761617	C₂×C₄	1	4	yes	—
42	C₂×#11761612	—	2	1	no	—
43	C₂²×C₆₇⋊C₈	—	2	1	no	—
44	C₂×C₈×D₆₇	—	2	2	no	—
45	C₂×C₈×D₆₇	—	2	2	no	—
46	C₂×C₈×D₆₇	—	4	4	no	—
47	#14524043	C₂×C₄	1	1	yes	characteristic
48	#14524043	C₂×C₄	1	1	yes	characteristic
49	#14524035	—	2	1	no	—
50	#14524022	—	2	1	no	—
51	#14524022	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₃₄	—	2	1	no	—
53	C₄×Dic₁₃₄	C₂³	1	2	yes	—
54	C₄×D₂₆₈	—	2	1	no	—
55	C₄×D₂₆₈	—	2	1	no	—
56	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
57	C₄×D₂₆₈	C₂³	1	2	yes	—
58	#14524002	—	2	1	no	—
59	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
61	#14523975	C₂×C₄	1	4	yes	—
62	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
63	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	#14523968	C₂×C₄	1	4	yes	—
69	#14523965	C₂×C₄	1	4	yes	—
70	(C₂×D₆₇)⋊C₈	—	2	2	no	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	#14523953	—	2	2	no	—
73	C₈×Dic₆₇	—	2	2	no	—
74	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	1	no	—
78	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	2	yes	—
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×D₆₇	—	2	1	no	—
88	C₂×Dic₁₃₄	—	2	1	no	—
89	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
90	C₂×D₂₆₈	—	2	1	no	—
91	C₂×D₂₆₈	—	2	4	no	—
92	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₆₇	—	2	1	no	—
94	C₂×C₄×D₆₇	—	2	2	no	—
95	C₂×C₄×D₆₇	—	2	2	no	—
96	C₂×C₄×D₆₇	—	4	2	no	—
97	C₂×C₄×D₆₇	—	4	2	no	—
98	#11761617	—	2	8	no	—
99	#11761612	—	4	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	—	4	2	no	—
103	C₈×D₆₇	—	4	4	no	—
104	C₈×D₆₇	—	4	4	no	—
105	C₄⋊Dic₆₇	—	2	1	no	—
106	C₄⋊Dic₆₇	—	2	1	no	—
107	C₄⋊Dic₆₇	—	2	1	no	—
108	C₄⋊Dic₆₇	—	2	1	no	—
109	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
111	Dic₆₇⋊C₄	—	2	2	no	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	C₂²⋊Dic₆₇	—	2	2	no	—
114	(C₂×D₆₇)⋊C₄	—	2	2	no	—
115	(C₂×D₆₇)⋊C₄	—	2	2	no	—
116	C₄×Dic₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₆₇	—	2	4	no	—
134	C₂²×D₆₇	—	2	2	no	—
135	C₂²×D₆₇	—	4	2	no	—
136	C₂²×D₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₆₇	#137	1	2	yes	—
144	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	4	2	no	—
151	D₂₆₈	—	2	4	no	—
152	D₂₆₈	—	2	2	no	—
153	D₂₆₈	#131	1	4	yes	—
154	C₄×D₆₇	—	4	2	no	—
155	C₄×D₆₇	—	4	2	no	—
156	C₄×D₆₇	—	4	4	no	—
157	Dic₁₃₄	—	2	4	no	—
158	Dic₁₃₄	—	4	2	no	—
159	C₈×C₆₇	—	2	4	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₆₇⋊C₈	—	4	2	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	#440	1	2	yes	—
170	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₆₇	—	2	4	no	—
172	C₂×D₆₇	—	4	8	no	—
173	C₂×D₆₇	—	4	2	no	—
174	C₂×D₆₇	—	4	2	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	#440	1	2	yes	—
179	C₄×C₆₇	C₂×#131	1	2	yes	—
180	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
182	Dic₆₇	—	4	2	no	—
183	Dic₆₇	—	4	2	no	—
184	Dic₆₇	—	4	2	no	—
185	#17905	—	67	1	no	maximal, Sylow
186	C₂×C₆₇	—	2	2	no	—
187	C₂×C₆₇	#2929	1	1	yes	characteristic
188	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
189	C₂×C₆₇	#2955	1	1	yes	characteristic
190	D₆₇	—	4	4	no	—
191	#3101	—	67	1	no	—
192	#2960	—	67	1	no	—
193	C₂×C₈×D₄	—	67	1	no	—
194	#1585	—	67	4	no	—
195	SD₁₆⋊C₈	—	67	4	no	—
196	#1571	—	67	1	no	—
197	#1486	—	67	1	no	—
198	#1481	—	67	1	no	—
199	#1452	D₆₇	1	1	yes	characteristic, complemented
200	C₆₇	#17905	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₈	—	134	1	no	—
202	#533	—	67	1	no	—
203	C₂×C₄×D₄	—	67	1	no	—
204	#483	—	67	2	no	—
205	#476	—	67	2	no	—
206	#461	—	67	2	no	—
207	#450	—	67	2	no	—
208	#446	—	67	2	no	—
209	#446	—	134	1	no	—
210	C₈×Q₈	—	67	2	no	—
211	C₈×Q₈	—	134	1	no	—
212	C₄×SD₁₆	—	67	4	no	—
213	C₈×D₄	—	67	2	no	—
214	C₈×D₄	—	67	2	no	—
215	C₈×D₄	—	134	1	no	—
216	C₈×D₄	—	134	1	no	—
217	#431	C₂×D₆₇	1	1	yes	characteristic
218	#431	C₂×D₆₇	1	1	yes	characteristic
219	#424	—	134	1	no	—
220	#423	—	67	1	no	—
221	C₂×C₄⋊C₈	—	67	1	no	—
222	C₂×C₂²⋊C₈	—	134	1	no	—
223	#405	—	134	1	no	—
224	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
225	#335	—	67	4	no	—
226	Q₈⋊C₈	—	67	4	no	—
227	D₄⋊C₈	—	67	4	no	—
228	#322	C₂×D₆₇	1	2	yes	complemented
229	#322	C₂×D₆₇	1	2	yes	complemented
230	C₂²×D₄	—	67	1	no	—
231	C₂³×C₄	—	134	1	no	—
232	C₂×SD₁₆	—	67	4	no	—
233	C₂×M₁₆	—	134	1	no	—
234	C₂²×C₈	—	268	4	no	—
235	C₂²×C₈	—	134	1	no	—
236	C₂²×C₈	—	134	2	no	—
237	C₂²×C₈	—	134	2	no	—
238	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
239	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
240	C₄⋊Q₈	—	67	1	no	—
241	C₄⋊Q₈	—	67	1	no	—
242	#125	—	134	1	no	—
243	C₂²⋊Q₈	—	67	2	no	—
244	C₂²⋊Q₈	—	134	1	no	—
245	C₄×Q₈	—	67	2	no	—
246	C₄×Q₈	—	134	1	no	—
247	C₄×D₄	—	67	2	no	—
248	C₄×D₄	—	67	2	no	—
249	C₄×D₄	—	134	1	no	—
250	C₄×D₄	—	134	1	no	—
251	#117	—	134	1	no	—
252	C₂×C₄⋊C₄	—	67	1	no	—
253	C₂×C₂²⋊C₄	—	134	1	no	—
254	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
255	#106	—	67	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	67	2	no	—
258	C₄⋊C₈	—	67	2	no	—
259	C₄⋊C₈	—	134	2	no	—
260	C₄⋊C₈	—	134	2	no	—
261	C₄⋊C₈	—	134	1	no	—
262	C₄⋊C₈	—	134	1	no	—
263	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
264	#103	—	67	4	no	—
265	#102	—	67	4	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	134	2	no	—
268	C₂²⋊C₈	—	134	2	no	—
269	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
270	#97	—	134	2	no	—
271	C₄×C₈	—	134	2	no	—
272	C₄×C₈	D₂₆₈	1	1	yes	characteristic
273	C₄×C₈	D₂₆₈	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₆₇	1	2	yes	—
275	C₄×C₈	C₂²×D₆₇	1	2	yes	—
276	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
278	C₂⁴	—	134	1	no	—
279	C₂×Q₈	—	67	2	no	—
280	C₂×Q₈	—	134	1	no	—
281	C₂×D₄	—	67	2	no	—
282	C₂×D₄	—	134	4	no	—
283	C₂×D₄	—	134	1	no	—
284	C₂²×C₄	—	268	2	no	—
285	C₂²×C₄	—	268	2	no	—
286	C₂²×C₄	—	134	1	no	—
287	C₂²×C₄	—	134	2	no	—
288	C₂²×C₄	—	134	2	no	—
289	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
291	C₂²×C₄	#11761628	1	1	yes	characteristic
292	SD₁₆	—	134	8	no	—
293	M₁₆	—	268	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	268	4	no	—
298	C₂×C₈	—	268	4	no	—
299	C₂×C₈	—	268	2	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
303	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
306	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
307	C₄⋊C₄	—	67	2	no	—
308	C₄⋊C₄	—	67	2	no	—
309	C₄⋊C₄	—	134	1	no	—
310	C₄⋊C₄	—	134	1	no	—
311	C₄⋊C₄	—	134	1	no	—
312	C₄⋊C₄	—	134	2	no	—
313	C₄⋊C₄	—	134	1	no	—
314	C₄⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₂²⋊C₄	—	134	2	no	—
317	C₂²⋊C₄	—	134	2	no	—
318	C₄²	—	134	2	no	—
319	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
320	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
321	C₄²	#11761628	1	1	yes	characteristic
322	C₄²	C₂³×D₆₇	1	1	yes	characteristic
323	C₂³	—	268	2	no	—
324	C₂³	—	268	2	no	—
325	C₂³	—	134	4	no	—
326	C₂³	—	134	2	no	—
327	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
328	Q₈	—	268	2	no	—
329	Q₈	—	134	4	no	—
330	D₄	—	67	4	no	—
331	D₄	—	134	4	no	—
332	D₄	—	134	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	268	2	no	—
336	C₂×C₄	—	268	4	no	—
337	C₂×C₄	—	268	2	no	—
338	C₂×C₄	—	268	2	no	—
339	C₂×C₄	—	134	2	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	—	134	2	no	—
342	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
343	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
344	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
345	C₂×C₄	#14524071	1	2	yes	—
346	C₂×C₄	#14524076	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
349	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	4	2	no	—
356	C₈	—	268	2	no	—
357	C₈	—	268	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	268	8	no	—
361	C₂²	—	268	2	no	—
362	C₂²	—	268	2	no	—
363	C₂²	—	134	4	no	—
364	C₂²	#16863498	1	2	yes	—
365	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	268	2	no	—
370	C₄	—	268	2	no	—
371	C₄	—	268	2	no	—
372	C₄	#16863498	1	2	yes	—
373	C₄	#131×D₆₇	1	2	yes	—
374	C₄	C₂×#14524071	1	1	yes	characteristic
375	C₄	C₂×#14524076	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	268	4	no	—
378	C₂	#22568905	1	1	yes	characteristic
379	C₂	C₂×#16863498	1	1	yes	characteristic
380	C₂	#22569113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#17905
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #22569143	C₂	semidirect
4. #22559068	C₂	semidirect
5. #22559023	C₂	semidirect
6. #22558984	C₂	semidirect
8. #22558613	C₂	semidirect
9. C₆₇×#1452	C₂	semidirect
18. #16863526	C₂²	semidirect
20. C₈×Dic₁₃₄	C₂²	semidirect
31. #16863149	C₂²	semidirect
32. #16863128	C₂²	semidirect
37. #322×C₆₇	C₂²	semidirect
38. #322×C₆₇	C₂²	semidirect
199. #1452	D₆₇	semidirect
200. C₆₇	#17905	semidirect
228. #322	C₂×D₆₇	semidirect
229. #322	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22570218	2
#22569143	2
C₂×C₈×D₂₆₈	2
#22559068	2
#22559023	2
#22558984	2
#22558635	2
#22558613	2
C₆₇×#1452	2
#17905	67