Subgroups — #30746609

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14523965	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863125	C₂	1	1	yes	maximal, characteristic
5	#22558049	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558044	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	—	2	1	no	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	C₈×D₂₆₈	—	2	2	no	—
11	C₈×D₂₆₈	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14523965	C₄	1	2	yes	—
15	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
16	C₂×C₈×Dic₆₇	—	2	1	no	—
17	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863159	C₄	1	2	yes	—
19	#16863150	—	2	1	no	—
20	#16863125	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#322×C₆₇	—	2	2	no	—
25	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₆₇	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₈×D₆₇	—	2	4	no	—
30	C₂×C₈×D₆₇	—	4	2	no	—
31	C₂×C₈×D₆₇	—	4	2	no	—
32	C₄×D₂₆₈	—	2	1	no	—
33	C₄×D₂₆₈	—	2	1	no	—
34	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
35	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
38	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₆₇	—	2	1	no	—
40	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
44	#14523965	C₈	1	8	yes	complemented
45	(C₂×D₆₇)⋊C₈	—	2	4	no	—
46	C₈×Dic₆₇	—	4	2	no	—
47	C₆₇⋊#95	—	2	1	no	—
48	C₂²×C₈×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₇	—	2	1	no	—
54	C₄×C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₇	D₄	1	2	yes	—
59	C₂³×D₆₇	—	2	1	no	—
60	C₂×D₂₆₈	—	2	1	no	—
61	C₂×D₂₆₈	—	2	2	no	—
62	C₂×D₂₆₈	C₄²	1	2	yes	—
63	C₂×D₂₆₈	C₂×C₈	1	4	yes	—
64	C₂²×Dic₆₇	—	2	2	no	—
65	C₂²×Dic₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	—	2	2	no	—
67	C₂×C₄×D₆₇	—	2	2	no	—
68	C₂×C₄×D₆₇	—	4	1	no	—
69	C₂×C₄×D₆₇	—	4	1	no	—
70	C₂×C₄×D₆₇	—	4	1	no	—
71	C₂×C₄×D₆₇	—	4	1	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	4	1	no	—
75	C₂×C₆₇⋊C₈	—	4	1	no	—
76	C₂×C₆₇⋊C₈	—	4	1	no	—
77	C₂×C₆₇⋊C₈	—	4	1	no	—
78	C₈×D₆₇	—	4	8	no	—
79	C₄⋊Dic₆₇	—	2	1	no	—
80	C₄⋊Dic₆₇	—	2	2	no	—
81	C₄⋊Dic₆₇	C₄²	1	2	yes	—
82	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	complemented
83	(C₂×D₆₇)⋊C₄	—	2	2	no	—
84	(C₂×D₆₇)⋊C₄	—	2	2	no	—
85	C₄×Dic₆₇	—	4	1	no	—
86	C₄×Dic₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	1	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	4	2	no	—
98	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	complemented
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₆₇	#41	1	1	yes	characteristic
105	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₆₇	—	2	2	no	—
107	C₂²×D₆₇	—	2	4	no	—
108	C₂²×D₆₇	—	4	1	no	—
109	C₂²×D₆₇	—	4	1	no	—
110	C₂²×D₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₆₇	#131	1	2	yes	—
120	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	2	4	no	—
126	C₂×Dic₆₇	—	4	1	no	—
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	4	1	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	D₂₆₈	—	2	2	no	—
131	D₂₆₈	—	2	4	no	—
132	D₂₆₈	C₄×C₈	1	4	yes	complemented
133	C₄×D₆₇	—	4	2	no	—
134	C₄×D₆₇	—	4	2	no	—
135	C₄×D₆₇	—	4	2	no	—
136	C₄×D₆₇	—	4	2	no	—
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	—	2	2	no	—
139	C₈×C₆₇	—	2	4	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₆₇⋊C₈	—	4	2	no	—
142	C₆₇⋊C₈	—	4	2	no	—
143	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₆₇	C₈×D₄	1	2	yes	—
145	C₂²×C₆₇	#440	1	1	yes	characteristic
146	C₂²×C₆₇	#442	1	1	yes	characteristic
147	C₂²×C₆₇	#444	1	2	yes	—
148	C₂×D₆₇	—	2	4	no	—
149	C₂×D₆₇	—	4	4	no	—
150	C₂×D₆₇	—	4	4	no	—
151	C₂×D₆₇	—	4	2	no	—
152	C₂×D₆₇	—	4	2	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
156	C₄×C₆₇	#405	1	1	yes	characteristic
157	C₄×C₆₇	#417	1	1	yes	characteristic
158	C₄×C₆₇	#418	1	1	yes	characteristic
159	C₄×C₆₇	C₈×D₄	1	2	yes	—
160	C₄×C₆₇	#440	1	1	yes	characteristic
161	C₄×C₆₇	#442	1	1	yes	characteristic
162	Dic₆₇	—	4	4	no	—
163	Dic₆₇	—	4	2	no	—
164	#8200	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	SD₁₆⋊C₈	1	2	yes	—
166	C₂×C₆₇	#1580	1	2	yes	—
167	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₆₇	#1762	1	1	yes	characteristic
169	C₂×C₆₇	#1765	1	1	yes	characteristic
170	D₆₇	—	4	4	no	—
171	C₂×C₈×D₄	—	67	1	no	—
172	C₈×C₄⋊C₄	—	67	1	no	—
173	C₄×#102	—	67	1	no	—
174	C₂×D₄⋊C₈	—	67	1	no	—
175	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
176	#1279	—	67	1	no	—
177	#1277	—	67	1	no	—
178	C₆₇	#8200	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	134	1	no	—
180	C₂×C₄×D₄	—	67	1	no	—
181	C₈×D₄	—	67	4	no	—
182	C₈×D₄	—	134	2	no	—
183	C₂×C₄⋊C₈	—	67	1	no	—
184	C₂×C₄⋊C₈	—	67	1	no	—
185	C₂×#102	—	67	2	no	—
186	C₂×C₂²⋊C₈	—	134	1	no	—
187	C₂×C₄×C₈	—	134	1	no	—
188	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	67	1	no	—
192	#340	—	67	2	no	—
193	#336	—	134	1	no	—
194	D₄⋊C₈	—	67	4	no	—
195	#322	—	2	2	no	—
196	C₂²×D₄	—	67	1	no	—
197	C₂³×C₄	—	134	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	268	2	no	—
200	C₂²×C₈	—	268	2	no	—
201	C₂²×C₈	—	134	1	no	—
202	C₂²×C₈	—	134	1	no	—
203	C₂²×C₈	—	134	4	no	—
204	C₂²×C₈	C₄×D₆₇	1	2	yes	—
205	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
206	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
207	C₄×D₄	—	67	2	no	—
208	C₄×D₄	—	67	2	no	—
209	C₄×D₄	—	134	1	no	—
210	C₄×D₄	—	134	1	no	—
211	C₂×C₄⋊C₄	—	67	1	no	—
212	C₂×C₄⋊C₄	—	67	2	no	—
213	C₂×C₂²⋊C₄	—	134	1	no	—
214	C₂×C₄²	—	134	1	no	—
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	C₄⋊C₈	—	67	2	no	—
217	C₄⋊C₈	—	67	2	no	—
218	C₄⋊C₈	—	134	1	no	—
219	C₄⋊C₈	—	134	1	no	—
220	#102	—	67	8	no	—
221	C₂²⋊C₈	—	134	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	268	2	no	—
226	C₄×C₈	C₄×D₆₇	1	2	yes	—
227	C₄×C₈	C₄×D₆₇	1	2	yes	—
228	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
229	#95	—	134	1	no	—
230	C₂⁴	—	134	1	no	—
231	C₂×D₄	—	67	2	no	—
232	C₂×D₄	—	67	4	no	—
233	C₂×D₄	—	134	1	no	—
234	C₂×D₄	—	134	2	no	—
235	C₂²×C₄	—	268	1	no	—
236	C₂²×C₄	—	268	1	no	—
237	C₂²×C₄	—	268	1	no	—
238	C₂²×C₄	—	268	1	no	—
239	C₂²×C₄	—	134	2	no	—
240	C₂²×C₄	—	134	1	no	—
241	C₂²×C₄	—	134	2	no	—
242	C₂²×C₄	—	134	2	no	—
243	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
245	C₂²×C₄	#11761628	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	268	8	no	—
255	C₂×C₈	—	268	1	no	—
256	C₂×C₈	—	268	1	no	—
257	C₂×C₈	—	268	1	no	—
258	C₂×C₈	—	268	1	no	—
259	C₂×C₈	—	134	2	no	—
260	C₂×C₈	—	134	2	no	—
261	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
262	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
263	C₂×C₈	C₈×D₆₇	1	4	yes	complemented
264	C₄⋊C₄	—	67	2	no	—
265	C₄⋊C₄	—	67	4	no	—
266	C₄⋊C₄	—	134	1	no	—
267	C₄⋊C₄	—	134	2	no	—
268	C₂²⋊C₄	—	134	2	no	—
269	C₂²⋊C₄	—	134	2	no	—
270	C₄²	—	268	1	no	—
271	C₄²	—	268	1	no	—
272	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₄²	#11761628	1	1	yes	characteristic
276	C₂³	—	268	1	no	—
277	C₂³	—	268	1	no	—
278	C₂³	—	268	2	no	—
279	C₂³	—	134	2	no	—
280	C₂³	—	134	4	no	—
281	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
282	D₄	—	67	4	no	—
283	D₄	—	134	2	no	—
284	D₄	—	134	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	268	1	no	—
287	C₂×C₄	—	268	2	no	—
288	C₂×C₄	—	268	1	no	—
289	C₂×C₄	—	268	2	no	—
290	C₂×C₄	—	268	2	no	—
291	C₂×C₄	—	268	2	no	—
292	C₂×C₄	—	268	2	no	—
293	C₂×C₄	—	268	2	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	—	134	4	no	—
296	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	#14524000	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
301	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
303	C₂×C₄	#14524049	1	2	yes	—
304	C₂×C₄	#14524071	1	1	yes	characteristic
305	C₂×C₄	#14524071	1	1	yes	characteristic
306	C₂×C₄	#14524071	1	1	yes	characteristic
307	C₂×C₄	#14524076	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	268	2	no	—
313	C₈	—	268	2	no	—
314	C₂²	—	268	2	no	—
315	C₂²	—	268	4	no	—
316	C₂²	—	268	2	no	—
317	C₂²	—	268	4	no	—
318	C₂²	—	134	4	no	—
319	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
320	C₂²	C₈×D₂₆₈	1	2	yes	—
321	C₂²	#16863498	1	1	yes	characteristic
322	C₂²	#16863509	1	1	yes	characteristic
323	C₂²	#16863519	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	268	2	no	—
327	C₄	—	268	4	no	—
328	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
329	C₄	#16863350	1	1	yes	characteristic
330	C₄	#16863398	1	1	yes	characteristic
331	C₄	#16863402	1	1	yes	characteristic
332	C₄	C₈×C₂²⋊D₆₇	1	2	yes	—
333	C₄	#16863498	1	1	yes	characteristic
334	C₄	#16863509	1	1	yes	characteristic
335	C₂	—	268	4	no	—
336	C₂	#22559023	1	2	yes	—
337	C₂	#22559034	1	2	yes	—
338	C₂	C₈×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
339	C₂	#22560035	1	1	yes	characteristic
340	C₂	#22560053	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8200
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₈×C₄⋊Dic₆₇	C₂	semidirect
5. #22558049	C₂	semidirect
6. #22558044	C₂	semidirect
7. C₂×#322×C₆₇	C₂	semidirect
11. C₈×D₂₆₈	C₄	semidirect
20. #16863125	C₄	semidirect
44. #14523965	C₈	semidirect
82. C₄⋊Dic₆₇	C₂×C₈	semidirect
100. C₂×C₈×C₆₇	C₂×C₈	semidirect
132. D₂₆₈	C₄×C₈	semidirect
175. C₂×#322	D₆₇	semidirect
178. C₆₇	#8200	semidirect
263. C₂×C₈	C₈×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₆₈	2
C₈×C₄⋊Dic₆₇	2
C₄×#14523965	2
C₂×#16863125	2
#22558049	2
#22558044	2
C₂×#322×C₆₇	2
#8200	67