Subgroups — #30746639

Order 17152.

570 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	C₄×#14523963	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863123	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558050	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1770	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×#14523963	C₄	1	2	yes	—
14	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	#16863158	C₄	1	2	yes	—
18	#16863123	C₄	1	4	yes	complemented
19	#16863118	—	2	2	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	—	2	1	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	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₆₇	—	2	4	no	—
31	C₂×C₈×D₆₇	—	4	2	no	—
32	C₂×C₈×D₆₇	—	4	2	no	—
33	C₄×D₂₆₈	—	2	1	no	—
34	C₄×D₂₆₈	—	2	1	no	—
35	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
36	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
41	#14523963	—	2	4	no	—
42	(C₂×D₆₇)⋊C₈	—	2	4	no	—
43	C₄×C₆₇⋊C₈	—	2	1	no	—
44	C₄×C₆₇⋊C₈	—	2	1	no	—
45	C₄×C₆₇⋊C₈	—	2	1	no	—
46	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	—	2	1	no	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	—	2	1	no	—
54	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
55	#97×C₆₇	—	4	2	no	—
56	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₆₇	D₄	1	2	yes	—
58	#95×C₆₇	—	2	1	no	—
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	M₁₆	1	4	yes	—
64	C₂²×Dic₆₇	—	2	1	no	—
65	C₂×C₄×D₆₇	—	2	2	no	—
66	C₂×C₄×D₆₇	—	2	2	no	—
67	C₂×C₄×D₆₇	—	4	1	no	—
68	C₂×C₄×D₆₇	—	4	1	no	—
69	C₂×C₄×D₆₇	—	4	1	no	—
70	C₂×C₄×D₆₇	—	4	1	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	4	1	no	—
78	C₂×C₆₇⋊C₈	—	4	1	no	—
79	C₈×D₆₇	—	4	8	no	—
80	C₄⋊Dic₆₇	—	2	1	no	—
81	C₄⋊Dic₆₇	C₄²	1	2	yes	—
82	(C₂×D₆₇)⋊C₄	—	2	2	no	—
83	(C₂×D₆₇)⋊C₄	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	4	1	no	—
94	C₂×C₈×C₆₇	—	4	1	no	—
95	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	4	1	no	—
100	C₄²×C₆₇	—	4	1	no	—
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	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₆₇	#131	1	2	yes	—
123	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	4	1	no	—
129	C₂×Dic₆₇	—	4	1	no	—
130	D₂₆₈	—	2	2	no	—
131	D₂₆₈	—	2	4	no	—
132	D₂₆₈	#97	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	—	4	2	no	—
140	C₆₇⋊C₈	—	2	4	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	#435	1	2	yes	—
145	C₂²×C₆₇	#440	1	1	yes	characteristic
146	C₂²×C₆₇	#442	1	1	yes	characteristic
147	C₂²×C₆₇	#443	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	—	8	2	no	—
156	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
157	C₄×C₆₇	#405	1	1	yes	characteristic
158	C₄×C₆₇	#417	1	1	yes	characteristic
159	C₄×C₆₇	#418	1	1	yes	characteristic
160	C₄×C₆₇	#435	1	2	yes	—
161	C₄×C₆₇	#440	1	1	yes	characteristic
162	C₄×C₆₇	#442	1	1	yes	characteristic
163	Dic₆₇	—	4	2	no	—
164	#8204	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	#1585	1	2	yes	—
166	C₂×C₆₇	#1586	1	2	yes	—
167	C₂×C₆₇	#1752	1	1	yes	characteristic
168	C₂×C₆₇	#1762	1	1	yes	characteristic
169	C₂×C₆₇	#1764	1	1	yes	characteristic
170	D₆₇	—	4	4	no	—
171	C₂×C₈×D₄	—	67	1	no	—
172	#1770	D₆₇	1	1	yes	characteristic, complemented
173	C₄×#102	—	67	1	no	—
174	C₂×D₄⋊C₈	—	67	1	no	—
175	C₂×#322	—	67	1	no	—
176	#1279	—	67	1	no	—
177	#1277	—	67	1	no	—
178	C₆₇	#8204	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	C₂×D₆₇	1	1	yes	characteristic
185	C₂×#102	—	67	2	no	—
186	C₂×C₂²⋊C₈	—	134	1	no	—
187	C₂×#97	—	2	1	no	—
188	C₂×C₄×C₈	—	67	1	no	—
189	C₂×C₄×C₈	—	67	1	no	—
190	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
192	#340	—	67	2	no	—
193	#336	—	2	1	no	—
194	D₄⋊C₈	—	67	4	no	—
195	#322	—	134	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	—	67	2	no	—
202	C₂²×C₈	—	134	1	no	—
203	C₂²×C₈	—	134	4	no	—
204	C₂²×C₈	—	134	1	no	—
205	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₆₇	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	C₄×D₆₇	1	2	yes	—
213	C₂×C₂²⋊C₄	—	134	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	67	2	no	—
218	C₄⋊C₈	—	134	1	no	—
219	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
220	#102	—	134	4	no	—
221	C₂²⋊C₈	—	134	4	no	—
222	#97	—	4	2	no	—
223	C₄×C₈	—	67	2	no	—
224	C₄×C₈	—	134	1	no	—
225	C₄×C₈	—	134	1	no	—
226	C₄×C₈	—	134	1	no	—
227	C₄×C₈	C₄×D₆₇	1	2	yes	—
228	C₄×C₈	D₂₆₈	1	2	yes	—
229	#95	—	2	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	—	2	2	no	—
236	C₂²×C₄	—	268	1	no	—
237	C₂²×C₄	—	268	1	no	—
238	C₂²×C₄	—	268	1	no	—
239	C₂²×C₄	—	268	1	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×C₂²⋊D₆₇	1	1	yes	characteristic
245	C₂²×C₄	#11761628	1	1	yes	characteristic
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	268	8	no	—
254	C₂×C₈	—	268	1	no	—
255	C₂×C₈	—	268	1	no	—
256	C₂×C₈	—	134	2	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	2	no	—
260	C₂×C₈	—	134	4	no	—
261	C₂×C₈	—	134	2	no	—
262	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
263	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	67	2	no	—
267	C₄⋊C₄	—	134	1	no	—
268	C₂²⋊C₄	—	134	2	no	—
269	C₂²⋊C₄	—	134	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₄²	C₂×D₂₆₈	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×C₂²⋊D₆₇	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	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	268	1	no	—
290	C₂×C₄	—	268	1	no	—
291	C₂×C₄	—	268	2	no	—
292	C₂×C₄	—	268	2	no	—
293	C₂×C₄	—	268	2	no	—
294	C₂×C₄	—	268	2	no	—
295	C₂×C₄	—	134	2	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×D₂₆₈	1	1	yes	characteristic
301	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
303	C₂×C₄	#14524051	1	2	yes	—
304	C₂×C₄	#14524068	1	1	yes	characteristic
305	C₂×C₄	#14524068	1	1	yes	characteristic
306	C₂×C₄	#14524069	1	1	yes	characteristic
307	C₂×C₄	#14524074	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	268	2	no	—
312	C₈	—	268	2	no	—
313	C₈	—	134	4	no	—
314	C₂²	—	268	2	no	—
315	C₂²	—	268	2	no	—
316	C₂²	—	268	4	no	—
317	C₂²	—	268	4	no	—
318	C₂²	—	134	4	no	—
319	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
320	C₂²	#16863465	1	2	yes	—
321	C₂²	#16863494	1	1	yes	characteristic
322	C₂²	#16863506	1	1	yes	characteristic
323	C₂²	#16863511	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	268	2	no	—
328	C₄	#97×D₆₇	1	1	yes	characteristic
329	C₄	#16863353	1	1	yes	characteristic
330	C₄	#16863395	1	1	yes	characteristic
331	C₄	#16863400	1	1	yes	characteristic
332	C₄	#16863469	1	2	yes	—
333	C₄	#16863494	1	1	yes	characteristic
334	C₄	#16863506	1	1	yes	characteristic
335	C₂	—	268	4	no	—
336	C₂	#22559064	1	2	yes	—
337	C₂	#22559071	1	2	yes	—
338	C₂	#22559986	1	1	yes	characteristic
339	C₂	#22560037	1	1	yes	characteristic
340	C₂	#22560048	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1770	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1770	D₆₇
hypocenter	C₆₇	#8204
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
4. C₂×#16863118	C₂	semidirect
5. #22558050	C₂	semidirect
6. #22558043	C₂	semidirect
7. C₆₇×#1770	C₂	semidirect
11. C₈×D₂₆₈	C₄	semidirect
18. #16863123	C₄	semidirect
132. D₂₆₈	#97	semidirect
172. #1770	D₆₇	semidirect
178. C₆₇	#8204	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
C₄×#14523963	2
C₂×#16863123	2
C₂×#16863118	2
#22558050	2
#22558043	2
C₆₇×#1770	2
#8204	67