Subgroups — #30748766

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863473	C₂	1	1	yes	maximal, characteristic
2	#22560198	C₂	1	1	yes	maximal, characteristic
3	#22560198	C₂	1	1	yes	maximal, characteristic
4	#22558967	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558964	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761611	C₄	1	2	yes	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	#16863473	—	2	1	no	—
11	#16863473	—	2	1	no	—
12	#16863473	—	2	1	no	—
13	#16863473	—	2	1	no	—
14	#16863420	C₂²	1	1	yes	characteristic
15	#16863420	C₂²	1	1	yes	characteristic
16	C₂×#14523959	—	2	1	no	—
17	C₂×#14523959	—	2	1	no	—
18	C₂×(C₂×D₆₇)⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	#16863164	—	2	1	no	—
21	#16863164	—	2	1	no	—
22	#16863148	—	2	1	no	—
23	#16863148	—	2	1	no	—
24	#16863146	—	2	1	no	—
25	#16863146	—	2	1	no	—
26	#423×C₆₇	C₂²	1	1	yes	characteristic
27	#423×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₆₇	C₂×C₄	1	2	yes	—
31	C₂×#11761612	—	2	1	no	—
32	C₂×#11761612	—	2	1	no	—
33	C₂×#11761611	—	2	2	no	—
34	C₂×#11761611	—	2	2	no	—
35	C₂×#11761611	—	4	4	no	—
36	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×D₂₆₈	—	2	1	no	—
39	C₄×D₂₆₈	—	2	1	no	—
40	C₄×D₂₆₈	—	2	1	no	—
41	C₄×D₂₆₈	—	2	1	no	—
42	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
43	C₂×(C₂×D₆₇)⋊C₄	Q₈	1	2	yes	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
48	#14523959	—	4	2	no	—
49	#14523959	—	4	2	no	—
50	(C₂×D₆₇)⋊C₈	—	2	2	no	—
51	(C₂×D₆₇)⋊C₈	—	2	2	no	—
52	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
53	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
54	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
55	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₆₇	—	2	1	no	—
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	—	2	1	no	—
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₂³×D₆₇	C₄⋊C₄	1	2	yes	—
67	C₂×D₂₆₈	—	2	1	no	—
68	C₂×D₂₆₈	—	2	1	no	—
69	C₂×D₂₆₈	—	2	1	no	—
70	C₂×D₂₆₈	—	2	1	no	—
71	C₂×D₂₆₈	—	2	1	no	—
72	C₂×D₂₆₈	—	2	1	no	—
73	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×D₆₇	—	2	2	no	—
75	C₂×C₄×D₆₇	—	2	2	no	—
76	C₂×C₄×D₆₇	—	4	2	no	—
77	C₂×C₄×D₆₇	—	4	2	no	—
78	#11761612	—	4	1	no	—
79	#11761612	—	4	1	no	—
80	#11761612	—	4	1	no	—
81	#11761612	—	4	1	no	—
82	#11761611	—	4	4	no	—
83	#11761611	—	4	4	no	—
84	C₂×C₆₇⋊C₈	—	2	2	no	—
85	C₂×C₆₇⋊C₈	—	4	1	no	—
86	C₂×C₆₇⋊C₈	—	4	1	no	—
87	C₂×C₆₇⋊C₈	—	4	1	no	—
88	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄⋊Dic₆₇	—	2	1	no	—
93	C₄⋊Dic₆₇	—	2	1	no	—
94	(C₂×D₆₇)⋊C₄	—	2	2	no	—
95	(C₂×D₆₇)⋊C₄	—	2	2	no	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
99	M₁₆×C₆₇	—	4	1	no	—
100	M₁₆×C₆₇	—	4	1	no	—
101	M₁₆×C₆₇	—	4	1	no	—
102	M₁₆×C₆₇	—	4	1	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	4	2	no	—
105	C₂×C₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	#41	1	1	yes	characteristic
110	C₄²×C₆₇	#41	1	1	yes	characteristic
111	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×D₆₇	—	2	2	no	—
113	C₂²×D₆₇	—	2	2	no	—
114	C₂²×D₆₇	—	2	2	no	—
115	C₂²×D₆₇	—	4	2	no	—
116	C₂²×D₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
130	C₂×Dic₆₇	—	2	2	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	D₂₆₈	—	4	1	no	—
133	D₂₆₈	—	4	1	no	—
134	D₂₆₈	—	4	1	no	—
135	D₂₆₈	—	4	1	no	—
136	C₄×D₆₇	—	4	4	no	—
137	C₄×D₆₇	—	4	2	no	—
138	C₄×D₆₇	—	4	2	no	—
139	C₈×C₆₇	—	4	2	no	—
140	C₈×C₆₇	—	8	1	no	—
141	C₈×C₆₇	—	8	1	no	—
142	C₆₇⋊C₈	—	2	1	no	—
143	C₆₇⋊C₈	—	2	1	no	—
144	C₆₇⋊C₈	—	2	1	no	—
145	C₆₇⋊C₈	—	8	1	no	—
146	C₆₇⋊C₈	—	8	1	no	—
147	C₆₇⋊C₈	#100	1	2	yes	complemented
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	#380	1	1	yes	characteristic
151	C₂²×C₆₇	#428	1	1	yes	characteristic
152	C₂²×C₆₇	#428	1	1	yes	characteristic
153	C₂×D₆₇	—	4	2	no	—
154	C₂×D₆₇	—	4	2	no	—
155	C₂×D₆₇	—	4	2	no	—
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	8	2	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
162	C₄×C₆₇	#413	1	1	yes	characteristic
163	C₄×C₆₇	#468	1	1	yes	characteristic
164	C₄×C₆₇	#468	1	1	yes	characteristic
165	C₄×C₆₇	#469	1	1	yes	characteristic
166	C₄×C₆₇	#470	1	1	yes	characteristic
167	Dic₆₇	—	4	2	no	—
168	#8618	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	—	2	1	no	—
170	C₂×C₆₇	—	2	1	no	—
171	C₂×C₆₇	#1800	1	1	yes	characteristic
172	C₂×C₆₇	#1809	1	1	yes	characteristic
173	C₂×C₆₇	#1813	1	1	yes	characteristic
174	D₆₇	—	8	2	no	—
175	C₂×#436	—	67	1	no	—
176	#1800	—	67	1	no	—
177	#1800	—	67	1	no	—
178	#1567	—	67	1	no	—
179	#1566	—	67	1	no	—
180	#1292	—	67	1	no	—
181	#1292	D₆₇	1	1	yes	characteristic, complemented
182	C₆₇	#8618	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	67	2	no	—
184	C₂×C₄×D₄	—	67	1	no	—
185	#436	—	134	1	no	—
186	#436	—	134	1	no	—
187	#436	—	134	1	no	—
188	#436	—	134	1	no	—
189	#423	—	67	1	no	—
190	#423	—	67	1	no	—
191	#423	C₂×D₆₇	1	1	yes	characteristic
192	#423	C₂×D₆₇	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₂×#100	—	134	1	no	—
195	C₂×#100	—	134	1	no	—
196	C₂×C₂²⋊C₈	—	67	2	no	—
197	C₂×C₄×C₈	—	67	1	no	—
198	#343	—	134	1	no	—
199	#343	—	134	1	no	—
200	#335	—	134	1	no	—
201	#335	—	134	1	no	—
202	#334	—	134	1	no	—
203	#334	—	134	1	no	—
204	C₂²×D₄	—	67	1	no	—
205	C₂³×C₄	—	67	2	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	268	4	no	—
209	C₂×M₁₆	—	134	2	no	—
210	C₂×M₁₆	—	134	2	no	—
211	C₂×M₁₆	—	134	1	no	—
212	C₂×M₁₆	—	134	1	no	—
213	C₂²×C₈	—	67	1	no	—
214	C₂²×C₈	—	67	1	no	—
215	C₂²×C₈	C₄×D₆₇	1	2	yes	—
216	C₄×D₄	—	134	1	no	—
217	C₄×D₄	—	134	1	no	—
218	C₄×D₄	—	134	1	no	—
219	C₄×D₄	—	134	1	no	—
220	C₂×C₄⋊C₄	—	67	1	no	—
221	C₂×C₂²⋊C₄	—	67	2	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	134	1	no	—
230	C₄⋊C₈	—	134	1	no	—
231	C₄⋊C₈	—	134	1	no	—
232	C₄⋊C₈	—	134	1	no	—
233	#100	—	268	2	no	—
234	#100	—	268	2	no	—
235	C₂²⋊C₈	—	134	2	no	—
236	C₂²⋊C₈	—	134	2	no	—
237	C₄×C₈	—	67	1	no	—
238	C₄×C₈	—	67	1	no	—
239	C₄×C₈	—	67	1	no	—
240	C₄×C₈	—	67	1	no	—
241	C₂⁴	—	67	2	no	—
242	C₂×D₄	—	134	1	no	—
243	C₂×D₄	—	134	1	no	—
244	C₂×D₄	—	134	1	no	—
245	C₂×D₄	—	134	1	no	—
246	C₂×D₄	—	134	1	no	—
247	C₂×D₄	—	134	1	no	—
248	C₂²×C₄	—	268	2	no	—
249	C₂²×C₄	—	268	2	no	—
250	C₂²×C₄	—	67	2	no	—
251	C₂²×C₄	—	134	2	no	—
252	C₂²×C₄	—	134	2	no	—
253	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	268	4	no	—
261	M₁₆	—	268	4	no	—
262	M₁₆	—	268	1	no	—
263	M₁₆	—	268	1	no	—
264	M₁₆	—	268	1	no	—
265	M₁₆	—	268	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	268	1	no	—
271	C₂×C₈	—	268	1	no	—
272	C₂×C₈	—	268	1	no	—
273	C₂×C₈	—	67	1	no	—
274	C₂×C₈	—	67	1	no	—
275	C₂×C₈	—	67	1	no	—
276	C₂×C₈	—	67	1	no	—
277	C₂×C₈	—	134	2	no	—
278	C₄⋊C₄	—	134	1	no	—
279	C₄⋊C₄	—	134	1	no	—
280	C₂²⋊C₄	—	134	2	no	—
281	C₂²⋊C₄	—	134	2	no	—
282	C₄²	D₄×D₆₇	1	1	yes	characteristic
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₄²	#11761628	1	1	yes	characteristic
285	C₄²	#11761628	1	1	yes	characteristic
286	C₂³	—	268	2	no	—
287	C₂³	—	268	2	no	—
288	C₂³	—	134	2	no	—
289	C₂³	—	134	2	no	—
290	C₂³	—	134	2	no	—
291	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
292	D₄	—	268	1	no	—
293	D₄	—	268	1	no	—
294	D₄	—	268	1	no	—
295	D₄	—	268	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	268	4	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	134	2	no	—
304	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
305	C₂×C₄	#14524000	1	1	yes	characteristic
306	C₂×C₄	#14524016	1	1	yes	characteristic
307	C₂×C₄	#14524016	1	1	yes	characteristic
308	C₂×C₄	#14524021	1	1	yes	characteristic
309	C₂×C₄	#14524021	1	1	yes	characteristic
310	C₂×C₄	#14524067	1	1	yes	characteristic
311	C₂×C₄	#14524070	1	1	yes	characteristic
312	C₂×C₄	#14524072	1	1	yes	characteristic
313	C₂×C₄	#14524075	1	1	yes	characteristic
314	C₈	—	4	2	no	—
315	C₈	—	8	1	no	—
316	C₈	—	8	1	no	—
317	C₈	—	536	1	no	—
318	C₈	—	536	1	no	—
319	C₈	—	67	2	no	—
320	C₈	—	134	1	no	—
321	C₈	—	134	1	no	—
322	C₈	—	134	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	268	2	no	—
326	C₂²	—	268	2	no	—
327	C₂²	—	268	2	no	—
328	C₂²	—	536	2	no	—
329	C₂²	—	536	2	no	—
330	C₂²	#16863255	1	1	yes	characteristic
331	C₂²	#16863435	1	1	yes	characteristic
332	C₂²	#16863437	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	268	2	no	—
337	C₄	#100×D₆₇	1	1	yes	characteristic
338	C₄	#16863379	1	1	yes	characteristic
339	C₄	#16863641	1	1	yes	characteristic
340	C₄	#16863645	1	1	yes	characteristic
341	C₄	#16863648	1	1	yes	characteristic
342	C₄	#16863653	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	536	2	no	—
346	C₂	#22560200	1	1	yes	characteristic
347	C₂	#22560241	1	1	yes	characteristic
348	C₂	#22560258	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863255
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1292	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×D₂₆₈	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1292	D₆₇
hypocenter	C₆₇	#8618
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
4. #22558967	C₂	semidirect
5. #22558964	C₂	semidirect
6. #22558084	C₂	semidirect
7. C₆₇×#1292	C₂	semidirect
147. C₆₇⋊C₈	#100	semidirect
181. #1292	D₆₇	semidirect
182. C₆₇	#8618	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863473	2
#22560198	2
#22558967	2
#22558964	2
#22558084	2
C₆₇×#1292	2
#8618	67