Subgroups — #30826383

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570768	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic
4	#22559113	C₂	1	4	yes	maximal, complemented
5	#22559023	C₂	1	4	yes	maximal, complemented
6	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558665	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558661	C₂	1	1	yes	maximal, characteristic
9	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524049	—	2	1	no	—
11	#16863953	C₂²	1	1	yes	characteristic
12	#16863946	C₂²	1	1	yes	characteristic
13	#16863794	C₄	1	1	yes	characteristic
14	#16863769	C₄	1	1	yes	characteristic
15	#16863747	C₄	1	1	yes	characteristic
16	#16863743	C₄	1	1	yes	characteristic
17	#16863625	C₄	1	4	yes	—
18	#16863526	—	2	1	no	—
19	#16863526	C₂²	1	2	yes	—
20	C₈×Dic₁₃₄	—	2	1	no	—
21	C₈×Dic₁₃₄	C₂²	1	2	yes	—
22	C₄×#11761617	C₂²	1	4	yes	—
23	#16863475	—	2	1	no	—
24	#16863475	C₂²	1	2	yes	—
25	C₈×D₂₆₈	—	2	1	no	—
26	C₈×D₂₆₈	C₂²	1	2	yes	—
27	#16863422	—	2	1	no	—
28	#16863420	C₂²	1	1	yes	characteristic
29	#16863420	C₂²	1	1	yes	characteristic
30	#16863362	—	2	1	no	—
31	#16863351	—	2	1	no	—
32	#16863149	C₂²	1	4	yes	—
33	#16863128	C₂²	1	4	yes	—
34	#16863125	C₂²	1	4	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	#322×C₆₇	C₂²	1	2	yes	—
39	#322×C₆₇	C₂²	1	2	yes	—
40	C₂×#11761628	—	2	1	no	—
41	C₂×#11761617	C₂×C₄	1	4	yes	—
42	#14524049	—	4	4	no	—
43	C₂×#11761612	—	2	1	no	—
44	C₂×#11761612	—	2	1	no	—
45	C₂×#11761611	—	2	2	no	—
46	C₂×C₈×D₆₇	—	2	2	no	—
47	#14524043	C₂×C₄	1	1	yes	characteristic
48	#14524043	C₂×C₄	1	1	yes	characteristic
49	#14524043	C₂×C₄	1	1	yes	characteristic
50	#14524040	C₂×C₄	1	1	yes	characteristic
51	#14524035	—	2	1	no	—
52	#14524031	—	2	1	no	—
53	#14524022	—	2	1	no	—
54	#14524022	C₂×C₄	1	2	yes	—
55	#14524017	—	2	1	no	—
56	#14524017	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₃₄	—	2	1	no	—
58	C₄×Dic₁₃₄	C₂³	1	2	yes	—
59	C₄×D₂₆₈	—	2	1	no	—
60	C₄×D₂₆₈	C₂³	1	2	yes	—
61	#14524002	—	2	1	no	—
62	#14523975	C₂×C₄	1	4	yes	—
63	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	#14523968	C₂×C₄	1	4	yes	—
70	#14523965	C₂×C₄	1	4	yes	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	#14523953	—	2	2	no	—
74	C₈×Dic₆₇	—	2	2	no	—
75	C₂²×C₈×C₆₇	—	2	1	no	—
76	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	C₄×C₈×C₆₇	C₂³	1	2	yes	—
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	#11761628	—	4	2	no	—
90	#11761628	—	4	2	no	—
91	C₂×Dic₁₃₄	—	2	1	no	—
92	C₂×Dic₁₃₄	—	2	1	no	—
93	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
94	C₂×D₂₆₈	—	2	1	no	—
95	C₂×D₂₆₈	—	2	1	no	—
96	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
97	C₂×C₂²⋊D₆₇	—	2	2	no	—
98	C₂×C₄×D₆₇	—	2	2	no	—
99	#11761617	—	2	8	no	—
100	#11761612	—	4	2	no	—
101	#11761612	—	4	2	no	—
102	#11761611	—	4	4	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₈×D₆₇	—	4	4	no	—
106	C₄⋊Dic₆₇	—	2	1	no	—
107	C₄⋊Dic₆₇	—	2	1	no	—
108	C₄⋊Dic₆₇	—	2	1	no	—
109	C₄⋊Dic₆₇	—	2	1	no	—
110	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	Dic₆₇⋊C₄	—	2	2	no	—
114	C₂²⋊Dic₆₇	—	2	2	no	—
115	(C₂×D₆₇)⋊C₄	—	2	2	no	—
116	(C₂×D₆₇)⋊C₄	—	2	2	no	—
117	C₄×Dic₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	4	2	no	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	#41	1	1	yes	characteristic
135	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₆₇	#137	1	2	yes	—
145	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₆₇	—	4	4	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	—	2	2	no	—
152	D₂₆₈	—	2	4	no	—
153	D₂₆₈	—	4	2	no	—
154	D₂₆₈	—	4	2	no	—
155	C₄×D₆₇	—	4	2	no	—
156	C₄×D₆₇	—	4	2	no	—
157	Dic₁₃₄	—	2	4	no	—
158	Dic₁₃₄	—	4	2	no	—
159	Dic₁₃₄	—	4	2	no	—
160	C₈×C₆₇	—	2	2	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₈×C₆₇	#131	1	4	yes	complemented
167	C₆₇⋊C₈	—	4	2	no	—
168	C₆₇⋊C₈	—	4	2	no	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	#440	1	2	yes	—
172	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₆₇	—	4	2	no	—
174	C₂×D₆₇	—	4	2	no	—
175	C₂×D₆₇	—	4	2	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	#440	1	2	yes	—
179	C₄×C₆₇	#444	1	2	yes	—
180	C₄×C₆₇	C₂×#131	1	2	yes	—
181	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
183	Dic₆₇	—	4	2	no	—
184	Dic₆₇	—	4	2	no	—
185	Dic₆₇	—	4	2	no	—
186	#17960	—	67	1	no	maximal, Sylow
187	C₂×C₆₇	—	2	2	no	—
188	C₂×C₆₇	#2931	1	1	yes	characteristic
189	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
190	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
191	D₆₇	—	8	2	no	—
192	#3167	—	67	1	no	—
193	#2960	—	67	1	no	—
194	#2931	—	67	1	no	—
195	#1592	—	67	4	no	—
196	SD₁₆⋊C₈	—	67	4	no	—
197	#1567	—	67	1	no	—
198	#1493	—	67	1	no	—
199	#1492	—	67	1	no	—
200	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
201	C₆₇	#17960	1	1	yes	characteristic, complemented, Sylow
202	C₂×#131	—	134	1	no	—
203	#533	—	67	1	no	—
204	#532	—	67	1	no	—
205	#498	—	67	1	no	—
206	#492	—	67	1	no	—
207	#487	—	67	1	no	—
208	#486	—	67	1	no	—
209	#465	—	67	4	no	—
210	#446	—	67	2	no	—
211	#446	—	134	1	no	—
212	C₈×Q₈	—	67	2	no	—
213	C₈×Q₈	—	134	1	no	—
214	C₄×SD₁₆	—	67	4	no	—
215	#436	—	67	2	no	—
216	#436	—	134	1	no	—
217	C₈×D₄	—	67	2	no	—
218	C₈×D₄	—	134	1	no	—
219	#424	—	134	1	no	—
220	#423	—	67	1	no	—
221	#423	—	67	1	no	—
222	#408	—	134	1	no	—
223	#405	—	134	1	no	—
224	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
226	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
227	#335	—	67	4	no	—
228	Q₈⋊C₈	—	67	4	no	—
229	D₄⋊C₈	—	67	4	no	—
230	#322	C₂×D₆₇	1	2	yes	—
231	#322	C₂×D₆₇	1	2	yes	—
232	C₂×#41	—	134	1	no	—
233	C₂×SD₁₆	—	67	4	no	—
234	#131	—	268	4	no	—
235	C₂×M₁₆	—	134	1	no	—
236	C₂×M₁₆	—	134	1	no	—
237	C₂×M₁₆	—	134	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	134	2	no	—
240	C₂²×C₈	C₄×D₆₇	1	2	yes	—
241	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
242	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
243	C₄⋊Q₈	—	67	1	no	—
244	C₄⋊Q₈	—	67	1	no	—
245	C₄⋊Q₈	—	67	1	no	—
246	#127	—	67	1	no	—
247	#125	—	134	1	no	—
248	#124	—	134	1	no	—
249	C₂²⋊Q₈	—	67	2	no	—
250	C₂²⋊Q₈	—	134	1	no	—
251	#121	—	67	2	no	—
252	#121	—	134	1	no	—
253	C₄×Q₈	—	67	2	no	—
254	C₄×Q₈	—	134	1	no	—
255	C₄×D₄	—	67	2	no	—
256	C₄×D₄	—	134	1	no	—
257	#117	—	134	1	no	—
258	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
259	#106	—	67	4	no	—
260	C₄⋊C₈	—	67	2	no	—
261	C₄⋊C₈	—	67	2	no	—
262	C₄⋊C₈	—	134	2	no	—
263	C₄⋊C₈	—	134	2	no	—
264	C₄⋊C₈	—	134	1	no	—
265	C₄⋊C₈	—	134	1	no	—
266	#103	—	67	4	no	—
267	#102	—	67	4	no	—
268	C₂²⋊C₈	—	134	2	no	—
269	C₂²⋊C₈	—	134	2	no	—
270	#97	—	134	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	134	2	no	—
273	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
274	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
275	C₄×C₈	D₂₆₈	1	1	yes	characteristic
276	C₄×C₈	D₂₆₈	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₆₇	1	2	yes	—
278	C₄×C₈	C₂²×D₆₇	1	2	yes	—
279	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
281	#41	—	268	2	no	—
282	#41	—	268	2	no	—
283	C₂×Q₈	—	67	2	no	—
284	C₂×Q₈	—	134	1	no	—
285	C₂×Q₈	—	134	1	no	—
286	C₂×D₄	—	67	2	no	—
287	C₂×D₄	—	134	2	no	—
288	C₂×D₄	—	134	1	no	—
289	C₂×D₄	—	134	1	no	—
290	C₂²×C₄	—	134	2	no	—
291	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
293	C₂²×C₄	#11761628	1	1	yes	characteristic
294	SD₁₆	—	134	8	no	—
295	M₁₆	—	268	4	no	—
296	M₁₆	—	268	2	no	—
297	M₁₆	—	268	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	268	4	no	—
304	C₂×C₈	—	134	2	no	—
305	C₂×C₈	—	134	2	no	—
306	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
307	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
308	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
309	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
310	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
311	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
312	C₄⋊C₄	—	67	2	no	—
313	C₄⋊C₄	—	67	2	no	—
314	C₄⋊C₄	—	134	1	no	—
315	C₄⋊C₄	—	134	1	no	—
316	C₄⋊C₄	—	134	1	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	—	134	1	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₂²⋊C₄	—	134	2	no	—
321	C₂²⋊C₄	—	134	2	no	—
322	C₂²⋊C₄	—	134	2	no	—
323	C₄²	—	134	2	no	—
324	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
325	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
326	C₄²	#11761628	1	1	yes	characteristic
327	C₄²	C₂³×D₆₇	1	1	yes	characteristic
328	C₂³	—	134	2	no	—
329	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
330	Q₈	—	268	2	no	—
331	Q₈	—	268	2	no	—
332	Q₈	—	134	4	no	—
333	D₄	—	268	4	no	—
334	D₄	—	268	2	no	—
335	D₄	—	268	2	no	—
336	D₄	—	134	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	268	2	no	—
340	C₂×C₄	—	268	2	no	—
341	C₂×C₄	—	134	2	no	—
342	C₂×C₄	—	134	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
345	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
346	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
347	C₂×C₄	#14524071	1	2	yes	—
348	C₂×C₄	#14524076	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
351	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	268	2	no	—
359	C₈	—	268	2	no	—
360	C₈	#14524049	1	4	yes	complemented
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	268	2	no	—
364	C₂²	—	268	2	no	—
365	C₂²	—	268	2	no	—
366	C₂²	#16863498	1	2	yes	—
367	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	268	2	no	—
371	C₄	—	268	2	no	—
372	C₄	—	268	2	no	—
373	C₄	#16863498	1	2	yes	—
374	C₄	#16863519	1	2	yes	—
375	C₄	C₂×#14524049	1	2	yes	—
376	C₄	C₂×#14524071	1	1	yes	characteristic
377	C₄	C₂×#14524076	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	536	2	no	—
380	C₂	#22568922	1	1	yes	characteristic
381	C₂	C₂×#16863498	1	1	yes	characteristic
382	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#17960
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #22569143	C₂	semidirect
4. #22559113	C₂	semidirect
5. #22559023	C₂	semidirect
6. #22558968	C₂	semidirect
7. #22558665	C₂	semidirect
9. C₂×#322×C₆₇	C₂	semidirect
166. C₈×C₆₇	#131	semidirect
200. C₂×#322	D₆₇	semidirect
201. C₆₇	#17960	semidirect
360. C₈	#14524049	semidirect

Maximal subgroups

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

subgroup	index
#22570768	2
#22569143	2
#22568922	2
#22559113	2
#22559023	2
#22558968	2
#22558665	2
#22558661	2
C₂×#322×C₆₇	2
#17960	67