Subgroups — #30839874

Order 17152.

706 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic, complemented
2	#22570300	C₂	1	2	yes	maximal, complemented
3	#22559742	C₂	1	4	yes	maximal, complemented
4	#22559732	C₂	1	2	yes	maximal, complemented
5	#22559688	C₂	1	2	yes	maximal, complemented
6	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558902	C₂	1	2	yes	maximal, complemented
8	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863946	C₂²	1	2	yes	—
10	#16863801	C₂²	1	1	yes	characteristic, complemented
11	#16863773	C₂²	1	1	yes	characteristic, complemented
12	#16863743	C₂²	1	2	yes	complemented
13	#16863714	—	2	2	no	—
14	#16863686	C₂²	1	4	yes	—
15	#16863632	—	2	1	no	—
16	#16863632	—	2	1	no	—
17	#16863591	C₂²	1	4	yes	—
18	#16863530	—	2	2	no	—
19	#16863506	—	2	4	no	—
20	C₄×D₅₃₆	—	2	1	no	—
21	C₄×D₅₃₆	—	2	1	no	—
22	#16863453	C₂²	1	2	yes	—
23	#16863149	C₂²	1	2	yes	complemented
24	#16863147	C₂²	1	2	yes	complemented
25	#16863144	C₂²	1	4	yes	complemented
26	#16863139	C₂²	1	4	yes	complemented
27	C₂²×M₁₆×C₆₇	—	2	1	no	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	#436×C₆₇	—	2	1	no	—
30	#436×C₆₇	—	2	1	no	—
31	#436×C₆₇	C₂²	1	2	yes	—
32	#436×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#11761628	—	2	2	no	—
37	C₂×D₅₃₆	—	2	1	no	—
38	C₂×D₅₃₆	—	2	1	no	—
39	C₂×#11761613	—	2	4	no	—
40	#14524043	C₂³	1	2	yes	—
41	#14524040	C₂³	1	2	yes	—
42	#14524031	—	2	2	no	—
43	#14524017	—	2	2	no	—
44	#14524017	—	2	2	no	—
45	C₄×D₂₆₈	—	2	2	no	—
46	C₄×D₂₆₈	—	2	2	no	—
47	#14523977	—	2	1	no	—
48	#14523977	—	2	1	no	—
49	#14523974	—	2	4	no	—
50	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
51	#14523965	—	2	2	no	—
52	#14523965	—	2	2	no	—
53	#14523964	—	2	4	no	—
54	#14523963	—	2	4	no	—
55	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
56	#14523954	—	2	2	no	—
57	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	2	2	no	—
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	4	4	no	—
62	C₂²×C₈×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄×D₄×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	D₄	1	2	yes	—
68	C₄×D₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	—	2	1	no	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	C₂²⋊C₈×C₆₇	—	2	2	no	—
75	C₂²⋊C₈×C₆₇	—	2	2	no	—
76	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	#11761628	—	8	4	no	—
81	C₂×Dic₁₃₄	—	2	2	no	—
82	C₂×D₂₆₈	—	2	2	no	—
83	C₂×D₂₆₈	—	2	2	no	—
84	C₂×D₂₆₈	—	2	2	no	—
85	C₂×C₂²⋊D₆₇	—	4	2	no	—
86	C₂×C₄×D₆₇	—	4	2	no	—
87	D₅₃₆	—	4	2	no	—
88	D₅₃₆	—	4	2	no	—
89	#11761613	—	4	8	no	—
90	C₂×C₆₇⋊C₈	—	2	4	no	—
91	C₄⋊Dic₆₇	—	2	2	no	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	(C₂×D₆₇)⋊C₄	—	4	2	no	—
94	(C₂×D₆₇)⋊C₄	—	4	2	no	—
95	C₂⁴×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
107	M₁₆×C₆₇	—	4	4	no	—
108	M₁₆×C₆₇	—	4	4	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	—	4	2	no	—
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₆₇	#41	1	1	yes	characteristic
124	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	—	4	2	no	—
129	C₂³×C₆₇	#121	1	1	yes	characteristic
130	C₂²×D₆₇	—	4	2	no	—
131	D₄×C₆₇	—	2	4	no	—
132	D₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×D₄	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	—	8	4	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	D₂₆₈	—	4	2	no	—
151	D₂₆₈	—	4	4	no	—
152	D₂₆₈	—	4	2	no	—
153	D₂₆₈	—	8	2	no	—
154	C₄×D₆₇	—	8	4	no	—
155	Dic₁₃₄	—	8	2	no	—
156	C₈×C₆₇	—	2	1	no	—
157	C₈×C₆₇	—	2	1	no	—
158	C₈×C₆₇	—	2	1	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
161	C₆₇⋊C₈	—	4	4	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	8	4	no	—
168	C₂²×C₆₇	#472	1	2	yes	—
169	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
170	C₂×D₆₇	—	8	2	no	—
171	C₂×D₆₇	—	8	2	no	—
172	C₂×D₆₇	—	8	2	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	4	2	no	—
178	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₄×C₆₇	#575	1	1	yes	characteristic
180	C₄×C₆₇	#575	1	2	yes	—
181	Dic₆₇	—	8	2	no	—
182	#19283	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	—	2	2	no	—
184	C₂×C₆₇	—	8	2	no	—
185	C₂×C₆₇	#3036	1	1	yes	characteristic
186	C₂×C₆₇	#3054	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
188	D₆₇	—	16	2	no	—
189	#3169	—	67	1	no	—
190	#3109	—	67	2	no	—
191	C₂×#436	D₆₇	1	1	yes	characteristic, complemented
192	#1697	—	67	4	no	—
193	#1695	—	67	2	no	—
194	#1687	—	67	2	no	—
195	#1568	—	67	1	no	—
196	#1551	—	67	2	no	—
197	C₆₇	#19283	1	1	yes	characteristic, complemented, Sylow
198	C₂²×M₁₆	—	2	1	no	—
199	#532	—	67	2	no	—
200	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
201	#500	—	67	1	no	—
202	#493	—	67	1	no	—
203	#486	—	67	2	no	—
204	#481	—	134	2	no	—
205	#476	—	67	4	no	—
206	#466	—	134	1	no	—
207	#466	—	134	1	no	—
208	#459	—	67	4	no	—
209	#447	—	134	2	no	—
210	#442	—	134	4	no	—
211	C₄×D₈	—	134	1	no	—
212	C₄×D₈	—	134	1	no	—
213	#436	—	2	1	no	—
214	#436	—	2	1	no	—
215	#436	C₂×D₆₇	1	2	yes	—
216	#436	C₂×D₆₇	1	2	yes	—
217	#432	—	67	2	no	—
218	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
219	C₂×C₂²⋊C₈	—	2	1	no	—
220	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
221	#335	—	67	2	no	—
222	#334	—	67	2	no	—
223	#333	—	67	4	no	—
224	#331	—	67	4	no	—
225	C₂×#41	—	134	2	no	—
226	C₂²×D₄	D₂₆₈	1	1	yes	characteristic
227	C₂³×C₄	—	2	1	no	—
228	C₂×D₈	—	134	4	no	—
229	C₂×D₈	—	134	1	no	—
230	C₂×D₈	—	134	1	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	4	4	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
237	C₄⋊Q₈	—	67	2	no	—
238	#127	—	67	2	no	—
239	#124	—	134	2	no	—
240	#121	—	134	2	no	—
241	#121	—	134	2	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	—	134	2	no	—
245	C₄×D₄	—	134	2	no	—
246	C₄×D₄	D₂₆₈	1	2	yes	—
247	C₄×D₄	C₂²×D₆₇	1	2	yes	—
248	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	#107	—	134	1	no	—
252	#107	—	134	1	no	—
253	#106	—	134	4	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	67	4	no	—
256	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
257	#102	—	134	4	no	—
258	#102	—	134	2	no	—
259	#102	—	134	2	no	—
260	#102	—	134	4	no	—
261	C₂²⋊C₈	—	2	2	no	—
262	C₂²⋊C₈	—	2	2	no	—
263	C₂²⋊C₈	—	134	2	no	—
264	#97	—	134	2	no	—
265	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
266	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
269	C₂⁴	—	2	1	no	—
270	#41	—	536	4	no	—
271	C₂×Q₈	—	134	2	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	268	2	no	—
275	C₂×D₄	—	134	2	no	—
276	C₂×D₄	—	134	2	no	—
277	C₂×D₄	—	134	2	no	—
278	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	268	2	no	—
285	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
287	C₂²×C₄	#11761628	1	1	yes	characteristic
288	D₈	—	268	2	no	—
289	D₈	—	268	8	no	—
290	D₈	—	268	2	no	—
291	M₁₆	—	4	4	no	—
292	M₁₆	—	4	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	134	4	no	—
298	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
301	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	134	2	no	—
304	C₄⋊C₄	—	134	2	no	—
305	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	268	2	no	—
309	C₂²⋊C₄	—	268	2	no	—
310	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
311	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
312	C₄²	#11761628	1	1	yes	characteristic
313	C₄²	C₂³×D₆₇	1	1	yes	characteristic
314	C₂³	—	2	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	268	2	no	—
319	C₂³	#14524017	1	1	yes	characteristic
320	Q₈	—	536	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	268	2	no	—
324	D₄	—	268	2	no	—
325	D₄	—	268	4	no	—
326	D₄	—	536	2	no	—
327	D₄	—	536	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	268	2	no	—
338	C₂×C₄	—	536	4	no	—
339	C₂×C₄	#14524017	1	1	yes	characteristic
340	C₂×C₄	#14524017	1	1	yes	characteristic
341	C₂×C₄	#14524017	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	268	4	no	—
350	C₈	#14524069	1	2	yes	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	8	4	no	—
357	C₂²	—	536	2	no	—
358	C₂²	—	536	2	no	—
359	C₂²	—	536	2	no	—
360	C₂²	#16863666	1	2	yes	—
361	C₂²	C₂×#14524017	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	536	2	no	—
368	C₄	C₂×#14524069	1	1	yes	characteristic
369	C₄	#16864194	1	1	yes	characteristic
370	C₄	#16864196	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	8	2	no	—
373	C₂	—	1072	2	no	—
374	C₂	#22569673	1	1	yes	characteristic
375	C₂	#22569818	1	1	yes	characteristic
376	C₂	C₂×#16863666	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524017
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#436×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
hypercenter	C₂×#436	D₆₇
hypocenter	C₆₇	#19283
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22570788	C₂	semidirect
2. #22570300	C₂	semidirect
3. #22559742	C₂	semidirect
4. #22559732	C₂	semidirect
5. #22559688	C₂	semidirect
6. #22558972	C₂	semidirect
7. #22558902	C₂	semidirect
8. C₂×#436×C₆₇	C₂	semidirect
10. #16863801	C₂²	semidirect
11. #16863773	C₂²	semidirect
12. #16863743	C₂²	semidirect
23. #16863149	C₂²	semidirect
24. #16863147	C₂²	semidirect
25. #16863144	C₂²	semidirect
26. #16863139	C₂²	semidirect
191. C₂×#436	D₆₇	semidirect
197. C₆₇	#19283	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22570300	2
#22559742	2
#22559732	2
#22559688	2
#22558972	2
#22558902	2
C₂×#436×C₆₇	2
#19283	67