Subgroups — #31012636

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573359	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573347	C₂	1	1	yes	maximal, characteristic, complemented
3	#22571997	C₂	1	2	yes	maximal, complemented
4	#22562470	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562335	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562330	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562278	C₂	1	2	yes	maximal, complemented
8	#22562181	C₂	1	2	yes	maximal, complemented
9	#22562174	C₂	1	2	yes	maximal, complemented
10	#22562047	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864210	C₂²	1	2	yes	complemented
13	#16864208	—	2	2	no	—
14	#16864198	C₂²	1	2	yes	—
15	#16863836	C₂²	1	2	yes	complemented
16	#16863835	—	2	2	no	—
17	#16863833	—	2	2	no	—
18	#16863831	C₂²	1	2	yes	—
19	#16863829	—	2	1	no	—
20	#16863825	—	2	1	no	—
21	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic, complemented
22	Q₃₂×D₆₇	—	2	1	no	—
23	C₂×#14523988	C₂²	1	2	yes	complemented
24	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
25	D₁₆×D₆₇	—	2	1	no	—
26	#16863812	C₂²	1	1	yes	characteristic
27	#16863812	C₂²	1	1	yes	characteristic, complemented
28	#16863780	C₂²	1	2	yes	complemented
29	#16863672	C₂²	1	2	yes	—
30	#16863667	C₂²	1	2	yes	complemented
31	#16863659	—	2	2	no	—
32	#16863575	—	2	2	no	—
33	#16863570	—	2	2	no	—
34	#16863514	C₂²	1	1	yes	characteristic
35	#16863217	C₂²	1	1	yes	characteristic, complemented
36	#16863211	C₂²	1	2	yes	complemented
37	#16863208	C₂²	1	2	yes	complemented
38	#16863179	C₂²	1	1	yes	characteristic
39	#16863179	C₂²	1	1	yes	characteristic, complemented
40	#508×C₆₇	C₂²	1	1	yes	characteristic
41	#443×C₆₇	C₂²	1	2	yes	complemented
42	#368×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#359×C₆₇	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#14524090	—	2	2	no	—
46	#14524089	—	2	2	no	—
47	#14524076	—	2	2	no	—
48	#14524076	—	4	2	no	—
49	#14524073	—	2	2	no	—
50	#14524072	—	4	2	no	—
51	#14524071	—	2	2	no	—
52	#14524071	—	4	2	no	—
53	#14524070	—	4	2	no	—
54	#14524068	—	2	2	no	—
55	#14524066	—	2	2	no	—
56	#14524065	—	4	2	no	—
57	#14524065	D₄	1	1	yes	characteristic
58	#14524062	—	2	2	no	—
59	C₂×Dic₂₆₈	C₂³	1	2	yes	—
60	Q₁₆×D₆₇	—	2	2	no	—
61	C₂×#11761617	—	2	2	no	—
62	C₂×#11761617	—	4	2	no	—
63	C₂×D₅₃₆	C₂³	1	2	yes	—
64	D₈×D₆₇	—	2	2	no	—
65	#14524049	C₂³	1	1	yes	characteristic
66	#14524049	C₂³	1	1	yes	characteristic
67	#14524031	—	2	2	no	—
68	Dic₅₃₆	—	2	1	no	—
69	Dic₅₃₆	—	2	1	no	—
70	#14523989	—	2	2	no	—
71	#14523988	—	4	2	no	—
72	#14523987	—	2	2	no	—
73	#14523986	—	2	2	no	—
74	D₁₀₇₂	—	2	1	no	—
75	D₁₀₇₂	—	2	1	no	—
76	#14523984	—	2	2	no	—
77	#14523983	C₂³	1	1	yes	characteristic
78	#14523983	C₂³	1	1	yes	characteristic
79	#14523982	—	2	1	no	—
80	#14523982	—	2	1	no	—
81	C₁₆×D₆₇	—	2	1	no	—
82	C₁₆×D₆₇	—	2	1	no	—
83	#14523979	D₄	1	1	yes	characteristic
84	#14523971	D₄	1	1	yes	characteristic
85	#14523971	D₄	1	1	yes	characteristic
86	#14523961	—	2	2	no	—
87	#14523959	—	2	2	no	—
88	#135×C₆₇	C₂³	1	2	yes	—
89	#131×C₆₇	—	2	2	no	—
90	Q₃₂×C₆₇	D₄	1	1	yes	characteristic
91	SD₃₂×C₆₇	D₄	1	2	yes	—
92	D₁₆×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
95	#108×C₆₇	C₂³	1	2	yes	—
96	#104×C₆₇	—	2	2	no	—
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	#11761629	—	4	2	no	—
99	#11761628	—	4	2	no	—
100	#11761628	—	4	2	no	—
101	#11761628	C₂×D₄	1	1	yes	characteristic
102	#11761628	C₂×D₄	1	1	yes	characteristic
103	#11761627	—	4	2	no	—
104	#11761627	—	8	2	no	—
105	C₂×Dic₁₃₄	—	2	2	no	—
106	C₂×Dic₁₃₄	—	4	2	no	—
107	Q₈×D₆₇	—	4	2	no	—
108	C₂×D₂₆₈	—	2	2	no	—
109	C₂×D₂₆₈	—	4	2	no	—
110	D₄×D₆₇	—	4	2	no	—
111	D₄×D₆₇	—	8	2	no	—
112	Dic₂₆₈	—	2	2	no	—
113	Dic₂₆₈	—	2	2	no	—
114	Dic₂₆₈	—	4	2	no	—
115	Dic₂₆₈	D₈	1	1	yes	characteristic
116	#11761618	—	4	2	no	—
117	#11761617	—	4	2	no	—
118	#11761617	—	4	2	no	—
119	#11761617	—	4	2	no	—
120	#11761617	—	4	2	no	—
121	#11761617	D₈	1	1	yes	characteristic
122	#11761617	D₈	1	1	yes	characteristic
123	#11761616	—	4	2	no	—
124	#11761615	—	4	2	no	—
125	D₅₃₆	—	2	2	no	—
126	D₅₃₆	—	2	2	no	—
127	D₅₃₆	—	4	2	no	—
128	D₅₃₆	D₈	1	1	yes	characteristic
129	#11761613	—	4	2	no	—
130	#11761612	C₂×D₄	1	1	yes	characteristic
131	#11761612	C₂×D₄	1	1	yes	characteristic
132	#11761611	—	2	1	no	—
133	#11761611	—	2	1	no	—
134	C₈×D₆₇	—	2	1	no	—
135	C₈×D₆₇	—	2	1	no	—
136	(C₂×D₆₇)⋊C₄	—	8	2	no	—
137	C₆₇⋊C₁₆	—	2	1	no	—
138	C₆₇⋊C₁₆	—	2	1	no	—
139	#41×C₆₇	—	2	2	no	—
140	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
141	SD₁₆×C₆₇	—	2	2	no	—
142	D₈×C₆₇	C₂×D₄	1	2	yes	—
143	M₁₆×C₆₇	—	2	2	no	—
144	M₁₆×C₆₇	—	4	2	no	—
145	C₂×C₈×C₆₇	—	4	2	no	—
146	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₆₇	—	2	1	no	—
150	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×D₆₇	—	4	2	no	—
153	C₂²×D₆₇	—	8	2	no	—
154	Q₈×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₆₇	—	2	1	no	—
161	C₂²⋊D₆₇	—	2	1	no	—
162	C₂²⋊D₆₇	—	8	2	no	—
163	C₂²⋊D₆₇	—	8	2	no	—
164	C₂×Dic₆₇	—	4	2	no	—
165	C₂×Dic₆₇	—	8	2	no	—
166	D₂₆₈	—	4	2	no	—
167	D₂₆₈	—	4	2	no	—
168	D₂₆₈	—	8	2	no	—
169	D₂₆₈	C₂×D₈	1	1	yes	characteristic
170	D₂₆₈	C₂×D₈	1	1	yes	characteristic
171	C₄×D₆₇	—	2	1	no	—
172	C₄×D₆₇	—	2	1	no	—
173	C₄×D₆₇	—	8	2	no	—
174	C₄×D₆₇	—	8	2	no	—
175	Dic₁₃₄	—	4	2	no	—
176	Dic₁₃₄	—	4	2	no	—
177	Dic₁₃₄	—	8	2	no	—
178	Dic₁₃₄	C₂×D₈	1	1	yes	characteristic
179	Dic₁₃₄	C₂×D₈	1	1	yes	characteristic
180	C₈×C₆₇	—	4	2	no	—
181	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
182	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
183	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
185	C₆₇⋊C₈	—	2	1	no	—
186	C₆₇⋊C₈	—	2	1	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	#576	1	1	yes	characteristic
189	C₂×D₆₇	—	2	1	no	—
190	C₂×D₆₇	—	2	1	no	—
191	C₂×D₆₇	—	8	2	no	—
192	C₂×D₆₇	—	16	2	no	—
193	C₂×D₆₇	—	32	2	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	D₄²	1	1	yes	characteristic
197	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
198	Dic₆₇	—	2	1	no	—
199	Dic₆₇	—	2	1	no	—
200	Dic₆₇	—	8	2	no	—
201	#33859	—	67	1	no	maximal, Sylow
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	—	8	2	no	—
204	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
205	D₆₇	—	4	1	no	—
206	D₆₇	—	4	1	no	—
207	D₆₇	—	16	2	no	—
208	#3417	—	67	1	no	—
209	#3415	—	67	1	no	—
210	#3290	—	67	2	no	—
211	#2249	—	67	1	no	—
212	#2222	—	67	1	no	—
213	#2221	—	67	1	no	—
214	#2212	—	67	2	no	—
215	#2194	—	67	2	no	—
216	#2193	—	67	2	no	—
217	#2178	D₆₇	1	1	yes	characteristic, complemented
218	#2170	—	67	1	no	—
219	C₆₇	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	—	67	2	no	—
221	#577	—	134	2	no	—
222	#576	—	67	2	no	—
223	#510	—	67	2	no	—
224	#510	—	134	2	no	—
225	#509	—	67	2	no	—
226	#509	—	134	2	no	—
227	#508	—	134	1	no	—
228	#508	—	134	1	no	—
229	#508	C₂×D₆₇	1	1	yes	characteristic
230	C₂×Q₃₂	—	67	1	no	—
231	C₂×Q₃₂	—	134	1	no	—
232	C₂×SD₃₂	—	67	2	no	—
233	C₂×D₁₆	—	67	1	no	—
234	C₂×D₁₆	—	134	1	no	—
235	#504	—	67	1	no	—
236	#504	—	67	1	no	—
237	#495	—	67	2	no	—
238	#473	—	67	2	no	—
239	#472	—	67	2	no	—
240	#471	—	134	2	no	—
241	#455	—	134	2	no	—
242	#454	—	134	2	no	—
243	#443	—	67	1	no	—
244	#443	C₂×D₆₇	1	2	yes	complemented
245	#368	C₂×D₆₇	1	1	yes	characteristic, complemented
246	#364	—	67	1	no	—
247	#362	—	67	2	no	—
248	#361	—	67	2	no	—
249	#359	C₂×D₆₇	1	2	yes	complemented
250	#350	—	67	1	no	—
251	#350	—	67	1	no	—
252	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
253	2₋¹⁺⁴	—	134	2	no	—
254	H(5,𝔽₂)	—	134	2	no	—
255	#137	—	268	2	no	—
256	#137	—	268	2	no	—
257	#137	—	134	2	no	—
258	#137	—	134	2	no	—
259	AGL(1,ℤ/8)	—	268	2	no	—
260	AGL(1,ℤ/8)	—	268	2	no	—
261	AGL(1,ℤ/8)	—	134	2	no	—
262	AGL(1,ℤ/8)	—	134	2	no	—
263	#135	—	268	2	no	—
264	#135	—	67	1	no	—
265	#135	—	134	2	no	—
266	#135	—	134	2	no	—
267	#135	C₂²×D₆₇	1	2	yes	—
268	C₂×Q₁₆	—	67	2	no	—
269	C₂×Q₁₆	—	134	2	no	—
270	C₂×SD₁₆	—	268	2	no	—
271	C₂×SD₁₆	—	134	2	no	—
272	C₂×D₈	—	67	2	no	—
273	C₂×D₈	—	134	2	no	—
274	#131	—	2	2	no	—
275	#131	—	67	1	no	—
276	#131	—	67	1	no	—
277	#124	—	134	2	no	—
278	Q₃₂	—	134	2	no	—
279	Q₃₂	—	134	1	no	—
280	Q₃₂	—	134	1	no	—
281	Q₃₂	D₂₆₈	1	1	yes	characteristic
282	SD₃₂	—	268	2	no	—
283	SD₃₂	—	134	2	no	—
284	SD₃₂	—	134	2	no	—
285	SD₃₂	D₂₆₈	1	2	yes	—
286	D₁₆	—	134	2	no	—
287	D₁₆	—	134	1	no	—
288	D₁₆	—	134	1	no	—
289	D₁₆	D₂₆₈	1	1	yes	characteristic
290	M₃₂	—	67	1	no	—
291	M₃₂	—	67	1	no	—
292	M₃₂	—	134	1	no	—
293	M₃₂	—	134	1	no	—
294	C₂×C₁₆	—	134	1	no	—
295	C₂×C₁₆	—	134	1	no	—
296	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
297	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
298	#108	—	67	1	no	—
299	#108	C₂²×D₆₇	1	2	yes	—
300	#104	—	2	2	no	—
301	#104	—	67	1	no	—
302	#104	—	67	1	no	—
303	#101	—	134	2	no	—
304	#100	—	134	2	no	—
305	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
306	#41	—	2	2	no	—
307	#41	—	268	2	no	—
308	#41	—	268	2	no	—
309	#41	—	268	2	no	—
310	#41	—	268	2	no	—
311	#41	—	536	2	no	—
312	#41	—	67	1	no	—
313	#41	—	67	1	no	—
314	C₂×Q₈	—	268	2	no	—
315	C₂×Q₈	—	268	2	no	—
316	C₂×Q₈	—	134	2	no	—
317	C₂×D₄	—	268	2	no	—
318	C₂×D₄	—	268	2	no	—
319	C₂×D₄	—	536	2	no	—
320	C₂×D₄	—	134	2	no	—
321	Q₁₆	—	268	2	no	—
322	Q₁₆	—	268	2	no	—
323	Q₁₆	—	67	1	no	—
324	Q₁₆	—	134	2	no	—
325	Q₁₆	—	134	2	no	—
326	Q₁₆	C₂×D₂₆₈	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	268	2	no	—
329	SD₁₆	—	268	2	no	—
330	SD₁₆	—	268	2	no	—
331	SD₁₆	—	268	2	no	—
332	SD₁₆	—	268	2	no	—
333	SD₁₆	—	268	2	no	—
334	SD₁₆	—	67	1	no	—
335	SD₁₆	—	67	1	no	—
336	D₈	—	268	2	no	—
337	D₈	—	268	2	no	—
338	D₈	—	67	1	no	—
339	D₈	—	134	2	no	—
340	D₈	—	134	2	no	—
341	D₈	C₂×D₂₆₈	1	2	yes	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	4	2	no	—
344	M₁₆	—	67	1	no	—
345	M₁₆	—	67	1	no	—
346	M₁₆	—	134	1	no	—
347	M₁₆	—	134	1	no	—
348	C₂×C₈	—	4	2	no	—
349	C₂×C₈	—	134	1	no	—
350	C₂×C₈	—	134	1	no	—
351	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
352	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
353	C₂²⋊C₄	—	536	2	no	—
354	C₄²	D₄×D₆₇	1	1	yes	characteristic
355	C₁₆	—	2	1	no	—
356	C₁₆	—	134	1	no	—
357	C₁₆	—	134	1	no	—
358	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
359	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
360	C₂³	—	268	2	no	—
361	C₂³	—	536	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	268	2	no	—
364	Q₈	—	268	2	no	—
365	Q₈	—	536	2	no	—
366	Q₈	—	67	1	no	—
367	Q₈	—	67	1	no	—
368	D₄	—	2	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	268	2	no	—
371	D₄	—	268	2	no	—
372	D₄	—	536	2	no	—
373	D₄	—	536	2	no	—
374	D₄	—	536	2	no	—
375	D₄	—	67	1	no	—
376	D₄	—	67	1	no	—
377	D₄	—	134	1	no	—
378	D₄	—	134	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	268	2	no	—
382	C₂×C₄	—	536	2	no	—
383	C₂×C₄	—	536	2	no	—
384	C₂×C₄	—	536	2	no	—
385	C₂×C₄	—	134	1	no	—
386	C₂×C₄	—	134	1	no	—
387	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
388	C₈	—	4	2	no	—
389	C₈	—	134	1	no	—
390	C₈	—	134	1	no	—
391	C₈	D₈×D₆₇	1	1	yes	characteristic
392	C₈	D₈×D₆₇	1	1	yes	characteristic
393	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
394	C₈	#14524089	1	1	yes	characteristic
395	C₂²	—	4	2	no	—
396	C₂²	—	536	2	no	—
397	C₂²	—	1072	2	no	—
398	C₂²	—	2144	2	no	—
399	C₂²	—	134	1	no	—
400	C₂²	—	134	1	no	—
401	C₂²	#16864198	1	1	yes	characteristic
402	C₄	—	4	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	536	2	no	—
405	C₄	—	134	1	no	—
406	C₄	—	134	1	no	—
407	C₄	D₄×D₂₆₈	1	1	yes	characteristic
408	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	8	2	no	—
411	C₂	—	268	1	no	—
412	C₂	—	268	1	no	—
413	C₂	—	1072	2	no	—
414	C₂	D₈×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₆₈
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2178	C₂
socle	C₂×C₆₇	D₄×D₈
hypercenter	#2178	D₆₇
hypocenter	C₆₇	#33859
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
1. #22573359	C₂	semidirect
2. #22573347	C₂	semidirect
3. #22571997	C₂	semidirect
4. #22562470	C₂	semidirect
5. #22562335	C₂	semidirect
6. #22562330	C₂	semidirect
7. #22562278	C₂	semidirect
8. #22562181	C₂	semidirect
9. #22562174	C₂	semidirect
10. #22562047	C₂	semidirect
11. C₆₇×#2178	C₂	semidirect
12. #16864210	C₂²	semidirect
15. #16863836	C₂²	semidirect
21. C₂×Dic₅₃₆	C₂²	semidirect
23. C₂×#14523988	C₂²	semidirect
27. #16863812	C₂²	semidirect
28. #16863780	C₂²	semidirect
30. #16863667	C₂²	semidirect
35. #16863217	C₂²	semidirect
36. #16863211	C₂²	semidirect
37. #16863208	C₂²	semidirect
39. #16863179	C₂²	semidirect
41. #443×C₆₇	C₂²	semidirect
42. #368×C₆₇	C₂²	semidirect
43. #359×C₆₇	C₂²	semidirect
44. C₄×C₁₆×C₆₇	C₂²	semidirect
217. #2178	D₆₇	semidirect
219. C₆₇	#33859	semidirect
244. #443	C₂×D₆₇	semidirect
245. #368	C₂×D₆₇	semidirect
249. #359	C₂×D₆₇	semidirect
252. C₄×C₁₆	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573359	2
#22573347	2
#22571997	2
#22562470	2
#22562335	2
#22562330	2
#22562278	2
#22562181	2
#22562174	2
#22562047	2
C₆₇×#2178	2
#33859	67