Subgroups — #31012629

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	#22573347	C₂	1	1	yes	maximal, characteristic, complemented
2	#22571995	C₂	1	2	yes	maximal, complemented
3	#22562471	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562331	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562329	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562280	C₂	1	2	yes	maximal, complemented
7	#22562177	C₂	1	2	yes	maximal, complemented
8	#22562173	C₂	1	2	yes	maximal, complemented
9	#22562090	C₂	1	1	yes	maximal, characteristic, complemented
10	#22562044	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864204	—	2	2	no	—
13	#16864198	C₂²	1	2	yes	—
14	#16863833	—	2	2	no	—
15	#16863831	C₂²	1	2	yes	—
16	#16863829	—	2	1	no	—
17	C₂×#14523986	C₂²	1	2	yes	complemented
18	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
19	D₁₆×D₆₇	—	2	1	no	—
20	#16863812	C₂²	1	1	yes	characteristic
21	#16863781	C₂²	1	2	yes	complemented
22	#16863668	C₂²	1	2	yes	—
23	#16863666	C₂²	1	2	yes	complemented
24	#16863661	—	2	2	no	—
25	#16863572	—	2	2	no	—
26	#16863569	—	2	2	no	—
27	#16863513	C₂²	1	2	yes	complemented
28	#16863511	C₂²	1	1	yes	characteristic
29	#16863226	C₂²	1	1	yes	characteristic, complemented
30	#16863216	C₂²	1	1	yes	characteristic, complemented
31	#16863212	C₂²	1	2	yes	complemented
32	#16863210	C₂²	1	2	yes	complemented
33	#16863203	C₂²	1	2	yes	complemented
34	#16863181	C₂²	1	1	yes	characteristic
35	#16863180	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
37	#578×C₆₇	C₂²	1	2	yes	complemented
38	#510×C₆₇	—	2	2	no	—
39	#510×C₆₇	C₂²	1	2	yes	complemented
40	#508×C₆₇	—	2	1	no	—
41	#508×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₆₇	—	2	1	no	—
43	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#504×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#14524089	—	2	2	no	—
46	#14524074	—	2	2	no	—
47	#14524074	—	4	2	no	—
48	#14524070	—	4	2	no	—
49	#14524068	—	2	2	no	—
50	#14524068	—	2	2	no	—
51	#14524068	—	4	2	no	—
52	#14524066	—	2	2	no	—
53	#14524063	—	4	2	no	—
54	#14524063	D₄	1	1	yes	characteristic
55	C₂×#11761615	—	2	2	no	—
56	C₂×#11761615	—	4	2	no	—
57	C₂×D₅₃₆	C₂³	1	2	yes	—
58	D₈×D₆₇	—	2	2	no	—
59	#14524050	—	2	2	no	—
60	#14524049	C₂³	1	1	yes	characteristic
61	#14524032	—	2	2	no	—
62	#14523986	—	2	2	no	—
63	#14523986	—	4	2	no	—
64	D₁₀₇₂	—	2	1	no	—
65	D₁₀₇₂	—	2	1	no	—
66	#14523984	—	2	2	no	—
67	#14523983	C₂³	1	1	yes	characteristic
68	#14523982	—	2	1	no	—
69	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₁₆×D₆₇	—	2	1	no	—
71	#14523979	C₂³	1	2	yes	—
72	#14523978	D₄	1	1	yes	characteristic
73	#14523970	—	2	2	no	—
74	#14523970	D₄	1	1	yes	characteristic
75	#14523969	D₄	1	1	yes	characteristic
76	#14523962	—	2	2	no	—
77	#14523959	—	2	2	no	—
78	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₆₇	—	2	2	no	—
80	#137×C₆₇	—	2	2	no	—
81	#137×C₆₇	—	4	2	no	—
82	#135×C₆₇	—	2	2	no	—
83	#135×C₆₇	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₆₇	—	2	2	no	—
85	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
86	#131×C₆₇	C₂³	1	1	yes	characteristic
87	Q₃₂×C₆₇	—	2	2	no	—
88	Q₃₂×C₆₇	—	2	1	no	—
89	Q₃₂×C₆₇	—	2	1	no	—
90	Q₃₂×C₆₇	D₄	1	1	yes	characteristic
91	SD₃₂×C₆₇	—	2	2	no	—
92	SD₃₂×C₆₇	D₄	1	2	yes	—
93	D₁₆×C₆₇	D₄	1	1	yes	characteristic
94	M₃₂×C₆₇	—	2	1	no	—
95	M₃₂×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₆₇	—	2	1	no	—
97	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
98	#11761629	—	4	2	no	—
99	#11761628	—	4	2	no	—
100	#11761628	C₂×D₄	1	1	yes	characteristic
101	C₂×D₂₆₈	—	2	2	no	—
102	C₂×D₂₆₈	—	4	2	no	—
103	D₄×D₆₇	—	4	2	no	—
104	D₄×D₆₇	—	8	2	no	—
105	#11761618	—	4	2	no	—
106	#11761618	D₈	1	1	yes	characteristic
107	#11761616	D₈	1	1	yes	characteristic
108	#11761615	—	4	2	no	—
109	#11761615	—	4	2	no	—
110	#11761615	—	4	2	no	—
111	#11761615	—	4	2	no	—
112	#11761615	—	4	2	no	—
113	#11761615	D₈	1	1	yes	characteristic
114	D₅₃₆	—	2	2	no	—
115	D₅₃₆	—	2	2	no	—
116	#11761613	—	4	2	no	—
117	#11761613	—	4	2	no	—
118	#11761613	D₈	1	1	yes	characteristic
119	#11761612	—	2	2	no	—
120	#11761612	—	4	2	no	—
121	#11761612	C₂×D₄	1	1	yes	characteristic
122	#11761611	—	2	1	no	—
123	C₂×C₆₇⋊C₈	—	4	2	no	—
124	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₈×D₆₇	—	2	1	no	—
126	(C₂×D₆₇)⋊C₄	—	8	2	no	—
127	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
128	C₆₇⋊C₁₆	—	2	1	no	—
129	C₆₇⋊C₁₆	—	2	1	no	—
130	#41×C₆₇	—	2	2	no	—
131	#41×C₆₇	—	4	2	no	—
132	#41×C₆₇	—	4	2	no	—
133	#41×C₆₇	—	8	2	no	—
134	#41×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	4	2	no	—
137	C₂×Q₈×C₆₇	—	4	2	no	—
138	Q₁₆×C₆₇	—	2	2	no	—
139	Q₁₆×C₆₇	—	2	2	no	—
140	Q₁₆×C₆₇	—	4	2	no	—
141	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
142	SD₁₆×C₆₇	—	2	2	no	—
143	SD₁₆×C₆₇	—	4	2	no	—
144	D₈×C₆₇	C₂×D₄	1	2	yes	—
145	M₁₆×C₆₇	—	2	1	no	—
146	M₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₂×C₈×C₆₇	C₂⁴	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	Q₈×C₆₇	—	4	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	8	2	no	—
158	Q₈×C₆₇	C₂×D₈	1	1	yes	characteristic
159	D₄×C₆₇	—	2	1	no	—
160	D₄×C₆₇	—	2	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	8	2	no	—
163	D₄×C₆₇	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	8	2	no	—
168	C₂×C₄×C₆₇	—	8	2	no	—
169	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₆₇	—	2	1	no	—
171	C₂²⋊D₆₇	—	8	2	no	—
172	C₂×Dic₆₇	—	2	1	no	—
173	D₂₆₈	—	4	2	no	—
174	D₂₆₈	—	4	2	no	—
175	D₂₆₈	—	8	2	no	—
176	D₂₆₈	C₂×D₈	1	1	yes	characteristic
177	C₄×D₆₇	—	2	1	no	—
178	C₄×D₆₇	—	8	2	no	—
179	Dic₁₃₄	C₂×D₈	1	1	yes	characteristic
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
183	C₆₇⋊C₈	—	2	1	no	—
184	C₆₇⋊C₈	—	4	2	no	—
185	C₆₇⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₆₇⋊C₈	C₂×D₈	1	1	yes	characteristic
187	C₂²×C₆₇	—	2	1	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	#576	1	1	yes	characteristic
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	—	2	1	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	8	2	no	—
197	C₄×C₆₇	D₄²	1	1	yes	characteristic
198	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
199	Dic₆₇	—	2	1	no	—
200	Dic₆₇	—	4	1	no	—
201	#33859	—	67	1	no	maximal, Sylow
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	—	4	1	no	—
204	C₂×C₆₇	—	8	2	no	—
205	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
206	D₆₇	—	4	1	no	—
207	D₆₇	—	16	2	no	—
208	#3417	D₆₇	1	1	yes	characteristic, complemented
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	—	67	1	no	—
218	#2170	—	67	1	no	—
219	C₆₇	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	C₂×D₆₇	1	2	yes	complemented
221	#577	—	134	2	no	—
222	#576	—	67	2	no	—
223	#510	—	2	2	no	—
224	#510	C₂×D₆₇	1	2	yes	complemented
225	#509	—	67	2	no	—
226	#509	—	134	2	no	—
227	#508	—	2	1	no	—
228	#508	—	134	1	no	—
229	#508	C₂×D₆₇	1	1	yes	characteristic
230	C₂×Q₃₂	—	2	1	no	—
231	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic, complemented
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	C₂×D₆₇	1	1	yes	characteristic, complemented
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	—	67	2	no	—
245	#368	—	67	1	no	—
246	#364	—	67	1	no	—
247	#362	—	67	2	no	—
248	#361	—	67	2	no	—
249	#359	—	67	2	no	—
250	#350	—	67	1	no	—
251	#350	—	67	1	no	—
252	C₄×C₁₆	—	67	1	no	—
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	134	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	268	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	—	2	2	no	—
264	#135	—	268	2	no	—
265	#135	—	67	1	no	—
266	#135	—	134	2	no	—
267	#135	C₂²×D₆₇	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
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	—	67	1	no	—
275	#131	—	134	2	no	—
276	#131	C₂²×D₆₇	1	1	yes	characteristic
277	#124	—	134	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₆₇	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	268	2	no	—
284	SD₃₂	—	134	2	no	—
285	SD₃₂	C₂²⋊D₆₇	1	2	yes	—
286	D₁₆	—	134	2	no	—
287	D₁₆	—	134	1	no	—
288	D₁₆	—	134	1	no	—
289	D₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	67	1	no	—
292	M₃₂	—	134	1	no	—
293	M₃₂	C₂²×D₆₇	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	67	1	no	—
296	C₂×C₁₆	—	134	1	no	—
297	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
298	#108	—	67	1	no	—
299	#108	—	67	2	no	—
300	#104	—	67	1	no	—
301	#104	—	67	1	no	—
302	#104	—	134	2	no	—
303	#101	—	134	2	no	—
304	#100	—	134	2	no	—
305	C₄×C₈	—	67	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	268	2	no	—
311	#41	—	268	2	no	—
312	#41	—	67	1	no	—
313	#41	D₄×D₆₇	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	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	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	268	2	no	—
325	Q₁₆	—	67	1	no	—
326	Q₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	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×C₂²⋊D₆₇	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	268	2	no	—
344	M₁₆	—	67	1	no	—
345	M₁₆	—	134	2	no	—
346	M₁₆	—	134	1	no	—
347	M₁₆	D₄×D₆₇	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	268	2	no	—
350	C₂×C₈	—	67	1	no	—
351	C₂×C₈	—	134	1	no	—
352	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
353	C₂²⋊C₄	—	536	2	no	—
354	C₄²	—	67	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	134	1	no	—
357	C₁₆	—	134	1	no	—
358	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
360	C₂³	—	268	2	no	—
361	C₂³	—	536	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	67	1	no	—
367	Q₈	D₈×D₆₇	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	268	2	no	—
373	D₄	—	268	2	no	—
374	D₄	—	536	2	no	—
375	D₄	—	536	2	no	—
376	D₄	—	67	1	no	—
377	D₄	—	134	1	no	—
378	D₄	D₈×D₆₇	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	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	—	2	1	no	—
389	C₈	—	268	2	no	—
390	C₈	—	67	1	no	—
391	C₈	—	67	1	no	—
392	C₈	—	134	1	no	—
393	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
394	C₈	#14524088	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	536	2	no	—
398	C₂²	—	1072	2	no	—
399	C₂²	—	2144	2	no	—
400	C₂²	—	134	1	no	—
401	C₂²	#16864197	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	268	1	no	—
406	C₄	—	134	1	no	—
407	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
408	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	268	1	no	—
413	C₂	—	1072	2	no	—
414	C₂	D₈×C₂²⋊D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3417	C₂
socle	C₂×C₆₇	D₄×D₈
hypercenter	#3417	D₆₇
hypocenter	C₆₇	#33859
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #22573347	C₂	semidirect
2. #22571995	C₂	semidirect
3. #22562471	C₂	semidirect
4. #22562331	C₂	semidirect
5. #22562329	C₂	semidirect
6. #22562280	C₂	semidirect
7. #22562177	C₂	semidirect
8. #22562173	C₂	semidirect
9. #22562090	C₂	semidirect
10. #22562044	C₂	semidirect
11. C₆₇×#3417	C₂	semidirect
17. C₂×#14523986	C₂²	semidirect
21. #16863781	C₂²	semidirect
23. #16863666	C₂²	semidirect
27. #16863513	C₂²	semidirect
29. #16863226	C₂²	semidirect
30. #16863216	C₂²	semidirect
31. #16863212	C₂²	semidirect
32. #16863210	C₂²	semidirect
33. #16863203	C₂²	semidirect
35. #16863180	C₂²	semidirect
36. C₁₆×Dic₆₇	C₂²	semidirect
37. #578×C₆₇	C₂²	semidirect
39. #510×C₆₇	C₂²	semidirect
43. C₂×Q₃₂×C₆₇	C₂²	semidirect
44. #504×C₆₇	C₂²	semidirect
208. #3417	D₆₇	semidirect
219. C₆₇	#33859	semidirect
220. #578	C₂×D₆₇	semidirect
224. #510	C₂×D₆₇	semidirect
231. C₂×Q₃₂	C₂×D₆₇	semidirect
236. #504	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573347	2
#22571995	2
#22562471	2
#22562331	2
#22562329	2
#22562280	2
#22562177	2
#22562173	2
#22562090	2
#22562044	2
C₆₇×#3417	2
#33859	67