Subgroups — #31012643

Order 17152.

636 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573362	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573358	C₂	1	1	yes	maximal, characteristic, complemented
3	#22572012	C₂	1	2	yes	maximal, complemented
4	#22562460	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562335	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562289	C₂	1	2	yes	maximal, complemented
7	#22562190	C₂	1	2	yes	maximal, complemented
8	#22562185	C₂	1	2	yes	maximal, complemented
9	#22562089	C₂	1	1	yes	maximal, characteristic, complemented
10	#22562044	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2222	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864213	C₂²	1	2	yes	—
13	#16864212	—	2	2	no	—
14	#16864209	C₂²	1	2	yes	—
15	#16863839	—	2	2	no	—
16	#16863839	C₂²	1	2	yes	—
17	#16863838	—	2	2	no	—
18	#16863837	C₂²	1	2	yes	—
19	#16863829	—	2	1	no	—
20	#16863828	—	2	1	no	—
21	#16863828	C₂²	1	1	yes	characteristic
22	C₂×Dic₅₃₆	—	2	1	no	—
23	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
24	C₂×#14523989	C₂²	1	2	yes	—
25	Q₃₂×D₆₇	—	2	1	no	—
26	#16863812	C₂²	1	1	yes	characteristic
27	#16863774	C₂²	1	2	yes	—
28	#16863672	C₂²	1	2	yes	—
29	#16863670	—	2	2	no	—
30	#16863581	—	2	2	no	—
31	#16863577	—	2	2	no	—
32	#16863512	C₂²	1	2	yes	—
33	#16863511	C₂²	1	1	yes	characteristic
34	#16863216	C₂²	1	1	yes	characteristic
35	#16863211	C₂²	1	2	yes	—
36	#16863202	C₂²	1	2	yes	—
37	#16863181	C₂²	1	1	yes	characteristic
38	#16863180	C₂²	1	1	yes	characteristic
39	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
40	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
41	#504×C₆₇	C₂²	1	1	yes	characteristic
42	#473×C₆₇	C₂²	1	2	yes	—
43	#368×C₆₇	C₂²	1	1	yes	characteristic
44	#362×C₆₇	C₂²	1	2	yes	—
45	#14524091	—	2	2	no	—
46	#14524090	—	2	2	no	—
47	#14524076	—	2	2	no	—
48	#14524076	—	2	2	no	—
49	#14524076	—	4	2	no	—
50	#14524075	—	4	2	no	—
51	#14524075	—	4	2	no	—
52	#14524074	—	2	2	no	—
53	#14524072	—	4	2	no	—
54	#14524066	—	2	2	no	—
55	#14524065	—	2	2	no	—
56	#14524065	C₂³	1	2	yes	—
57	#14524064	—	4	2	no	—
58	#14524063	D₄	1	1	yes	characteristic
59	C₂×Dic₂₆₈	—	2	2	no	—
60	C₂×Dic₂₆₈	C₂³	1	2	yes	—
61	C₂×#11761618	—	2	2	no	—
62	Q₁₆×D₆₇	—	2	2	no	—
63	Q₁₆×D₆₇	—	4	2	no	—
64	#14524051	—	2	2	no	—
65	#14524049	C₂³	1	1	yes	characteristic
66	#14524041	—	2	2	no	—
67	Dic₅₃₆	—	2	2	no	—
68	Dic₅₃₆	—	2	1	no	—
69	Dic₅₃₆	—	2	1	no	—
70	Dic₅₃₆	D₄	1	1	yes	characteristic
71	#14523989	—	2	2	no	—
72	#14523989	—	4	2	no	—
73	#14523988	—	2	2	no	—
74	#14523988	D₄	1	2	yes	—
75	#14523986	—	2	2	no	—
76	D₁₀₇₂	D₄	1	1	yes	characteristic
77	#14523983	C₂³	1	1	yes	characteristic
78	#14523982	—	2	1	no	—
79	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
80	C₁₆×D₆₇	—	2	1	no	—
81	#14523978	D₄	1	1	yes	characteristic
82	#14523970	D₄	1	1	yes	characteristic
83	#14523969	—	2	2	no	—
84	#14523969	D₄	1	1	yes	characteristic
85	#14523961	—	2	2	no	—
86	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
87	#137×C₆₇	—	2	2	no	—
88	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
89	#131×C₆₇	C₂³	1	1	yes	characteristic
90	Q₃₂×C₆₇	—	2	1	no	—
91	Q₃₂×C₆₇	—	2	1	no	—
92	M₃₂×C₆₇	—	2	1	no	—
93	M₃₂×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₆₇	—	2	1	no	—
95	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
96	#108×C₆₇	C₂³	1	2	yes	—
97	#101×C₆₇	—	2	2	no	—
98	#11761629	—	4	2	no	—
99	#11761629	—	4	2	no	—
100	#11761628	—	2	2	no	—
101	#11761628	—	4	2	no	—
102	#11761628	—	8	2	no	—
103	#11761628	C₂×D₄	1	1	yes	characteristic
104	#11761627	—	4	2	no	—
105	#11761627	—	8	2	no	—
106	C₂×Dic₁₃₄	—	2	2	no	—
107	C₂×Dic₁₃₄	—	4	2	no	—
108	Q₈×D₆₇	—	4	2	no	—
109	Q₈×D₆₇	—	4	2	no	—
110	Q₈×D₆₇	—	4	2	no	—
111	Dic₂₆₈	—	2	2	no	—
112	Dic₂₆₈	—	2	2	no	—
113	Dic₂₆₈	—	4	2	no	—
114	Dic₂₆₈	—	4	2	no	—
115	Dic₂₆₈	C₂×D₄	1	2	yes	—
116	#11761618	—	4	2	no	—
117	#11761618	—	4	2	no	—
118	#11761618	—	4	2	no	—
119	#11761618	D₈	1	1	yes	characteristic
120	#11761617	—	2	2	no	—
121	#11761617	—	4	2	no	—
122	#11761617	—	4	2	no	—
123	#11761616	D₈	1	1	yes	characteristic
124	#11761615	—	4	2	no	—
125	#11761615	D₈	1	1	yes	characteristic
126	D₅₃₆	C₂×D₄	1	2	yes	—
127	#11761613	D₈	1	1	yes	characteristic
128	#11761612	C₂×D₄	1	1	yes	characteristic
129	#11761611	—	2	1	no	—
130	#11761611	—	4	2	no	—
131	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₈×D₆₇	—	2	1	no	—
133	C₈×D₆₇	—	4	2	no	—
134	Dic₆₇⋊C₄	—	8	2	no	—
135	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
136	C₆₇⋊C₁₆	—	2	1	no	—
137	C₆₇⋊C₁₆	—	2	1	no	—
138	#41×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	Q₁₆×C₆₇	—	2	2	no	—
141	Q₁₆×C₆₇	—	2	2	no	—
142	Q₁₆×C₆₇	—	4	2	no	—
143	SD₁₆×C₆₇	—	4	2	no	—
144	M₁₆×C₆₇	—	2	1	no	—
145	M₁₆×C₆₇	—	2	2	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	Q₈×C₆₇	—	4	2	no	—
153	Q₈×C₆₇	—	4	2	no	—
154	Q₈×C₆₇	C₂×D₈	1	1	yes	characteristic
155	D₄×C₆₇	—	2	1	no	—
156	D₄×C₆₇	C₂×D₈	1	1	yes	characteristic
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	—	4	2	no	—
162	C₂²⋊D₆₇	—	8	2	no	—
163	C₂×Dic₆₇	—	2	1	no	—
164	C₂×Dic₆₇	—	4	2	no	—
165	C₂×Dic₆₇	—	8	2	no	—
166	D₂₆₈	—	2	2	no	—
167	D₂₆₈	—	8	2	no	—
168	D₂₆₈	C₂×D₈	1	1	yes	characteristic
169	C₄×D₆₇	—	2	1	no	—
170	C₄×D₆₇	—	4	2	no	—
171	C₄×D₆₇	—	8	2	no	—
172	C₄×D₆₇	—	8	2	no	—
173	C₄×D₆₇	—	8	2	no	—
174	Dic₁₃₄	—	2	2	no	—
175	Dic₁₃₄	—	4	2	no	—
176	Dic₁₃₄	—	4	2	no	—
177	Dic₁₃₄	—	8	2	no	—
178	Dic₁₃₄	—	8	2	no	—
179	Dic₁₃₄	C₂×D₈	1	1	yes	characteristic
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
184	C₆₇⋊C₈	—	2	1	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	#576	1	1	yes	characteristic
189	C₂×D₆₇	—	2	1	no	—
190	C₂×D₆₇	—	4	2	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	8	2	no	—
193	C₄×C₆₇	D₄²	1	1	yes	characteristic
194	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
195	Dic₆₇	—	2	2	no	—
196	Dic₆₇	—	2	1	no	—
197	Dic₆₇	—	4	2	no	—
198	Dic₆₇	—	8	2	no	—
199	#33860	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	—	2	1	no	—
201	C₂×C₆₇	—	4	1	no	—
202	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
203	D₆₇	—	4	1	no	—
204	D₆₇	—	8	2	no	—
205	#3417	—	67	1	no	—
206	#3417	—	67	1	no	—
207	#3293	—	67	2	no	—
208	#2247	—	67	1	no	—
209	#2222	—	67	1	no	—
210	#2222	D₆₇	1	1	yes	characteristic, complemented
211	#2214	—	67	2	no	—
212	#2195	—	67	2	no	—
213	#2195	—	67	2	no	—
214	#2178	—	67	1	no	—
215	#2170	—	67	1	no	—
216	C₆₇	#33860	1	1	yes	characteristic, complemented, Sylow
217	#578	—	67	2	no	—
218	#578	—	67	2	no	—
219	#578	—	134	2	no	—
220	#510	—	67	2	no	—
221	#510	—	67	2	no	—
222	#510	—	134	2	no	—
223	#510	—	134	2	no	—
224	#508	—	67	1	no	—
225	#508	—	134	1	no	—
226	#508	—	134	1	no	—
227	C₂×Q₃₂	—	67	2	no	—
228	C₂×Q₃₂	—	67	1	no	—
229	C₂×Q₃₂	—	134	1	no	—
230	C₂×Q₃₂	—	134	1	no	—
231	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
232	#504	—	67	1	no	—
233	#504	C₂×D₆₇	1	1	yes	characteristic
234	#494	—	67	2	no	—
235	#473	—	67	2	no	—
236	#473	—	134	2	no	—
237	#473	C₂×D₆₇	1	2	yes	—
238	#456	—	134	2	no	—
239	#456	—	134	2	no	—
240	#443	—	67	1	no	—
241	#443	—	67	2	no	—
242	#368	C₂×D₆₇	1	1	yes	characteristic
243	#364	—	67	1	no	—
244	#362	—	67	2	no	—
245	#362	C₂×D₆₇	1	2	yes	—
246	#359	—	67	2	no	—
247	#350	—	67	1	no	—
248	#350	—	67	1	no	—
249	C₄×C₁₆	—	67	1	no	—
250	2₋¹⁺⁴	—	134	2	no	—
251	2₋¹⁺⁴	—	134	2	no	—
252	#137	—	2	2	no	—
253	#137	—	268	2	no	—
254	#137	—	268	2	no	—
255	#137	—	268	2	no	—
256	#137	—	268	2	no	—
257	#137	—	134	2	no	—
258	#137	—	134	2	no	—
259	#137	—	134	2	no	—
260	#135	—	268	2	no	—
261	#135	—	67	1	no	—
262	#135	—	67	2	no	—
263	#135	—	134	2	no	—
264	#135	—	134	2	no	—
265	C₂×Q₁₆	—	268	2	no	—
266	C₂×Q₁₆	—	67	2	no	—
267	C₂×Q₁₆	—	134	2	no	—
268	C₂×Q₁₆	—	134	2	no	—
269	C₂×Q₁₆	—	134	2	no	—
270	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
271	#131	—	67	1	no	—
272	#131	—	134	2	no	—
273	#131	C₂²×D₆₇	1	1	yes	characteristic
274	C₄⋊Q₈	—	134	2	no	—
275	Q₃₂	—	2	1	no	—
276	Q₃₂	—	2	1	no	—
277	Q₃₂	—	268	2	no	—
278	Q₃₂	—	67	1	no	—
279	Q₃₂	—	134	2	no	—
280	Q₃₂	—	134	2	no	—
281	Q₃₂	—	134	1	no	—
282	Q₃₂	—	134	1	no	—
283	SD₃₂	—	67	2	no	—
284	SD₃₂	—	134	2	no	—
285	SD₃₂	—	134	2	no	—
286	D₁₆	—	67	1	no	—
287	M₃₂	—	2	1	no	—
288	M₃₂	—	67	1	no	—
289	M₃₂	—	134	1	no	—
290	M₃₂	C₂²×D₆₇	1	1	yes	characteristic
291	C₂×C₁₆	—	2	1	no	—
292	C₂×C₁₆	—	67	1	no	—
293	C₂×C₁₆	—	134	1	no	—
294	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
295	#108	—	67	1	no	—
296	#108	C₂²×D₆₇	1	2	yes	—
297	#104	—	67	1	no	—
298	#104	—	67	1	no	—
299	#104	—	134	2	no	—
300	#101	—	2	2	no	—
301	#101	—	134	2	no	—
302	C₄×C₈	—	67	1	no	—
303	#41	—	268	2	no	—
304	#41	—	268	2	no	—
305	#41	—	268	2	no	—
306	#41	—	268	2	no	—
307	#41	—	536	2	no	—
308	#41	—	536	2	no	—
309	#41	—	67	1	no	—
310	#41	—	134	2	no	—
311	#41	D₄×D₆₇	1	1	yes	characteristic
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	268	2	no	—
314	C₂×Q₈	—	268	2	no	—
315	C₂×Q₈	—	268	2	no	—
316	C₂×Q₈	—	268	2	no	—
317	C₂×Q₈	—	134	2	no	—
318	Q₁₆	—	2	2	no	—
319	Q₁₆	—	2	2	no	—
320	Q₁₆	—	4	2	no	—
321	Q₁₆	—	268	2	no	—
322	Q₁₆	—	268	2	no	—
323	Q₁₆	—	268	2	no	—
324	Q₁₆	—	268	2	no	—
325	Q₁₆	—	268	2	no	—
326	Q₁₆	—	67	1	no	—
327	Q₁₆	—	67	2	no	—
328	Q₁₆	—	134	2	no	—
329	Q₁₆	—	134	2	no	—
330	SD₁₆	—	4	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	SD₁₆	—	134	2	no	—
337	D₈	—	67	1	no	—
338	D₈	—	67	2	no	—
339	M₁₆	—	2	1	no	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	268	2	no	—
342	M₁₆	—	67	1	no	—
343	M₁₆	—	134	1	no	—
344	M₁₆	D₄×D₆₇	1	1	yes	characteristic
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	268	2	no	—
347	C₂×C₈	—	67	1	no	—
348	C₂×C₈	—	134	1	no	—
349	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
350	C₄⋊C₄	—	536	2	no	—
351	C₄²	—	67	1	no	—
352	C₁₆	—	2	1	no	—
353	C₁₆	—	134	1	no	—
354	C₁₆	—	134	1	no	—
355	C₁₆	D₄×D₆₇	1	1	yes	characteristic
356	C₁₆	D₄×D₆₇	1	1	yes	characteristic
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	268	2	no	—
360	Q₈	—	268	2	no	—
361	Q₈	—	536	2	no	—
362	Q₈	—	536	2	no	—
363	Q₈	—	67	1	no	—
364	Q₈	—	134	2	no	—
365	Q₈	D₈×D₆₇	1	1	yes	characteristic
366	D₄	—	2	1	no	—
367	D₄	—	268	2	no	—
368	D₄	—	536	2	no	—
369	D₄	—	536	2	no	—
370	D₄	—	67	1	no	—
371	D₄	—	134	1	no	—
372	D₄	—	134	2	no	—
373	D₄	D₈×D₆₇	1	1	yes	characteristic
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	268	2	no	—
377	C₂×C₄	—	268	2	no	—
378	C₂×C₄	—	536	2	no	—
379	C₂×C₄	—	536	2	no	—
380	C₂×C₄	—	536	2	no	—
381	C₂×C₄	—	536	2	no	—
382	C₂×C₄	—	134	1	no	—
383	C₂×C₄	—	134	1	no	—
384	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
385	C₈	—	2	1	no	—
386	C₈	—	4	2	no	—
387	C₈	—	67	1	no	—
388	C₈	—	67	1	no	—
389	C₈	—	134	1	no	—
390	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
391	C₈	#14524089	1	1	yes	characteristic
392	C₂²	—	2	1	no	—
393	C₂²	—	268	2	no	—
394	C₂²	—	134	1	no	—
395	C₂²	#16864198	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	8	2	no	—
398	C₄	—	268	2	no	—
399	C₄	—	536	2	no	—
400	C₄	—	134	2	no	—
401	C₄	—	134	1	no	—
402	C₄	#16864030	1	1	yes	characteristic
403	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	4	1	no	—
406	C₂	—	268	1	no	—
407	C₂	—	536	2	no	—
408	C₂	#22571883	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22571883
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2222	C₂
socle	C₂×C₆₇	D₄×D₈
hypercenter	#2222	D₆₇
hypocenter	C₆₇	#33860
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2222

Complements

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

N	H ≅ G/N	type
1. #22573362	C₂	semidirect
2. #22573358	C₂	semidirect
3. #22572012	C₂	semidirect
4. #22562460	C₂	semidirect
5. #22562335	C₂	semidirect
6. #22562289	C₂	semidirect
7. #22562190	C₂	semidirect
8. #22562185	C₂	semidirect
9. #22562089	C₂	semidirect
10. #22562044	C₂	semidirect
11. C₆₇×#2222	C₂	semidirect
210. #2222	D₆₇	semidirect
216. C₆₇	#33860	semidirect

Maximal subgroups

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

subgroup	index
#22573362	2
#22573358	2
#22572012	2
#22562460	2
#22562335	2
#22562289	2
#22562190	2
#22562185	2
#22562089	2
#22562044	2
C₆₇×#2222	2
#33860	67