Subgroups — #31010539

Order 17152.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570749	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569083	C₂	1	1	yes	maximal, characteristic
3	#22562376	C₂	1	2	yes	maximal, complemented
4	#22562365	C₂	1	2	yes	maximal, complemented
5	#22562360	C₂	1	2	yes	maximal, complemented
6	#22562345	C₂	1	2	yes	maximal, complemented
7	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561969	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561922	C₂	1	1	yes	maximal, characteristic
10	#22561914	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	#16864024	—	2	1	no	—
14	#16863877	—	2	1	no	—
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863803	—	2	2	no	—
17	#16863800	C₂²	1	2	yes	—
18	#16863797	C₂²	1	2	yes	—
19	#16863495	—	2	2	no	—
20	#16863490	C₂²	1	2	yes	—
21	#16863478	C₂²	1	2	yes	—
22	#16863439	—	2	1	no	—
23	C₂×#14523977	C₂²	1	1	yes	characteristic
24	#16863395	—	2	1	no	—
25	#16863351	C₂²	1	1	yes	characteristic
26	#16863221	C₂²	1	2	yes	—
27	#16863221	C₂²	1	2	yes	—
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
30	#16863201	C₂²	1	2	yes	—
31	#16863199	C₂²	1	1	yes	characteristic
32	#16863199	C₂²	1	1	yes	characteristic
33	#16863198	C₂²	1	2	yes	—
34	#16863196	C₂²	1	1	yes	characteristic
35	#16863196	C₂²	1	1	yes	characteristic
36	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
37	#427×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
39	#367×C₆₇	C₂²	1	2	yes	—
40	#366×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	—	2	1	no	—
42	#14524065	—	2	2	no	—
43	#14524065	Q₈	1	4	yes	complemented
44	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
45	C₂×#11761617	—	2	1	no	—
46	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
47	C₂×#11761612	D₄	1	1	yes	characteristic
48	#14524043	—	4	1	no	—
49	#14524042	—	2	2	no	—
50	#14524035	—	4	1	no	—
51	#14524034	—	2	2	no	—
52	#14524020	—	4	2	no	—
53	#14524008	—	2	2	no	—
54	#14524005	—	2	2	no	—
55	C₄×C₂²⋊D₆₇	—	4	2	no	—
56	#14524002	D₄	1	1	yes	characteristic
57	#14524001	—	4	1	no	—
58	C₂×C₄⋊Dic₆₇	—	2	1	no	—
59	C₄⋊C₄×D₆₇	—	4	1	no	—
60	C₂×C₆₇⋊C₁₆	—	2	2	no	—
61	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
63	#14523977	—	2	1	no	—
64	#14523977	C₂³	1	2	yes	—
65	#14523976	—	2	2	no	—
66	#14523974	—	2	2	no	—
67	#14523966	—	2	2	no	—
68	#14523963	—	2	2	no	—
69	#14523953	D₄	1	2	yes	—
70	C₈×Dic₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	#117×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	M₃₂×C₆₇	—	2	2	no	—
75	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
76	#107×C₆₇	—	2	1	no	—
77	#107×C₆₇	C₂³	1	2	yes	—
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	C₂³	1	1	yes	characteristic
80	#106×C₆₇	—	2	1	no	—
81	#11761628	—	2	4	no	—
82	#11761628	—	4	2	no	—
83	C₂×Dic₁₃₄	—	2	1	no	—
84	C₂×Dic₁₃₄	—	4	2	no	—
85	C₂×D₂₆₈	—	2	1	no	—
86	C₂×C₂²⋊D₆₇	—	4	1	no	—
87	C₂²×Dic₆₇	—	4	1	no	—
88	C₂×C₄×D₆₇	—	4	1	no	—
89	C₂×C₄×D₆₇	—	4	2	no	—
90	Dic₂₆₈	—	2	1	no	—
91	Dic₂₆₈	C₂×Q₈	1	2	yes	—
92	#11761617	—	2	4	no	—
93	D₅₃₆	—	2	1	no	—
94	D₅₃₆	C₂×Q₈	1	2	yes	—
95	#11761612	—	2	2	no	—
96	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
97	C₄⋊Dic₆₇	—	2	2	no	—
98	C₄⋊Dic₆₇	—	4	1	no	—
99	C₄⋊Dic₆₇	—	4	2	no	—
100	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
101	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
102	Dic₆₇⋊C₄	—	4	2	no	—
103	Dic₆₇⋊C₄	—	4	2	no	—
104	Dic₆₇⋊C₄	—	4	2	no	—
105	C₂²⋊Dic₆₇	D₈	1	2	yes	—
106	(C₂×D₆₇)⋊C₄	—	4	2	no	—
107	C₄×Dic₆₇	—	4	2	no	—
108	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
109	C₆₇⋊C₁₆	—	2	1	no	—
110	C₆₇⋊C₁₆	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₆₇	#41	1	2	yes	—
116	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₂²⋊C₄×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₁₆×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	16	1	no	—
130	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#137	1	2	yes	—
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₆₇	—	4	4	no	—
135	C₂×Dic₆₇	—	4	1	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	C₂×Dic₆₇	—	16	1	no	—
138	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
139	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
140	D₂₆₈	—	2	2	no	—
141	D₂₆₈	—	4	1	no	—
142	C₄×D₆₇	—	4	4	no	—
143	C₄×D₆₇	—	16	2	no	—
144	Dic₁₃₄	—	2	2	no	—
145	Dic₁₃₄	—	4	1	no	—
146	Dic₁₃₄	—	16	2	no	—
147	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
148	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₆₇⋊C₈	—	2	2	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	#474	1	2	yes	—
154	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₆₇	—	4	2	no	—
156	C₂×D₆₇	—	8	1	no	—
157	C₂×D₆₇	—	8	1	no	—
158	C₄×C₆₇	—	8	1	no	—
159	C₄×C₆₇	—	8	2	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	#474	1	2	yes	—
162	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
164	Dic₆₇	—	2	2	no	—
165	Dic₆₇	—	2	2	no	—
166	Dic₆₇	—	4	2	no	—
167	Dic₆₇	—	8	2	no	—
168	#33633	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	—	2	2	no	—
170	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
171	C₂×C₆₇	#3415	1	1	yes	characteristic
172	C₂×C₆₇	#3417	1	1	yes	characteristic
173	D₆₇	—	8	2	no	—
174	#3164	—	67	1	no	—
175	#2951	—	67	1	no	—
176	#2229	—	67	2	no	—
177	#2228	—	67	2	no	—
178	#2227	—	67	2	no	—
179	#2224	—	67	2	no	—
180	#2159	D₆₇	1	1	yes	characteristic, complemented
181	C₂×#366	—	67	1	no	—
182	#2151	—	67	1	no	—
183	#2139	—	67	1	no	—
184	#2138	—	67	1	no	—
185	C₆₇	#33633	1	1	yes	characteristic, complemented, Sylow
186	C₂×#135	—	67	1	no	—
187	#544	—	134	1	no	—
188	#520	—	134	1	no	—
189	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
190	C₂²×C₁₆	—	67	1	no	—
191	#501	—	134	2	no	—
192	#500	—	67	2	no	—
193	#499	—	67	2	no	—
194	#440	—	134	2	no	—
195	C₄×Q₁₆	—	67	2	no	—
196	C₄×D₈	—	67	2	no	—
197	#428	—	134	1	no	—
198	#427	C₂×D₆₇	1	1	yes	characteristic
199	C₂×#107	—	67	1	no	—
200	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
201	#417	—	134	1	no	—
202	#405	—	67	1	no	—
203	#367	C₂×D₆₇	1	2	yes	—
204	#366	—	67	2	no	—
205	#366	—	67	2	no	—
206	#366	C₂×D₆₇	1	2	yes	—
207	C₄⋊C₁₆	—	67	2	no	—
208	C₄⋊C₁₆	—	67	2	no	—
209	#358	—	67	2	no	—
210	#358	—	67	1	no	—
211	#358	—	67	1	no	—
212	#357	—	67	2	no	—
213	#357	—	67	1	no	—
214	#357	—	67	1	no	—
215	C₂×#41	—	134	1	no	—
216	#135	—	67	4	no	—
217	#135	—	134	2	no	—
218	C₂×Q₁₆	—	67	1	no	—
219	C₂×SD₁₆	—	134	1	no	—
220	C₂×D₈	—	67	1	no	—
221	C₂×M₁₆	—	67	1	no	—
222	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₄⋊Q₈	—	268	1	no	—
224	C₄⋊Q₈	—	134	2	no	—
225	#125	—	268	1	no	—
226	#125	—	134	2	no	—
227	C₂²⋊Q₈	—	268	2	no	—
228	C₄×Q₈	—	134	2	no	—
229	C₄×D₄	—	268	2	no	—
230	C₄×D₄	—	134	2	no	—
231	#117	—	2	1	no	—
232	#117	—	268	1	no	—
233	#117	—	67	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	268	1	no	—
236	C₂×C₄⋊C₄	—	134	1	no	—
237	M₃₂	—	2	2	no	—
238	C₂×C₁₆	—	67	1	no	—
239	C₂×C₁₆	—	67	1	no	—
240	C₂×C₁₆	—	134	2	no	—
241	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
242	#107	—	2	1	no	—
243	#107	—	67	2	no	—
244	#107	—	134	2	no	—
245	#107	—	134	1	no	—
246	#107	C₂²×D₆₇	1	1	yes	characteristic
247	#107	C₂²×D₆₇	1	1	yes	characteristic
248	#107	C₂²×D₆₇	1	2	yes	—
249	#106	—	2	1	no	—
250	#106	—	134	2	no	—
251	#103	—	134	2	no	—
252	#102	—	134	2	no	—
253	#97	—	67	2	no	—
254	C₄×C₈	—	67	2	no	—
255	#41	—	268	2	no	—
256	#41	—	134	4	no	—
257	C₂×Q₈	—	268	2	no	—
258	C₂×Q₈	—	134	1	no	—
259	C₂×D₄	—	268	1	no	—
260	C₂×D₄	—	134	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	268	2	no	—
263	C₂²×C₄	—	268	1	no	—
264	C₂²×C₄	—	268	1	no	—
265	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
266	Q₁₆	—	67	2	no	—
267	Q₁₆	—	134	1	no	—
268	SD₁₆	—	134	4	no	—
269	D₈	—	67	2	no	—
270	D₈	—	134	1	no	—
271	M₁₆	—	134	2	no	—
272	C₂×C₈	—	67	2	no	—
273	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₆₇	1	2	yes	—
275	C₂×C₈	#11761628	1	2	yes	—
276	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	268	2	no	—
282	C₄⋊C₄	—	268	2	no	—
283	C₄⋊C₄	—	268	2	no	—
284	C₄⋊C₄	—	268	2	no	—
285	C₄⋊C₄	—	268	1	no	—
286	C₄⋊C₄	—	67	1	no	—
287	C₄⋊C₄	—	67	1	no	—
288	C₄⋊C₄	—	134	2	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	268	2	no	—
291	C₂²⋊C₄	—	67	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	268	2	no	—
294	C₄²	—	67	2	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	134	1	no	—
297	C₁₆	—	134	1	no	—
298	C₂³	—	268	1	no	—
299	C₂³	D₈×D₆₇	1	1	yes	characteristic
300	Q₈	—	268	1	no	—
301	Q₈	—	1072	2	no	—
302	Q₈	—	134	2	no	—
303	D₄	—	268	4	no	—
304	D₄	—	268	1	no	—
305	D₄	—	134	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	268	1	no	—
310	C₂×C₄	—	268	4	no	—
311	C₂×C₄	—	268	2	no	—
312	C₂×C₄	—	16	1	no	—
313	C₂×C₄	—	1072	2	no	—
314	C₂×C₄	—	1072	1	no	—
315	C₂×C₄	—	67	2	no	—
316	C₂×C₄	—	67	2	no	—
317	C₂×C₄	#14524021	1	2	yes	—
318	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
319	C₂×C₄	#14524075	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
321	C₈	—	134	2	no	—
322	C₈	#14524021	1	2	yes	—
323	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
324	C₈	C₂×#11761628	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	268	2	no	—
328	C₂²	—	536	1	no	—
329	C₂²	—	536	1	no	—
330	C₂²	#16863675	1	2	yes	—
331	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	268	2	no	—
336	C₄	—	536	2	no	—
337	C₄	—	134	2	no	—
338	C₄	—	134	2	no	—
339	C₄	#16863675	1	2	yes	—
340	C₄	C₂×#14524021	1	1	yes	characteristic
341	C₄	C₂×#14524075	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	536	2	no	—
344	C₂	C₂×#16863675	1	1	yes	characteristic
345	C₂	#22573349	1	1	yes	characteristic
346	C₂	#22573361	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2159	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#14524065	C₂²
hypercenter	#2159	D₆₇
hypocenter	C₆₇	#33633
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
1. #22570749	C₂	semidirect
3. #22562376	C₂	semidirect
4. #22562365	C₂	semidirect
5. #22562360	C₂	semidirect
6. #22562345	C₂	semidirect
7. C₂×#16863221	C₂	semidirect
8. #22561969	C₂	semidirect
11. C₆₇×#2159	C₂	semidirect
43. #14524065	Q₈	semidirect
180. #2159	D₆₇	semidirect
185. C₆₇	#33633	semidirect

Maximal subgroups

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

subgroup	index
#22570749	2
#22569083	2
#22562376	2
#22562365	2
#22562360	2
#22562345	2
C₂×#16863221	2
#22561969	2
#22561922	2
#22561914	2
C₆₇×#2159	2
#33633	67