Subgroups — #31012755

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572639	C₂	1	1	yes	maximal, characteristic, complemented
2	#22571930	C₂	1	1	yes	maximal, characteristic
3	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562419	C₂	1	2	yes	maximal, complemented
5	#22562301	C₂	1	2	yes	maximal, complemented
6	#22562260	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562150	C₂	1	2	yes	maximal, complemented
8	C₄×#14523987	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562028	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561913	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524062	C₂²	1	2	yes	—
13	#16864120	—	2	1	no	—
14	#16864050	—	2	1	no	—
15	C₂×Dic₅₃₆	C₂²	1	2	yes	—
16	C₂×#14523987	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
18	#16863776	C₂²	1	1	yes	characteristic
19	#16863739	—	2	2	no	—
20	#16863678	—	2	1	no	—
21	#16863638	C₂²	1	2	yes	—
22	#16863638	C₂²	1	2	yes	—
23	#16863601	—	2	1	no	—
24	#16863559	—	2	2	no	—
25	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
26	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
27	#16863433	C₂²	1	2	yes	—
28	#16863223	C₂²	1	1	yes	characteristic
29	#16863201	C₂²	1	2	yes	—
30	#16863199	C₂²	1	1	yes	characteristic
31	#16863199	C₂²	1	2	yes	—
32	#16863199	C₂²	1	2	yes	—
33	#16863197	C₂²	1	1	yes	characteristic
34	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
36	#500×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
38	#366×C₆₇	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	#357×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761627	—	2	2	no	—
42	#14524062	—	2	4	no	—
43	#14524062	—	4	2	no	—
44	C₂×Dic₂₆₈	—	2	1	no	—
45	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
46	C₂×Dic₂₆₈	C₂³	1	2	yes	—
47	C₂×#11761616	—	2	2	no	—
48	C₂×C₈×D₆₇	C₂³	1	2	yes	—
49	#14524043	—	2	1	no	—
50	#14524038	—	4	2	no	—
51	#14524032	—	4	1	no	—
52	#14524027	—	4	2	no	—
53	#14524023	—	2	2	no	—
54	#14524023	—	2	2	no	—
55	#14524017	—	4	1	no	—
56	Q₈×Dic₆₇	—	4	1	no	—
57	C₄×Dic₁₃₄	—	2	1	no	—
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	#14524001	—	2	2	no	—
60	C₂×C₄⋊Dic₆₇	—	4	1	no	—
61	Dic₅₃₆	—	4	2	no	—
62	#14523987	—	2	2	no	—
63	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₁₆×D₆₇	—	4	2	no	—
66	#14523977	C₂³	1	1	yes	characteristic
67	#14523974	—	2	2	no	—
68	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
69	#14523968	—	2	1	no	—
70	#14523966	—	2	2	no	—
71	#14523966	—	2	2	no	—
72	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
73	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₆₇	—	2	1	no	—
75	C₄×D₄×C₆₇	—	2	1	no	—
76	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
77	#107×C₆₇	—	2	1	no	—
78	#107×C₆₇	C₂³	1	2	yes	—
79	#107×C₆₇	C₂³	1	1	yes	characteristic
80	#102×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#11761627	—	4	4	no	—
83	#11761627	—	8	2	no	—
84	C₂×Dic₁₃₄	—	2	1	no	—
85	C₂×Dic₁₃₄	—	2	2	no	—
86	C₂×D₂₆₈	D₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₆₇	—	4	2	no	—
88	C₂²×Dic₆₇	—	4	2	no	—
89	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
90	Dic₂₆₈	—	2	1	no	—
91	Dic₂₆₈	—	2	2	no	—
92	Dic₂₆₈	—	2	1	no	—
93	Dic₂₆₈	—	2	2	no	—
94	Dic₂₆₈	—	8	1	no	—
95	#11761616	—	4	4	no	—
96	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
97	C₈×D₆₇	—	2	2	no	—
98	C₈×D₆₇	—	2	2	no	—
99	C₄⋊Dic₆₇	—	2	1	no	—
100	C₄⋊Dic₆₇	—	4	1	no	—
101	C₄⋊Dic₆₇	—	4	1	no	—
102	C₄⋊Dic₆₇	—	4	2	no	—
103	C₄⋊Dic₆₇	—	4	2	no	—
104	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
105	Dic₆₇⋊C₄	—	4	1	no	—
106	(C₂×D₆₇)⋊C₄	—	4	2	no	—
107	(C₂×D₆₇)⋊C₄	—	4	2	no	—
108	(C₂×D₆₇)⋊C₄	—	4	2	no	—
109	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
110	C₄×Dic₆₇	—	4	1	no	—
111	C₄×Dic₆₇	—	4	2	no	—
112	C₆₇⋊C₁₆	—	2	1	no	—
113	C₆₇⋊C₁₆	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	4	1	no	—
115	C₂×D₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	4	1	no	—
117	D₈×C₆₇	—	2	1	no	—
118	D₈×C₆₇	#41	1	2	yes	—
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₂²⋊C₄×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	4	1	no	—
128	C₂²×D₆₇	C₂×D₈	1	2	yes	—
129	Q₈×C₆₇	—	8	2	no	—
130	D₄×C₆₇	—	2	2	no	—
131	D₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	8	2	no	—
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂²⋊D₆₇	—	8	4	no	—
139	C₂×Dic₆₇	—	4	1	no	—
140	C₂×Dic₆₇	—	4	1	no	—
141	C₂×Dic₆₇	—	4	2	no	—
142	C₂×Dic₆₇	—	8	4	no	—
143	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
144	D₂₆₈	—	2	2	no	—
145	C₄×D₆₇	—	2	2	no	—
146	C₄×D₆₇	—	2	2	no	—
147	Dic₁₃₄	—	4	2	no	—
148	Dic₁₃₄	—	4	1	no	—
149	Dic₁₃₄	—	4	2	no	—
150	Dic₁₃₄	—	4	1	no	—
151	C₈×C₆₇	—	2	1	no	—
152	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₆₇⋊C₈	—	2	2	no	—
155	C₂²×C₆₇	—	4	2	no	—
156	C₂²×C₆₇	—	8	1	no	—
157	C₂²×C₆₇	—	8	1	no	—
158	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₆₇	—	2	2	no	—
160	C₂×D₆₇	—	2	2	no	—
161	C₂×D₆₇	—	2	2	no	—
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	8	1	no	—
164	C₄×C₆₇	—	8	2	no	—
165	C₄×C₆₇	#508	1	2	yes	—
166	C₄×C₆₇	#546	1	1	yes	characteristic
167	C₄×C₆₇	#577	1	1	yes	characteristic
168	Dic₆₇	—	2	2	no	—
169	Dic₆₇	—	8	1	no	—
170	Dic₆₇	—	8	1	no	—
171	Dic₆₇	—	8	2	no	—
172	#33869	—	67	1	no	maximal, Sylow
173	C₂×C₆₇	—	8	2	no	—
174	C₂×C₆₇	#3294	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
176	C₂×C₆₇	#3417	1	1	yes	characteristic
177	D₆₇	—	4	2	no	—
178	#3342	—	67	1	no	—
179	#3283	—	67	1	no	—
180	#2241	—	67	1	no	—
181	#2237	—	67	2	no	—
182	#2224	D₆₇	1	1	yes	characteristic, complemented
183	#2216	—	67	2	no	—
184	#2209	—	67	1	no	—
185	#2189	—	67	2	no	—
186	C₄×SD₃₂	—	67	1	no	—
187	D₄×C₁₆	—	67	1	no	—
188	#2138	—	67	2	no	—
189	C₆₇	#33869	1	1	yes	characteristic, complemented, Sylow
190	C₂×#135	—	67	2	no	—
191	#556	—	134	1	no	—
192	#547	—	134	1	no	—
193	C₂×Q₃₂	—	67	2	no	—
194	C₂×SD₃₂	—	67	1	no	—
195	C₂²×C₁₆	—	67	2	no	—
196	#500	C₂×D₆₇	1	1	yes	characteristic
197	#494	—	67	1	no	—
198	#485	—	134	2	no	—
199	#475	—	134	1	no	—
200	#467	—	67	2	no	—
201	#467	—	67	2	no	—
202	#461	—	134	1	no	—
203	#452	—	134	2	no	—
204	C₄×Q₁₆	—	67	1	no	—
205	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
206	C₈×D₄	—	67	1	no	—
207	#427	—	67	2	no	—
208	#367	—	67	1	no	—
209	#366	C₂×D₆₇	1	2	yes	—
210	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
211	#358	—	67	2	no	—
212	#358	—	67	2	no	—
213	#358	—	67	1	no	—
214	#358	—	67	2	no	—
215	#357	—	67	1	no	—
216	#357	C₂×D₆₇	1	2	yes	—
217	C₂²⋊C₁₆	—	67	2	no	—
218	C₄×C₁₆	—	67	1	no	—
219	C₂×#41	—	134	2	no	—
220	#135	—	268	2	no	—
221	#135	—	134	4	no	—
222	C₂×Q₁₆	—	67	2	no	—
223	C₂×Q₁₆	—	67	1	no	—
224	C₂×Q₁₆	—	134	1	no	—
225	C₂×SD₁₆	—	134	2	no	—
226	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
227	C₂²×C₈	—	67	2	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	134	1	no	—
230	#126	—	268	2	no	—
231	#124	—	268	1	no	—
232	#123	—	268	2	no	—
233	C₂²⋊Q₈	—	134	2	no	—
234	C₂²⋊Q₈	—	134	2	no	—
235	#121	—	268	1	no	—
236	C₄×Q₈	—	268	1	no	—
237	C₄×Q₈	—	134	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	67	1	no	—
240	#117	—	134	2	no	—
241	C₂×C₄⋊C₄	—	268	1	no	—
242	Q₃₂	—	268	2	no	—
243	SD₃₂	—	134	2	no	—
244	C₂×C₁₆	—	268	2	no	—
245	C₂×C₁₆	—	67	1	no	—
246	C₂×C₁₆	—	67	1	no	—
247	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	—	67	1	no	—
250	#107	C₂²×D₆₇	1	2	yes	—
251	#107	C₂²×D₆₇	1	1	yes	characteristic
252	#106	—	134	2	no	—
253	C₄⋊C₈	—	67	1	no	—
254	#103	—	134	2	no	—
255	#103	—	134	2	no	—
256	#103	—	134	1	no	—
257	#102	—	2	1	no	—
258	C₂²⋊C₈	—	67	2	no	—
259	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
260	#41	—	268	4	no	—
261	#41	—	536	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	134	1	no	—
264	C₂×Q₈	—	134	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	268	2	no	—
267	C₂×D₄	—	67	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	268	2	no	—
270	C₂²×C₄	—	67	2	no	—
271	Q₁₆	—	536	1	no	—
272	Q₁₆	—	134	1	no	—
273	Q₁₆	—	134	2	no	—
274	Q₁₆	—	134	1	no	—
275	Q₁₆	—	134	2	no	—
276	SD₁₆	—	268	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#11761629	1	2	yes	—
279	C₂×C₈	—	67	2	no	—
280	C₂×C₈	—	134	2	no	—
281	C₂×C₈	—	134	2	no	—
282	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
283	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	268	2	no	—
288	C₄⋊C₄	—	268	2	no	—
289	C₄⋊C₄	—	268	1	no	—
290	C₄⋊C₄	—	268	1	no	—
291	C₄⋊C₄	—	268	1	no	—
292	C₄⋊C₄	—	67	1	no	—
293	C₄⋊C₄	—	134	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	268	2	no	—
296	C₂²⋊C₄	—	268	2	no	—
297	C₂²⋊C₄	—	268	2	no	—
298	C₂²⋊C₄	—	67	2	no	—
299	C₄²	—	268	2	no	—
300	C₄²	—	268	1	no	—
301	C₄²	D₄×D₆₇	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	—	134	1	no	—
304	C₁₆	—	134	1	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	67	2	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	268	2	no	—
309	Q₈	—	268	1	no	—
310	Q₈	—	268	2	no	—
311	Q₈	—	268	1	no	—
312	D₄	—	2	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	536	4	no	—
315	D₄	—	134	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	268	1	no	—
320	C₂×C₄	—	268	1	no	—
321	C₂×C₄	—	268	2	no	—
322	C₂×C₄	—	536	4	no	—
323	C₂×C₄	—	67	2	no	—
324	C₂×C₄	—	134	2	no	—
325	C₂×C₄	—	134	2	no	—
326	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
327	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	134	2	no	—
331	C₈	C₂×#11761629	1	1	yes	characteristic
332	C₈	#14524089	1	1	yes	characteristic
333	C₂²	—	4	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	134	2	no	—
337	C₂²	—	134	2	no	—
338	C₂²	—	134	2	no	—
339	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	536	1	no	—
344	C₄	—	536	2	no	—
345	C₄	—	536	1	no	—
346	C₄	—	134	2	no	—
347	C₄	#16863825	1	2	yes	—
348	C₄	#16864037	1	1	yes	characteristic
349	C₄	#16864205	1	1	yes	characteristic
350	C₂	—	8	2	no	—
351	C₂	—	268	2	no	—
352	C₂	#22572019	1	1	yes	characteristic
353	C₂	C₂×#16863825	1	1	yes	characteristic
354	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×#2224	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22571930	C₂
hypercenter	#2224	D₆₇
hypocenter	C₆₇	#33869
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2224

Complements

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

N	H ≅ G/N	type
1. #22572639	C₂	semidirect
3. #22562438	C₂	semidirect
4. #22562419	C₂	semidirect
5. #22562301	C₂	semidirect
6. #22562260	C₂	semidirect
7. #22562150	C₂	semidirect
8. C₄×#14523987	C₂	semidirect
9. #22562028	C₂	semidirect
10. #22561913	C₂	semidirect
11. C₆₇×#2224	C₂	semidirect
182. #2224	D₆₇	semidirect
189. C₆₇	#33869	semidirect

Maximal subgroups

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

subgroup	index
#22572639	2
#22571930	2
#22562438	2
#22562419	2
#22562301	2
#22562260	2
#22562150	2
C₄×#14523987	2
#22562028	2
#22561913	2
C₆₇×#2224	2
#33869	67