Subgroups — #30825610

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	2	yes	maximal, complemented
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22559124	C₂	1	2	yes	maximal, complemented
4	#22559119	C₂	1	2	yes	maximal, complemented
5	#22559113	C₂	1	4	yes	maximal, complemented
6	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558592	C₂	1	2	yes	maximal, complemented
8	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₄	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	2	yes	—
11	#16863526	C₂²	1	4	yes	—
12	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
13	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
14	C₄×#11761617	C₄	1	2	yes	—
15	C₄×#11761617	C₂²	1	2	yes	—
16	C₄×D₅₃₆	C₄	1	1	yes	characteristic
17	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
18	#16863475	C₂²	1	4	yes	—
19	#16863467	—	2	4	no	—
20	#16863449	C₂²	1	2	yes	—
21	#16863447	—	2	2	no	—
22	#16863434	C₄	1	1	yes	characteristic
23	#16863393	C₄	1	2	yes	—
24	C₄×#11761611	—	2	2	no	—
25	#16863149	C₂²	1	2	yes	—
26	#16863147	C₂²	1	2	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#16863125	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#322×C₆₇	C₂²	1	2	yes	—
33	#322×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761628	C₂×C₄	1	2	yes	—
35	#14524065	—	2	4	no	—
36	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#11761617	C₂×C₄	1	2	yes	—
38	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#11761611	—	2	4	no	—
40	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₃₄	C₂³	1	2	yes	—
42	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
43	C₄×D₂₆₈	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₆₇	—	2	2	no	—
45	C₄×C₂²⋊D₆₇	—	2	2	no	—
46	#14524002	C₂×C₄	1	2	yes	—
47	#14524000	—	2	2	no	—
48	C₄²×D₆₇	—	2	2	no	—
49	#14523977	C₂×C₄	1	1	yes	characteristic
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	#14523975	C₂×C₄	1	2	yes	—
52	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
53	Dic₆₇⋊C₈	—	2	4	no	—
54	#14523968	C₂×C₄	1	2	yes	—
55	#14523968	C₂×C₄	1	2	yes	—
56	#14523965	C₂×C₄	1	2	yes	—
57	#14523965	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
59	(C₂×D₆₇)⋊C₈	—	2	4	no	—
60	#14523953	—	2	4	no	—
61	C₄×C₆₇⋊C₈	—	2	2	no	—
62	C₂²×C₈×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	D₄	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	D₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂³	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	2	yes	—
75	#11761628	—	2	4	no	—
76	#11761628	—	2	4	no	—
77	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
78	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
79	C₂×C₂²⋊D₆₇	—	2	2	no	—
80	C₂×C₄×D₆₇	—	2	2	no	—
81	C₂×C₄×D₆₇	—	2	2	no	—
82	C₂×C₄×D₆₇	—	2	2	no	—
83	Dic₂₆₈	—	2	2	no	—
84	#11761617	—	2	4	no	—
85	D₅₃₆	—	2	2	no	—
86	#11761611	—	4	8	no	—
87	C₂×C₆₇⋊C₈	—	2	4	no	—
88	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
89	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
90	Dic₆₇⋊C₄	—	2	2	no	—
91	Dic₆₇⋊C₄	—	2	2	no	—
92	C₂²⋊Dic₆₇	—	2	2	no	—
93	(C₂×D₆₇)⋊C₄	—	2	2	no	—
94	(C₂×D₆₇)⋊C₄	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	4	2	no	—
104	C₂×C₈×C₆₇	—	4	2	no	—
105	C₂×C₈×C₆₇	M₁₆	1	4	yes	complemented
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	#41	1	1	yes	characteristic
114	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂²×D₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₆₇	#131	1	2	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂²⋊D₆₇	—	4	4	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	2	2	no	—
131	D₂₆₈	—	2	2	no	—
132	D₂₆₈	—	2	2	no	—
133	C₄×D₆₇	—	4	4	no	—
134	C₄×D₆₇	—	4	4	no	—
135	C₄×D₆₇	—	4	4	no	—
136	Dic₁₃₄	—	2	2	no	—
137	Dic₁₃₄	—	2	2	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	4	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
143	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
144	C₆₇⋊C₈	—	4	4	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	#436	1	2	yes	—
148	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×D₆₇	—	4	2	no	—
150	C₂×D₆₇	—	4	2	no	—
151	C₂×D₆₇	—	4	2	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	#436	1	2	yes	—
155	C₄×C₆₇	#444	1	2	yes	—
156	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
158	C₄×C₆₇	#575	1	2	yes	—
159	Dic₆₇	—	4	2	no	—
160	Dic₆₇	—	4	2	no	—
161	Dic₆₇	—	4	2	no	—
162	#17890	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
165	C₂×C₆₇	#2943	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
167	D₆₇	—	8	2	no	—
168	#2965	—	67	2	no	—
169	C₄×#135	—	67	1	no	—
170	#1594	—	67	2	no	—
171	#1593	—	67	2	no	—
172	#1592	—	67	4	no	—
173	#1564	—	67	1	no	—
174	#1476	—	67	2	no	—
175	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
176	C₆₇	#17890	1	1	yes	characteristic, complemented, Sylow
177	C₂×#135	—	67	1	no	—
178	C₄×#41	—	67	2	no	—
179	#446	—	67	4	no	—
180	C₄×Q₁₆	—	67	1	no	—
181	C₄×Q₁₆	—	67	1	no	—
182	C₄×SD₁₆	—	67	2	no	—
183	C₄×SD₁₆	—	67	2	no	—
184	C₄×D₈	—	67	1	no	—
185	C₄×D₈	—	67	1	no	—
186	#436	—	67	4	no	—
187	#435	—	134	4	no	—
188	#431	—	67	2	no	—
189	#431	—	134	2	no	—
190	#427	—	67	1	no	—
191	#416	—	67	2	no	—
192	C₄×M₁₆	—	134	2	no	—
193	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
196	#335	—	67	2	no	—
197	#334	—	67	2	no	—
198	Q₈⋊C₈	—	67	4	no	—
199	D₄⋊C₈	—	67	4	no	—
200	#322	C₂×D₆₇	1	2	yes	—
201	#322	C₂×D₆₇	1	2	yes	—
202	C₂×#41	—	67	2	no	—
203	#135	—	134	4	no	—
204	C₂×Q₁₆	—	67	1	no	—
205	C₂×SD₁₆	—	67	2	no	—
206	C₂×D₈	—	67	1	no	—
207	C₂×M₁₆	—	134	4	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
211	C₂²×C₈	D₂₆₈	1	2	yes	—
212	C₄×Q₈	—	67	2	no	—
213	C₄×Q₈	—	67	2	no	—
214	C₄×D₄	—	67	2	no	—
215	C₄×D₄	—	67	2	no	—
216	C₄×D₄	—	134	2	no	—
217	C₄×D₄	—	134	2	no	—
218	#117	—	67	2	no	—
219	#117	—	134	2	no	—
220	C₂×C₄²	—	134	2	no	—
221	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
222	#107	—	67	1	no	—
223	#107	—	67	1	no	—
224	#106	—	67	2	no	—
225	C₄⋊C₈	—	67	4	no	—
226	C₄⋊C₈	—	134	4	no	—
227	#103	—	67	2	no	—
228	#103	—	67	2	no	—
229	#102	—	67	2	no	—
230	#102	—	67	2	no	—
231	C₂²⋊C₈	—	134	2	no	—
232	C₂²⋊C₈	—	134	4	no	—
233	#97	—	134	4	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	134	2	no	—
236	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
237	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
238	C₄×C₈	D₂₆₈	1	2	yes	—
239	C₄×C₈	C₂²×D₆₇	1	2	yes	—
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₆₇	1	2	yes	—
243	#41	—	134	4	no	—
244	#41	—	134	4	no	—
245	C₂×Q₈	—	67	2	no	—
246	C₂×D₄	—	67	2	no	—
247	C₂×D₄	—	134	2	no	—
248	C₂²×C₄	—	134	2	no	—
249	C₂²×C₄	—	134	2	no	—
250	C₂²×C₄	—	134	2	no	—
251	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
253	C₂²×C₄	#11761628	1	1	yes	characteristic
254	Q₁₆	—	134	2	no	—
255	SD₁₆	—	134	4	no	—
256	D₈	—	134	2	no	—
257	M₁₆	—	268	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	134	4	no	—
263	C₂×C₈	#11761611	1	4	yes	complemented
264	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
269	C₄⋊C₄	—	67	2	no	—
270	C₄⋊C₄	—	67	2	no	—
271	C₄⋊C₄	—	134	2	no	—
272	C₄⋊C₄	—	134	2	no	—
273	C₂²⋊C₄	—	134	2	no	—
274	C₂²⋊C₄	—	134	2	no	—
275	C₂²⋊C₄	—	134	2	no	—
276	C₄²	—	134	2	no	—
277	C₄²	—	134	2	no	—
278	C₄²	—	134	2	no	—
279	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
280	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
281	C₄²	#11761628	1	1	yes	characteristic
282	C₄²	C₂³×D₆₇	1	1	yes	characteristic
283	C₂³	—	134	2	no	—
284	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
285	Q₈	—	134	2	no	—
286	Q₈	—	134	2	no	—
287	D₄	—	268	4	no	—
288	D₄	—	134	2	no	—
289	D₄	—	134	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	268	4	no	—
293	C₂×C₄	—	268	4	no	—
294	C₂×C₄	—	268	4	no	—
295	C₂×C₄	—	134	2	no	—
296	C₂×C₄	—	134	2	no	—
297	C₂×C₄	—	134	2	no	—
298	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
299	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
301	C₂×C₄	C₂×#11761611	1	2	yes	—
302	C₂×C₄	#14524049	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	268	4	no	—
311	C₈	C₂×#11761611	1	2	yes	—
312	C₈	C₂×#11761611	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	268	2	no	—
316	C₂²	—	268	2	no	—
317	C₂²	—	268	2	no	—
318	C₂²	#16863475	1	2	yes	—
319	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	268	2	no	—
323	C₄	—	268	2	no	—
324	C₄	—	268	2	no	—
325	C₄	#16863475	1	2	yes	—
326	C₄	#16863519	1	2	yes	—
327	C₄	C₂²×#11761611	1	1	yes	characteristic
328	C₄	C₂×#14524049	1	1	yes	characteristic
329	C₄	#16864196	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	536	2	no	—
332	C₂	C₂×#16863475	1	1	yes	characteristic
333	C₂	#22569011	1	1	yes	characteristic
334	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#17890
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
1. #22569175	C₂	semidirect
3. #22559124	C₂	semidirect
4. #22559119	C₂	semidirect
5. #22559113	C₂	semidirect
6. #22558957	C₂	semidirect
7. #22558592	C₂	semidirect
8. C₂×#322×C₆₇	C₂	semidirect
105. C₂×C₈×C₆₇	M₁₆	semidirect
175. C₂×#322	D₆₇	semidirect
176. C₆₇	#17890	semidirect
263. C₂×C₈	#11761611	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
C₄×#14524065	2
#22559124	2
#22559119	2
#22559113	2
#22558957	2
#22558592	2
C₂×#322×C₆₇	2
#17890	67