Subgroups — #30827567

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569338	C₂	1	2	yes	maximal, complemented
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22559124	C₂	1	2	yes	maximal, complemented
4	#22559054	C₂	1	4	yes	maximal, complemented
5	#22559029	C₂	1	2	yes	maximal, complemented
6	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558649	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₄	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	2	yes	—
11	C₈×Dic₁₃₄	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	#16863471	—	2	4	no	—
20	#16863448	—	2	2	no	—
21	#16863434	C₄	1	1	yes	characteristic
22	#16863420	C₂²	1	2	yes	—
23	#16863418	—	2	2	no	—
24	#16863393	C₄	1	2	yes	—
25	#16863147	C₂²	1	2	yes	—
26	#16863147	C₂²	1	2	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#16863125	C₂²	1	4	yes	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×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×#11761612	—	2	2	no	—
40	C₂×#11761611	—	2	4	no	—
41	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₃₄	C₂³	1	2	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
44	C₄×D₂₆₈	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₆₇	—	2	2	no	—
46	C₄×C₂²⋊D₆₇	—	2	2	no	—
47	#14524002	C₂×C₄	1	2	yes	—
48	#14524000	—	2	2	no	—
49	C₄²×D₆₇	—	2	2	no	—
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	#14523977	C₂×C₄	1	1	yes	characteristic
52	#14523975	C₂×C₄	1	2	yes	—
53	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	#14523968	C₂×C₄	1	2	yes	—
57	#14523968	C₂×C₄	1	2	yes	—
58	#14523965	C₂×C₄	1	2	yes	—
59	#14523965	C₂×C₄	1	2	yes	—
60	(C₂×D₆₇)⋊C₈	—	2	4	no	—
61	C₈×Dic₆₇	—	2	4	no	—
62	C₂×M₁₆×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	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
67	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
68	#97×C₆₇	—	2	1	no	—
69	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
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	—	4	2	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	M₁₆	1	4	yes	complemented
84	#11761617	—	2	4	no	—
85	D₅₃₆	—	2	2	no	—
86	#11761612	—	8	2	no	—
87	#11761611	—	8	4	no	—
88	C₂×C₆₇⋊C₈	—	2	4	no	—
89	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
91	Dic₆₇⋊C₄	—	2	2	no	—
92	Dic₆₇⋊C₄	—	2	2	no	—
93	C₂²⋊Dic₆₇	—	2	2	no	—
94	(C₂×D₆₇)⋊C₄	—	2	2	no	—
95	(C₂×D₆₇)⋊C₄	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₄×Dic₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
102	M₁₆×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	4	1	no	—
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	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
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	2	no	—
135	C₄×D₆₇	—	4	2	no	—
136	C₄×D₆₇	—	4	4	no	—
137	Dic₁₃₄	—	2	2	no	—
138	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	4	no	—
143	C₈×C₆₇	—	4	2	no	—
144	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
145	C₆₇⋊C₈	—	4	4	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₆₇	—	4	2	no	—
151	C₂×D₆₇	—	4	2	no	—
152	C₂×D₆₇	—	4	2	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	#444	1	2	yes	—
157	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₆₇	#568	1	1	yes	characteristic
159	C₄×C₆₇	#577	1	2	yes	—
160	Dic₆₇	—	2	4	no	—
161	Dic₆₇	—	4	2	no	—
162	Dic₆₇	—	4	2	no	—
163	#18063	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	4	1	no	—
165	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
166	C₂×C₆₇	#2952	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
168	D₆₇	—	8	2	no	—
169	#2991	—	67	2	no	—
170	C₄×#135	—	67	1	no	—
171	#1594	—	67	2	no	—
172	#1583	—	67	4	no	—
173	#1579	—	67	2	no	—
174	#1566	—	67	1	no	—
175	#1489	—	67	2	no	—
176	#1451	D₆₇	1	1	yes	characteristic, complemented
177	C₆₇	#18063	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	—	67	1	no	—
179	C₄×#41	—	67	2	no	—
180	C₈×Q₈	—	67	4	no	—
181	C₄×Q₁₆	—	67	1	no	—
182	C₄×Q₁₆	—	67	1	no	—
183	C₄×SD₁₆	—	67	2	no	—
184	C₄×SD₁₆	—	67	2	no	—
185	C₄×D₈	—	67	1	no	—
186	C₄×D₈	—	67	1	no	—
187	#436	—	67	4	no	—
188	#436	—	134	4	no	—
189	#431	—	134	2	no	—
190	#431	C₂×D₆₇	1	1	yes	characteristic
191	#427	—	67	1	no	—
192	#423	—	67	2	no	—
193	#423	—	134	2	no	—
194	#416	—	67	2	no	—
195	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
197	#334	—	67	2	no	—
198	#334	—	67	2	no	—
199	Q₈⋊C₈	—	67	4	no	—
200	D₄⋊C₈	—	67	4	no	—
201	#322	C₂×D₆₇	1	2	yes	—
202	#322	C₂×D₆₇	1	2	yes	—
203	C₂×#41	—	67	2	no	—
204	#135	—	134	4	no	—
205	C₂×Q₁₆	—	67	1	no	—
206	C₂×SD₁₆	—	67	2	no	—
207	C₂×D₈	—	67	1	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	134	2	no	—
210	C₂×M₁₆	—	134	4	no	—
211	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
213	C₄×Q₈	—	67	2	no	—
214	C₄×Q₈	—	67	2	no	—
215	C₄×D₄	—	67	2	no	—
216	C₄×D₄	—	67	2	no	—
217	C₄×D₄	—	134	2	no	—
218	C₄×D₄	—	134	2	no	—
219	#117	—	67	2	no	—
220	#117	—	134	2	no	—
221	C₂×C₄²	—	134	2	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	#107	—	67	1	no	—
224	#107	—	67	1	no	—
225	#106	—	67	2	no	—
226	C₄⋊C₈	—	67	4	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	2	no	—
229	C₄⋊C₈	D₂₆₈	1	2	yes	—
230	#103	—	67	2	no	—
231	#103	—	67	2	no	—
232	#102	—	67	2	no	—
233	#102	—	67	2	no	—
234	C₂²⋊C₈	—	134	4	no	—
235	C₂²⋊C₈	D₂₆₈	1	2	yes	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	134	4	no	—
238	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
239	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₇	1	2	yes	—
241	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₆₇	1	2	yes	—
244	#41	—	268	2	no	—
245	#41	—	134	4	no	—
246	C₂×Q₈	—	67	2	no	—
247	C₂×D₄	—	67	2	no	—
248	C₂×D₄	—	134	2	no	—
249	C₂²×C₄	—	134	2	no	—
250	C₂²×C₄	—	134	2	no	—
251	C₂²×C₄	—	134	2	no	—
252	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
254	C₂²×C₄	#11761628	1	1	yes	characteristic
255	Q₁₆	—	67	4	no	—
256	SD₁₆	—	134	4	no	—
257	D₈	—	134	2	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	536	4	no	—
260	M₁₆	—	536	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	134	4	no	—
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×C₄×D₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
270	C₄⋊C₄	—	67	2	no	—
271	C₄⋊C₄	—	67	2	no	—
272	C₄⋊C₄	—	134	2	no	—
273	C₄⋊C₄	—	134	2	no	—
274	C₂²⋊C₄	—	134	2	no	—
275	C₂²⋊C₄	—	134	2	no	—
276	C₂²⋊C₄	—	134	2	no	—
277	C₄²	—	134	2	no	—
278	C₄²	—	134	2	no	—
279	C₄²	—	134	2	no	—
280	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
281	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
282	C₄²	#11761628	1	1	yes	characteristic
283	C₄²	C₂³×D₆₇	1	1	yes	characteristic
284	C₂³	—	134	2	no	—
285	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
286	Q₈	—	67	4	no	—
287	Q₈	—	134	2	no	—
288	D₄	—	268	4	no	—
289	D₄	—	134	2	no	—
290	D₄	—	134	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	268	4	no	—
296	C₂×C₄	—	268	4	no	—
297	C₂×C₄	—	268	2	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	134	2	no	—
300	C₂×C₄	—	134	2	no	—
301	C₂×C₄	—	134	2	no	—
302	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
303	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
304	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
307	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	268	4	no	—
314	C₈	M₁₆×D₆₇	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	268	2	no	—
319	C₂²	—	268	2	no	—
320	C₂²	—	268	2	no	—
321	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	268	2	no	—
326	C₄	—	268	2	no	—
327	C₄	—	134	4	no	—
328	C₄	#16863519	1	2	yes	—
329	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
330	C₄	#16864162	1	1	yes	characteristic
331	C₄	#16864208	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	536	2	no	—
334	C₂	M₁₆×D₂₆₈	1	1	yes	characteristic
335	C₂	#22569094	1	1	yes	characteristic
336	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#18063
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #22569338	C₂	semidirect
3. #22559124	C₂	semidirect
4. #22559054	C₂	semidirect
5. #22559029	C₂	semidirect
6. #22558965	C₂	semidirect
7. #22558649	C₂	semidirect
8. C₆₇×#1451	C₂	semidirect
83. Dic₂₆₈	M₁₆	semidirect
176. #1451	D₆₇	semidirect
177. C₆₇	#18063	semidirect

Maximal subgroups

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

subgroup	index
#22569338	2
C₄×#14524065	2
#22559124	2
#22559054	2
#22559029	2
#22558965	2
#22558649	2
C₆₇×#1451	2
#18063	67