Subgroups — #30766903

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#22560238	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558218	C₂	1	4	yes	maximal, complemented
4	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
6	C₂×#14523977	—	2	1	no	—
7	#16863256	—	2	1	no	—
8	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	complemented
10	#16863159	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#121×C₆₇	—	2	1	no	—
14	#466×C₆₇	—	2	1	no	—
15	#466×C₆₇	—	2	1	no	—
16	#466×C₆₇	C₄	1	2	yes	complemented
17	#466×C₆₇	C₂²	1	2	yes	—
18	C₂×#107×C₆₇	C₄	1	1	yes	characteristic, complemented
19	C₂×#102×C₆₇	—	2	1	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	4	2	no	—
24	#14523977	—	4	2	no	—
25	C₆₇⋊#95	—	4	1	no	—
26	C₂²×D₄×C₆₇	—	2	1	no	—
27	C₂²×D₄×C₆₇	—	4	1	no	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂×D₈×C₆₇	—	2	1	no	—
30	C₂×D₈×C₆₇	—	4	2	no	—
31	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	—	4	2	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	#121×C₆₇	—	2	2	no	—
40	#121×C₆₇	—	2	2	no	—
41	#121×C₆₇	—	4	1	no	—
42	#121×C₆₇	—	4	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
45	#107×C₆₇	—	2	1	no	—
46	#107×C₆₇	C₂×C₄	1	2	yes	—
47	#102×C₆₇	—	2	2	no	—
48	#102×C₆₇	—	2	2	no	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	2	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₂²⋊Dic₆₇	—	8	2	no	—
53	C₆₇⋊C₁₆	—	4	2	no	—
54	C₂⁴×C₆₇	—	4	1	no	—
55	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂×D₄×C₆₇	—	4	2	no	—
60	C₂×D₄×C₆₇	—	4	2	no	—
61	C₂×D₄×C₆₇	—	4	1	no	—
62	C₂×D₄×C₆₇	—	8	2	no	—
63	C₂×D₄×C₆₇	—	8	2	no	—
64	C₂²×C₄×C₆₇	—	4	1	no	—
65	C₂²×C₄×C₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	—	4	1	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	D₈×C₆₇	—	4	2	no	—
74	D₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	4	1	no	—
79	C₂×C₈×C₆₇	—	4	1	no	—
80	C₂×C₈×C₆₇	—	4	1	no	—
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	4	1	no	—
87	C₂²⋊C₄×C₆₇	—	4	2	no	—
88	C₂²⋊C₄×C₆₇	—	4	2	no	—
89	C₂³×C₆₇	—	4	1	no	—
90	C₂³×C₆₇	—	4	1	no	—
91	C₂³×C₆₇	—	4	2	no	—
92	C₂³×C₆₇	—	8	2	no	—
93	C₂³×C₆₇	—	8	2	no	—
94	C₂³×C₆₇	—	8	2	no	—
95	C₂³×C₆₇	#102	1	1	yes	characteristic
96	C₂³×C₆₇	#102	1	1	yes	characteristic
97	C₂³×C₆₇	D₁₆	1	2	yes	—
98	C₂³×C₆₇	SD₃₂	1	2	yes	—
99	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
100	D₄×C₆₇	—	4	2	no	—
101	D₄×C₆₇	—	4	2	no	—
102	D₄×C₆₇	—	8	2	no	—
103	D₄×C₆₇	—	8	4	no	—
104	D₄×C₆₇	—	8	2	no	—
105	D₄×C₆₇	—	8	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	8	2	no	—
117	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×Dic₆₇	—	8	2	no	—
122	C₂×Dic₆₇	—	8	2	no	—
123	C₂×Dic₆₇	—	8	2	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	#99	1	2	yes	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	—	2	1	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	—	8	2	no	—
137	C₂²×C₆₇	—	8	2	no	—
138	C₂²×C₆₇	—	8	2	no	—
139	C₂²×C₆₇	—	16	4	no	—
140	C₂²×C₆₇	#357	1	2	yes	—
141	C₂²×C₆₇	#357	1	2	yes	—
142	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
145	C₄×C₆₇	—	2	1	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	8	2	no	—
149	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
150	C₄×C₆₇	#421	1	1	yes	characteristic
151	Dic₆₇	—	16	2	no	—
152	#11886	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	16	2	no	—
158	C₂×C₆₇	#1519	1	1	yes	characteristic
159	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
160	C₂×C₆₇	#2145	1	1	yes	characteristic
161	C₂×#466	D₆₇	1	1	yes	characteristic
162	C₂×C₂²⋊C₁₆	—	67	1	no	—
163	#1808	—	67	1	no	—
164	#1347	—	67	4	no	—
165	C₆₇	#11886	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₈	Dic₆₇	1	1	yes	characteristic, complemented
167	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#121	—	2	1	no	—
169	C₂²×C₁₆	—	134	1	no	—
170	#466	—	2	1	no	—
171	#466	—	2	1	no	—
172	#466	Dic₆₇	1	2	yes	complemented
173	#466	C₂×D₆₇	1	2	yes	—
174	C₂×#107	—	134	1	no	—
175	C₂×#107	Dic₆₇	1	1	yes	characteristic, complemented
176	C₂×#102	—	2	1	no	—
177	#380	—	134	1	no	—
178	C₂²⋊C₁₆	—	67	2	no	—
179	C₂²⋊C₁₆	—	134	1	no	—
180	#340	—	67	2	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	4	1	no	—
183	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
187	C₂²×C₈	—	4	2	no	—
188	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	#121	—	2	2	no	—
195	#121	—	2	2	no	—
196	#121	—	4	1	no	—
197	#121	—	4	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	134	2	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	C₂×C₂²⋊C₄	—	268	1	no	—
202	C₂×C₁₆	—	268	2	no	—
203	C₂×C₁₆	—	134	2	no	—
204	#107	—	2	1	no	—
205	#107	—	268	2	no	—
206	#107	C₂×Dic₆₇	1	2	yes	—
207	#102	—	2	2	no	—
208	#102	—	2	2	no	—
209	#95	—	268	1	no	—
210	C₂⁴	—	4	1	no	—
211	C₂⁴	#11761613	1	1	yes	characteristic
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	8	2	no	—
219	C₂×D₄	—	8	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	268	2	no	—
224	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₂²×C₄	#11761613	1	1	yes	characteristic
227	C₂²×C₄	#11761616	1	1	yes	characteristic
228	C₂²×C₄	#11761616	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	D₈	—	4	2	no	—
231	D₈	—	4	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	268	2	no	—
245	C₄⋊C₄	—	268	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	536	2	no	—
249	C₁₆	—	268	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	#14523964	1	1	yes	characteristic
257	C₂³	#14523964	1	1	yes	characteristic
258	C₂³	#14523984	1	2	yes	—
259	C₂³	#14523987	1	2	yes	—
260	C₂³	C₂×#11761613	1	1	yes	characteristic
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	536	2	no	—
279	C₂×C₄	—	536	2	no	—
280	C₂×C₄	—	536	2	no	—
281	C₂×C₄	#14523964	1	1	yes	characteristic
282	C₂×C₄	#14523964	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	#14523958	1	2	yes	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	16	4	no	—
301	C₂²	#16863197	1	2	yes	—
302	C₂²	#16863197	1	2	yes	—
303	C₂²	C₂×#14523964	1	1	yes	characteristic
304	C₂²	C₂×#14523984	1	1	yes	characteristic
305	C₂²	C₂×#14523987	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	1072	2	no	—
311	C₄	C₂×#14523958	1	1	yes	characteristic
312	C₄	#16863413	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	16	2	no	—
318	C₂	#22558777	1	1	yes	characteristic
319	C₂	C₂×#16863197	1	1	yes	characteristic
320	C₂	#22561945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₂×#466×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#466	D₆₇
hypocenter	C₆₇	#11886
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	semidirect
2. #22560238	C₂	semidirect
3. #22558218	C₂	semidirect
9. C₂²⋊(C₆₇⋊C₁₆)	C₂²	semidirect
10. #16863159	C₂²	semidirect
11. C₂²×D₈×C₆₇	C₄	semidirect
16. #466×C₆₇	C₄	semidirect
18. C₂×#107×C₆₇	C₄	semidirect
165. C₆₇	#11886	semidirect
166. C₂²×D₈	Dic₆₇	semidirect
172. #466	Dic₆₇	semidirect
175. C₂×#107	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
#22560238	2
#22558218	2
C₂×#466×C₆₇	2
#11886	67