Subgroups — #30772785

Order 17152.

530 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863200	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#22558300	C₂	1	2	yes	maximal, complemented
5	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524002	—	2	1	no	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#16863434	—	2	4	no	—
9	C₂×#14523977	C₂²	1	2	yes	—
10	C₂×#14523975	—	2	1	no	—
11	#16863200	—	2	4	no	—
12	C₂²⋊(C₆₇⋊C₁₆)	—	2	2	no	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
16	#467×C₆₇	—	2	2	no	—
17	#467×C₆₇	C₄	1	4	yes	complemented
18	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#103×C₆₇	—	2	1	no	—
20	#14524002	—	2	4	no	—
21	#14524002	—	4	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
24	C₂×C₄×Dic₆₇	—	4	1	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₆₇⋊C₁₆	—	2	2	no	—
28	#14523977	—	2	2	no	—
29	#14523977	D₄	1	4	yes	—
30	#14523975	—	2	4	no	—
31	C₂²×Q₈×C₆₇	—	2	1	no	—
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	1	no	—
35	C₂×Q₁₆×C₆₇	—	2	2	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₆₇	—	2	4	no	—
44	C₂²⋊Q₈×C₆₇	—	4	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
48	#107×C₆₇	—	2	1	no	—
49	#107×C₆₇	C₂×C₄	1	2	yes	—
50	#103×C₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	2	4	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₂²⋊Dic₆₇	—	4	4	no	—
55	C₄×Dic₆₇	—	4	4	no	—
56	C₆₇⋊C₁₆	—	4	2	no	—
57	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	—	4	1	no	—
60	C₂×Q₈×C₆₇	—	4	1	no	—
61	C₂×Q₈×C₆₇	—	4	1	no	—
62	C₂×Q₈×C₆₇	—	4	1	no	—
63	C₂×Q₈×C₆₇	—	4	1	no	—
64	C₂²×C₄×C₆₇	—	2	1	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	4	1	no	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₆₇	—	2	4	no	—
75	Q₁₆×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	—	2	1	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	#41	1	2	yes	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	4	4	no	—
87	C₄⋊C₄×C₆₇	—	4	1	no	—
88	C₂²⋊C₄×C₆₇	—	4	4	no	—
89	C₂³×C₆₇	—	2	1	no	—
90	C₂³×C₆₇	—	2	1	no	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	#102	1	2	yes	—
94	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₆₇	—	4	2	no	—
96	Q₈×C₆₇	—	4	2	no	—
97	Q₈×C₆₇	—	8	1	no	—
98	Q₈×C₆₇	—	8	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	8	1	no	—
108	C₂×C₄×C₆₇	—	8	1	no	—
109	C₂×C₄×C₆₇	—	8	2	no	—
110	C₂×C₄×C₆₇	#102	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
117	C₂×Dic₆₇	—	4	4	no	—
118	C₂×Dic₆₇	—	8	2	no	—
119	C₂×Dic₆₇	—	8	2	no	—
120	C₈×C₆₇	—	2	2	no	—
121	C₈×C₆₇	C₄×D₄	1	2	yes	—
122	C₈×C₆₇	#121	1	2	yes	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	2	no	—
128	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₆₇	#447	1	1	yes	characteristic
130	C₂²×C₆₇	#480	1	1	yes	characteristic
131	C₂²×C₆₇	#508	1	2	yes	—
132	C₂²×C₆₇	#510	1	2	yes	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	—	8	2	no	—
136	C₄×C₆₇	#386	1	1	yes	characteristic
137	C₄×C₆₇	#417	1	1	yes	characteristic
138	C₄×C₆₇	#448	1	1	yes	characteristic
139	C₄×C₆₇	#481	1	1	yes	characteristic
140	Dic₆₇	—	8	4	no	—
141	#12994	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	4	2	no	—
143	C₂×C₆₇	#1874	1	1	yes	characteristic
144	C₂×C₆₇	#2138	1	1	yes	characteristic
145	C₂×C₆₇	#2141	1	1	yes	characteristic
146	C₂×C₆₇	#2189	1	2	yes	—
147	C₂×C₆₇	#2237	1	2	yes	—
148	C₂×#467	D₆₇	1	1	yes	characteristic
149	C₂×#427	—	67	1	no	—
150	C₂×#358	—	67	2	no	—
151	C₂×C₂²⋊C₁₆	—	67	1	no	—
152	#1380	—	67	2	no	—
153	C₆₇	#12994	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
155	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
156	C₂×C₂²⋊Q₈	—	2	1	no	—
157	C₂×#117	—	134	1	no	—
158	C₂²×C₁₆	—	67	2	no	—
159	#467	—	2	2	no	—
160	#467	Dic₆₇	1	4	yes	complemented
161	#427	—	134	4	no	—
162	C₂×#107	—	67	2	no	—
163	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#106	—	134	1	no	—
165	C₂×#103	—	2	1	no	—
166	#358	—	134	4	no	—
167	C₂²⋊C₁₆	—	134	2	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	—	2	1	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
177	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
179	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
180	C₂²⋊Q₈	—	2	4	no	—
181	C₂²⋊Q₈	—	4	2	no	—
182	#117	—	268	2	no	—
183	#117	—	134	4	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄⋊C₄	—	134	2	no	—
187	C₂×C₂²⋊C₄	—	4	1	no	—
188	C₂×C₂²⋊C₄	—	268	1	no	—
189	C₂×C₄²	—	268	1	no	—
190	C₂×C₁₆	—	134	2	no	—
191	C₂×C₁₆	—	134	2	no	—
192	C₂×C₁₆	—	134	2	no	—
193	#107	—	2	1	no	—
194	#107	—	67	4	no	—
195	#107	—	134	2	no	—
196	#107	C₂×Dic₆₇	1	2	yes	—
197	#106	—	134	4	no	—
198	#103	—	2	4	no	—
199	C₂⁴	#11761613	1	1	yes	characteristic
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	268	2	no	—
213	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
214	C₂²×C₄	#11761613	1	1	yes	characteristic
215	C₂²×C₄	#11761616	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	4	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
224	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
225	C₂×C₈	D₄×D₆₇	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂×C₈	#11761627	1	2	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	268	2	no	—
232	C₄⋊C₄	—	134	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	268	4	no	—
235	C₄²	—	268	4	no	—
236	C₁₆	—	268	2	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	#14523964	1	2	yes	—
242	C₂³	C₂×#11761613	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	1	no	—
246	Q₈	—	8	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	—	268	4	no	—
259	C₂×C₄	—	536	2	no	—
260	C₂×C₄	—	536	2	no	—
261	C₂×C₄	#14523964	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
263	C₂×C₄	#14524011	1	1	yes	characteristic
264	C₂×C₄	#14524024	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
266	C₂×C₄	#14524068	1	1	yes	characteristic
267	C₂×C₄	#14524073	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	D₄×Dic₆₇	1	2	yes	—
270	C₈	#14524015	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂×#14523964	1	1	yes	characteristic
277	C₂²	#16863529	1	1	yes	characteristic
278	C₂²	#16863707	1	1	yes	characteristic
279	C₂²	#16863826	1	2	yes	—
280	C₂²	#16863837	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	536	4	no	—
285	C₄	#16863276	1	1	yes	characteristic
286	C₄	#16863397	1	1	yes	characteristic
287	C₄	#16863534	1	1	yes	characteristic
288	C₄	#16863712	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	#22560527	1	1	yes	characteristic
291	C₂	#22561916	1	1	yes	characteristic
292	C₂	#22561929	1	1	yes	characteristic
293	C₂	#22562147	1	2	yes	—
294	C₂	#22562414	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#467×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#467	D₆₇
hypocenter	C₆₇	#12994
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#16863200	C₂	semidirect
4. #22558300	C₂	semidirect
17. #467×C₆₇	C₄	semidirect
153. C₆₇	#12994	semidirect
160. #467	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863434	2
C₂×#16863200	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
#22558300	2
C₂×#467×C₆₇	2
#12994	67