Subgroups — #30748935

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560337	C₂	1	1	yes	maximal, characteristic, complemented
2	#22560209	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558093	C₂	1	2	yes	maximal, complemented
5	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14523964	—	2	2	no	—
8	#16863266	C₂²	1	2	yes	—
9	#16863159	—	2	1	no	—
10	#16863159	—	2	1	no	—
11	#16863158	—	2	2	no	—
12	#16863149	—	2	2	no	—
13	C₂²×M₁₆×C₆₇	—	2	1	no	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	#436×C₆₇	—	2	2	no	—
16	#436×C₆₇	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523964	—	4	4	no	—
27	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	—	2	1	no	—
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	—	4	4	no	—
31	C₂²×C₈×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₄×D₄×C₆₇	—	2	1	no	—
35	C₄×D₄×C₆₇	—	2	1	no	—
36	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₆₇	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	1	no	—
42	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₆₇	—	2	4	no	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₂²×Dic₆₇	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	2	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	8	2	no	—
55	C₂⁴×C₆₇	—	2	1	no	—
56	C₂×D₄×C₆₇	—	2	1	no	—
57	C₂×D₄×C₆₇	—	2	2	no	—
58	C₂×D₄×C₆₇	—	2	2	no	—
59	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
68	M₁₆×C₆₇	—	4	8	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²⋊C₄×C₆₇	—	2	2	no	—
78	C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₆₇	#41	1	1	yes	characteristic
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	—	4	2	no	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	—	4	1	no	—
87	C₂³×C₆₇	—	4	1	no	—
88	C₂³×C₆₇	#121	1	1	yes	characteristic
89	D₄×C₆₇	—	2	4	no	—
90	D₄×C₆₇	—	4	1	no	—
91	D₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#135	1	2	yes	—
106	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×C₄×C₆₇	#137	1	2	yes	—
109	C₂×Dic₆₇	—	8	2	no	—
110	C₂×Dic₆₇	—	8	2	no	—
111	C₂×Dic₆₇	—	8	2	no	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	—	2	2	no	—
114	C₈×C₆₇	—	4	2	no	—
115	C₆₇⋊C₈	—	8	2	no	—
116	C₂²×C₆₇	—	4	2	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	—	8	2	no	—
120	C₂²×C₆₇	—	8	2	no	—
121	C₂²×C₆₇	#380	1	1	yes	characteristic
122	C₂²×C₆₇	#463	1	2	yes	—
123	C₂²×C₆₇	#468	1	2	yes	—
124	C₂²×C₆₇	#472	1	1	yes	characteristic
125	C₂²×C₆₇	#473	1	1	yes	characteristic
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	4	2	no	—
129	C₄×C₆₇	#416	1	1	yes	characteristic
130	C₄×C₆₇	#418	1	1	yes	characteristic
131	C₄×C₆₇	#428	1	2	yes	—
132	C₄×C₆₇	#469	1	1	yes	characteristic
133	C₄×C₆₇	#470	1	1	yes	characteristic
134	C₄×C₆₇	#478	1	2	yes	—
135	Dic₆₇	—	16	2	no	—
136	#8654	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	8	2	no	—
138	C₂×C₆₇	#1697	1	2	yes	—
139	C₂×C₆₇	#1699	1	2	yes	—
140	C₂×C₆₇	#1804	1	1	yes	characteristic
141	C₂×C₆₇	#1809	1	1	yes	characteristic
142	C₂×C₆₇	#1816	1	1	yes	characteristic
143	C₂×#436	D₆₇	1	1	yes	characteristic
144	#1834	—	67	1	no	—
145	#1802	—	67	2	no	—
146	C₂×#335	—	67	1	no	—
147	#1296	—	67	2	no	—
148	C₆₇	#8654	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	2	1	no	—
150	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
151	#436	—	2	2	no	—
152	#436	Dic₆₇	1	4	yes	complemented
153	C₂×C₄⋊C₈	—	67	2	no	—
154	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂×#102	—	134	2	no	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	#384	—	67	2	no	—
159	#340	—	134	1	no	—
160	#340	—	134	1	no	—
161	#340	—	134	2	no	—
162	#335	—	134	2	no	—
163	C₂²×D₄	D₂₆₈	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	4	4	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	134	2	no	—
169	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	Dic₁₃₄	1	2	yes	—
174	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
175	C₂×C₄⋊C₄	—	134	2	no	—
176	C₂×C₄⋊C₄	—	134	2	no	—
177	C₂×C₄⋊C₄	—	134	2	no	—
178	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	134	2	no	—
183	C₄⋊C₈	—	134	2	no	—
184	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
185	#102	—	268	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
188	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
189	C₂⁴	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	268	2	no	—
200	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
202	C₂²×C₄	#11761628	1	1	yes	characteristic
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	268	2	no	—
209	C₂×C₈	—	268	2	no	—
210	C₂×C₈	—	268	2	no	—
211	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
212	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	268	2	no	—
215	C₄⋊C₄	—	268	2	no	—
216	C₄⋊C₄	—	268	2	no	—
217	C₄⋊C₄	—	268	2	no	—
218	C₄⋊C₄	—	536	2	no	—
219	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
224	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
225	C₄²	#11761628	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#14524017	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	536	2	no	—
243	C₂×C₄	—	536	2	no	—
244	C₂×C₄	—	536	2	no	—
245	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
247	C₂×C₄	#14524002	1	1	yes	characteristic
248	C₂×C₄	#14524017	1	1	yes	characteristic
249	C₂×C₄	#14524022	1	1	yes	characteristic
250	C₂×C₄	#14524022	1	1	yes	characteristic
251	C₂×C₄	#14524063	1	2	yes	—
252	C₂×C₄	#14524069	1	2	yes	—
253	C₂×C₄	#14524071	1	2	yes	—
254	C₂×C₄	#14524076	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	536	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	#16863256	1	1	yes	characteristic
265	C₂²	#16863611	1	2	yes	—
266	C₂²	#16863642	1	2	yes	—
267	C₂²	#16863666	1	1	yes	characteristic
268	C₂²	#16863671	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	1072	2	no	—
273	C₄	#16863392	1	1	yes	characteristic
274	C₄	#16863401	1	1	yes	characteristic
275	C₄	#16863438	1	2	yes	—
276	C₄	#16863649	1	1	yes	characteristic
277	C₄	#16863654	1	1	yes	characteristic
278	C₄	#16863696	1	2	yes	—
279	C₂	—	8	2	no	—
280	C₂	#22559742	1	2	yes	—
281	C₂	#22559756	1	2	yes	—
282	C₂	#22560219	1	1	yes	characteristic
283	C₂	#22560242	1	1	yes	characteristic
284	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#436×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂²×D₄×C₆₇	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#436	D₆₇
hypocenter	C₆₇	#8654
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22560337	C₂	semidirect
3. C₂×#16863149	C₂	semidirect
4. #22558093	C₂	semidirect
16. #436×C₆₇	C₄	semidirect
148. C₆₇	#8654	semidirect
152. #436	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560337	2
#22560209	2
C₂×#16863149	2
#22558093	2
C₂×#436×C₆₇	2
#8654	67