Subgroups — #30772383

Order 17152.

420 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863220	C₂	1	2	yes	maximal, complemented
2	#22561820	C₂	1	1	yes	maximal, characteristic
3	#22560234	C₂	1	1	yes	maximal, characteristic
4	#22558309	C₂	1	2	yes	maximal, complemented
5	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523975	C₂²	1	1	yes	characteristic
8	C₂×#14523975	C₂²	1	1	yes	characteristic
9	#16863256	—	2	1	no	—
10	#16863220	—	2	4	no	—
11	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	complemented
12	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	complemented
13	#16863159	—	2	1	no	—
14	C₂²×M₁₆×C₆₇	—	2	1	no	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₇	—	2	1	no	—
17	C₂×#108×C₆₇	—	2	4	no	—
18	C₂×#108×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₆₇	—	2	1	no	—
20	C₂×C₄⋊Dic₆₇	—	2	1	no	—
21	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
22	#14523983	—	2	2	no	—
23	#14523983	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
25	#14523975	—	2	2	no	—
26	#14523975	—	2	2	no	—
27	C₆₇⋊#95	—	4	1	no	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	4	2	no	—
32	C₂×M₁₆×C₆₇	—	4	1	no	—
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂²×C₈×C₆₇	—	4	1	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	#108×C₆₇	—	2	4	no	—
43	#108×C₆₇	—	2	2	no	—
44	#108×C₆₇	D₄	1	4	yes	—
45	C₂²×Dic₆₇	—	4	1	no	—
46	C₂²×Dic₆₇	—	4	1	no	—
47	C₄⋊Dic₆₇	—	4	1	no	—
48	C₄⋊Dic₆₇	—	4	1	no	—
49	C₄⋊Dic₆₇	—	4	1	no	—
50	C₄⋊Dic₆₇	—	4	1	no	—
51	C₂²⋊Dic₆₇	—	8	2	no	—
52	C₆₇⋊C₁₆	—	2	4	no	—
53	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	—	2	1	no	—
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₆₇	—	2	4	no	—
65	M₁₆×C₆₇	—	4	4	no	—
66	M₁₆×C₆₇	—	4	2	no	—
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	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	—	2	2	no	—
73	C₂×C₈×C₆₇	—	4	4	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₆₇	#41	1	2	yes	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	2	1	no	—
83	C₂³×C₆₇	—	2	1	no	—
84	C₂³×C₆₇	—	4	1	no	—
85	C₂³×C₆₇	#106	1	1	yes	characteristic
86	C₂³×C₆₇	#106	1	1	yes	characteristic
87	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
106	C₂×Dic₆₇	—	8	1	no	—
107	C₂×Dic₆₇	—	8	1	no	—
108	C₂×Dic₆₇	—	8	1	no	—
109	C₂×Dic₆₇	—	8	1	no	—
110	C₂×Dic₆₇	—	8	1	no	—
111	C₂×Dic₆₇	—	8	1	no	—
112	C₈×C₆₇	—	2	1	no	—
113	C₈×C₆₇	—	2	1	no	—
114	C₈×C₆₇	—	4	4	no	—
115	C₈×C₆₇	C₄×D₄	1	2	yes	—
116	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	—	2	1	no	—
120	C₂²×C₆₇	—	2	1	no	—
121	C₂²×C₆₇	—	2	1	no	—
122	C₂²×C₆₇	—	2	1	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	4	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	#365	1	2	yes	complemented
127	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
128	C₂²×C₆₇	#448	1	1	yes	characteristic
129	C₂²×C₆₇	#481	1	1	yes	characteristic
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	#385	1	1	yes	characteristic
133	C₄×C₆₇	#428	1	1	yes	characteristic
134	C₄×C₆₇	#450	1	1	yes	characteristic
135	C₄×C₆₇	#483	1	1	yes	characteristic
136	Dic₆₇	—	16	1	no	—
137	Dic₆₇	—	16	1	no	—
138	#12928	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	—	2	1	no	—
141	C₂×C₆₇	—	2	2	no	—
142	C₂×C₆₇	—	4	1	no	—
143	C₂×C₆₇	#1853	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#365	1	1	yes	characteristic
145	C₂×C₆₇	#2155	1	1	yes	characteristic
146	C₂²×#108	D₆₇	1	1	yes	characteristic
147	C₂×#365	—	67	2	no	—
148	#2112	—	67	1	no	—
149	#1807	—	67	1	no	—
150	#1383	—	67	2	no	—
151	C₆₇	#12928	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	—	2	1	no	—
153	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₂×M₃₂	—	67	2	no	—
155	C₂×#108	—	2	1	no	—
156	C₂×#108	—	2	4	no	—
157	C₂×#108	C₂×D₆₇	1	2	yes	—
158	C₂×#106	—	67	1	no	—
159	C₂×#106	—	67	1	no	—
160	#380	—	134	1	no	—
161	#365	—	134	4	no	—
162	C₂²⋊C₁₆	—	67	2	no	—
163	C₂²⋊C₁₆	—	67	2	no	—
164	#340	—	134	1	no	—
165	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	4	2	no	—
169	C₂×M₁₆	—	4	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	4	1	no	—
173	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
174	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
175	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
179	C₂×C₄⋊C₄	—	134	1	no	—
180	C₂×C₄⋊C₄	—	134	1	no	—
181	C₂×C₂²⋊C₄	—	268	1	no	—
182	M₃₂	—	134	2	no	—
183	M₃₂	—	134	2	no	—
184	C₂×C₁₆	—	67	4	no	—
185	#108	—	2	4	no	—
186	#108	—	2	2	no	—
187	#108	C₂²⋊D₆₇	1	4	yes	—
188	#106	—	134	2	no	—
189	#106	—	134	2	no	—
190	#95	—	268	1	no	—
191	C₂⁴	#11761615	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	268	1	no	—
197	C₂²×C₄	—	268	1	no	—
198	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	#11761615	1	1	yes	characteristic
201	C₂²×C₄	#11761616	1	1	yes	characteristic
202	C₂²×C₄	#11761616	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
204	M₁₆	—	2	4	no	—
205	M₁₆	—	4	4	no	—
206	M₁₆	—	4	2	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
216	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
219	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
220	C₂×C₈	#11761628	1	2	yes	—
221	C₄⋊C₄	—	268	1	no	—
222	C₄⋊C₄	—	268	1	no	—
223	C₄⋊C₄	—	268	1	no	—
224	C₄⋊C₄	—	268	1	no	—
225	C₂²⋊C₄	—	536	2	no	—
226	C₁₆	—	134	4	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#14523974	1	1	yes	characteristic
232	C₂³	#14523974	1	1	yes	characteristic
233	C₂³	C₂×#11761615	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	536	1	no	—
245	C₂×C₄	—	536	1	no	—
246	C₂×C₄	—	536	1	no	—
247	C₂×C₄	—	536	1	no	—
248	C₂×C₄	—	536	1	no	—
249	C₂×C₄	—	536	1	no	—
250	C₂×C₄	#14523974	1	1	yes	characteristic
251	C₂×C₄	#14523974	1	1	yes	characteristic
252	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
253	C₂×C₄	#14524012	1	1	yes	characteristic
254	C₂×C₄	#14524026	1	1	yes	characteristic
255	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
256	C₂×C₄	#14524069	1	1	yes	characteristic
257	C₂×C₄	#14524074	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	4	no	—
261	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
262	C₈	#14524022	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	#16863220	1	2	yes	complemented
273	C₂²	C₂×#14523974	1	1	yes	characteristic
274	C₂²	#16863535	1	1	yes	characteristic
275	C₂²	#16863714	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	1072	1	no	—
279	C₄	—	1072	1	no	—
280	C₄	#16863269	1	1	yes	characteristic
281	C₄	#16863436	1	1	yes	characteristic
282	C₄	#16863547	1	1	yes	characteristic
283	C₄	#16863726	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	#22560402	1	1	yes	characteristic
289	C₂	C₂×#16863220	1	1	yes	characteristic
290	C₂	#22561983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523974
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	#12928
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
1. C₂×#16863220	C₂	semidirect
4. #22558309	C₂	semidirect
11. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
12. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
126. C₂²×C₆₇	#365	semidirect
151. C₆₇	#12928	semidirect
272. C₂²	#16863220	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863220	2
#22561820	2
#22560234	2
#22558309	2
C₂²×#108×C₆₇	2
#12928	67