Subgroups — #30770713

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal, complemented
4	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	—	2	1	no	—
6	C₂×#14523983	—	2	2	no	—
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	C₄	1	4	yes	complemented
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863434	—	2	2	no	—
11	#16863434	C₄	1	4	yes	complemented
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523975	—	2	1	no	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	—	2	2	no	—
18	C₂×#107×C₆₇	C₄	1	4	yes	complemented
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
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	#14523983	—	2	8	no	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₆₇⋊C₁₆	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
29	#14523977	—	2	2	no	—
30	#14523977	C₂×C₄	1	4	yes	complemented
31	#14523975	—	2	4	no	—
32	C₂³×C₄×C₆₇	—	4	1	no	—
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₇	D₄	1	2	yes	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
46	#107×C₆₇	—	2	4	no	—
47	#107×C₆₇	—	2	2	no	—
48	#107×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	2	4	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₂²⋊Dic₆₇	—	4	4	no	—
53	C₄×Dic₆₇	—	4	4	no	—
54	C₆₇⋊C₁₆	—	2	4	no	—
55	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	4	4	no	—
57	C₂²×C₄×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	8	4	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₄²	1	4	yes	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
73	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₄⋊C₄×C₆₇	—	4	4	no	—
82	C₄⋊C₄×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	#102	1	2	yes	—
86	C₂³×C₆₇	#102	1	2	yes	—
87	C₂³×C₆₇	#107	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	8	8	no	—
91	C₂×C₄×C₆₇	—	8	4	no	—
92	C₂×C₄×C₆₇	#95	1	4	yes	—
93	C₂×C₄×C₆₇	#102	1	2	yes	—
94	C₂×C₄×C₆₇	#102	1	2	yes	—
95	C₂×C₄×C₆₇	#103	1	4	yes	—
96	C₂×C₄×C₆₇	#103	1	4	yes	—
97	C₂×C₄×C₆₇	#106	1	4	yes	—
98	C₂×C₄×C₆₇	#107	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×Dic₆₇	—	4	4	no	—
106	C₂×Dic₆₇	—	8	2	no	—
107	C₂×Dic₆₇	—	8	2	no	—
108	C₈×C₆₇	#95	1	4	yes	—
109	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
110	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	4	no	—
117	C₂²×C₆₇	#340	1	4	yes	—
118	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
121	C₂²×C₆₇	#509	1	2	yes	—
122	C₂²×C₆₇	#510	1	2	yes	—
123	C₄×C₆₇	—	8	4	no	—
124	C₄×C₆₇	#340	1	4	yes	—
125	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
126	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
127	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
128	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
129	Dic₆₇	—	8	4	no	—
130	#12557	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	2	4	no	—
132	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
133	C₂×C₆₇	#2139	1	2	yes	—
134	C₂×C₆₇	#2140	1	1	yes	characteristic
135	C₂×C₆₇	#2141	1	1	yes	characteristic
136	C₂×C₆₇	#2159	1	2	yes	—
137	C₂²×M₃₂	—	67	1	no	—
138	C₂×#427	—	67	1	no	—
139	C₂²×#107	D₆₇	1	1	yes	characteristic
140	#1380	—	67	4	no	—
141	C₆₇	#12557	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
143	C₂×#117	—	134	1	no	—
144	C₂²×C₄⋊C₄	—	2	1	no	—
145	C₂×M₃₂	—	67	4	no	—
146	C₂×M₃₂	—	134	2	no	—
147	C₂×M₃₂	—	134	2	no	—
148	C₂²×C₁₆	—	67	2	no	—
149	#427	—	67	4	no	—
150	#427	—	134	2	no	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	—	2	2	no	—
153	C₂×#107	—	67	2	no	—
154	C₂×#107	Dic₆₇	1	4	yes	complemented
155	C₂×#107	C₂×D₆₇	1	2	yes	—
156	C₂×#106	—	134	1	no	—
157	C₂³×C₄	—	4	1	no	—
158	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂²×C₈	Dic₁₃₄	1	2	yes	—
160	C₂²×C₈	C₄×D₆₇	1	2	yes	—
161	C₂²×C₈	C₄×D₆₇	1	2	yes	—
162	C₂²×C₈	D₂₆₈	1	2	yes	—
163	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
166	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
167	#117	—	268	2	no	—
168	#117	—	134	4	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	4	2	no	—
173	C₂×C₄⋊C₄	—	134	2	no	—
174	C₂×C₂²⋊C₄	—	268	1	no	—
175	C₂×C₄²	—	268	1	no	—
176	M₃₂	—	134	8	no	—
177	C₂×C₁₆	—	67	4	no	—
178	C₂×C₁₆	—	134	2	no	—
179	C₂×C₁₆	—	134	2	no	—
180	#107	—	2	4	no	—
181	#107	—	2	2	no	—
182	#107	—	67	4	no	—
183	#107	—	134	2	no	—
184	#107	C₂×Dic₆₇	1	4	yes	—
185	#106	—	134	4	no	—
186	C₂⁴	#11761613	1	1	yes	characteristic
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	8	4	no	—
190	C₂²×C₄	—	268	2	no	—
191	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
192	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
193	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
194	C₂²×C₄	#11761613	1	1	yes	characteristic
195	C₂²×C₄	#11761615	1	2	yes	—
196	C₂²×C₄	#11761616	1	2	yes	—
197	C₂²×C₄	#11761618	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
199	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
202	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
203	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
204	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
205	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
206	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
211	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	268	2	no	—
216	C₄⋊C₄	—	134	4	no	—
217	C₂²⋊C₄	—	268	4	no	—
218	C₄²	—	268	4	no	—
219	C₁₆	—	134	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	#14523963	1	2	yes	—
223	C₂³	#14523964	1	2	yes	—
224	C₂³	#14523976	1	2	yes	—
225	C₂³	C₂×#11761613	1	1	yes	characteristic
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	8	8	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	268	4	no	—
230	C₂×C₄	—	536	2	no	—
231	C₂×C₄	—	536	2	no	—
232	C₂×C₄	C₆₇⋊#95	1	4	yes	—
233	C₂×C₄	#14523963	1	2	yes	—
234	C₂×C₄	#14523964	1	2	yes	—
235	C₂×C₄	#14523966	1	4	yes	—
236	C₂×C₄	#14523967	1	4	yes	—
237	C₂×C₄	#14523974	1	4	yes	—
238	C₂×C₄	#14523976	1	2	yes	—
239	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
243	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
244	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
245	C₈	C₆₇⋊#95	1	4	yes	—
246	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
247	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
249	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#16863158	1	4	yes	—
255	C₂²	C₂×#14523963	1	1	yes	characteristic
256	C₂²	C₂×#14523964	1	1	yes	characteristic
257	C₂²	C₂×#14523976	1	1	yes	characteristic
258	C₂²	#16863832	1	2	yes	—
259	C₂²	#16863837	1	2	yes	—
260	C₄	—	8	4	no	—
261	C₄	—	536	4	no	—
262	C₄	#16863158	1	4	yes	—
263	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
264	C₄	C₂×#14523966	1	1	yes	characteristic
265	C₄	C₂×#14523967	1	1	yes	characteristic
266	C₄	C₂×#14523974	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	C₂×#16863158	1	1	yes	characteristic
269	C₂	#22561919	1	2	yes	—
270	C₂	#22561925	1	1	yes	characteristic
271	C₂	#22561929	1	1	yes	characteristic
272	C₂	#22561995	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	#12557
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #22558300	C₂	semidirect
8. C₂×#14523983	C₄	semidirect
11. #16863434	C₄	semidirect
18. C₂×#107×C₆₇	C₄	semidirect
28. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
30. #14523977	C₂×C₄	semidirect
141. C₆₇	#12557	semidirect
154. 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
C₂²×#14523983	2
C₂×#16863434	2
#22558300	2
C₂²×#107×C₆₇	2
#12557	67