Subgroups — #30745310

Order 17152.

574 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863453	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
3	#22558043	C₂	1	4	yes	maximal, complemented
4	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863453	—	2	2	no	—
6	#16863453	C₄	1	4	yes	—
7	#16863449	—	2	2	no	—
8	#16863449	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
12	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
13	C₂×#14523954	—	2	1	no	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#431×C₆₇	—	2	2	no	—
17	#431×C₆₇	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
20	C₂×C₄×C₈×C₆₇	—	2	1	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
28	#14523954	—	2	4	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²⋊C₈×C₆₇	—	2	4	no	—
46	C₄×C₈×C₆₇	—	2	4	no	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	2	8	no	—
50	C₂×C₆₇⋊C₈	—	4	4	no	—
51	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	—	2	4	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	4	4	no	—
69	C₄²×C₆₇	C₄²	1	2	yes	—
70	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	#95	1	2	yes	—
83	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	#95	1	2	yes	—
95	C₂×C₄×C₆₇	#95	1	2	yes	—
96	C₂×C₄×C₆₇	#95	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#131	1	4	yes	—
104	C₂×C₄×C₆₇	#135	1	4	yes	—
105	C₈×C₆₇	—	4	4	no	—
106	C₆₇⋊C₈	—	4	4	no	—
107	C₆₇⋊C₈	—	4	4	no	—
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	4	no	—
112	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
113	C₂²×C₆₇	#424	1	2	yes	—
114	C₂²×C₆₇	#427	1	2	yes	—
115	C₂²×C₆₇	#430	1	2	yes	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	#405	1	2	yes	—
122	C₄×C₆₇	#408	1	2	yes	—
123	C₄×C₆₇	#416	1	2	yes	—
124	C₄×C₆₇	#416	1	2	yes	—
125	#7878	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	4	2	no	—
127	C₂×C₆₇	#1572	1	4	yes	—
128	C₂×C₆₇	#1729	1	1	yes	characteristic
129	C₂×C₆₇	#1735	1	1	yes	characteristic
130	C₂×C₆₇	#1738	1	1	yes	characteristic
131	C₂×#432	—	67	1	no	—
132	C₂×#431	—	67	1	no	—
133	C₂×#431	D₆₇	1	1	yes	characteristic
134	#1277	—	67	4	no	—
135	C₆₇	#7878	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
137	#432	—	67	4	no	—
138	#432	—	134	2	no	—
139	#431	—	2	2	no	—
140	#431	—	134	2	no	—
141	#431	—	134	2	no	—
142	#431	Dic₆₇	1	4	yes	—
143	C₂×C₄⋊C₈	—	67	2	no	—
144	C₂×C₄⋊C₈	—	67	2	no	—
145	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₂²⋊C₈	—	67	2	no	—
148	C₂×C₂²⋊C₈	—	134	1	no	—
149	C₂×#97	—	134	1	no	—
150	C₂×C₄×C₈	—	2	1	no	—
151	C₂×C₄×C₈	—	67	2	no	—
152	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
153	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
154	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	67	4	no	—
157	C₂²×C₈	—	134	2	no	—
158	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
159	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₆₇	1	2	yes	—
161	C₂×C₄²	D₂₆₈	1	2	yes	—
162	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	67	4	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	—	134	2	no	—
170	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	134	4	no	—
173	C₂²⋊C₈	—	134	4	no	—
174	#97	—	134	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	134	2	no	—
177	C₄×C₈	—	134	2	no	—
178	C₂⁴	Dic₆₇⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
183	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
184	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
185	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
186	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
190	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	268	4	no	—
197	C₂×C₈	—	134	4	no	—
198	C₂×C₈	—	134	4	no	—
199	C₂×C₈	—	134	8	no	—
200	C₄²	C₄×Dic₆₇	1	2	yes	—
201	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
202	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
203	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
204	C₄²	Dic₆₇⋊C₄	1	2	yes	—
205	C₄²	C₄⋊Dic₆₇	1	2	yes	—
206	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
207	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
209	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₆₇⋊#95	1	2	yes	—
214	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₆₇⋊#95	1	2	yes	—
226	C₂×C₄	C₆₇⋊#95	1	2	yes	—
227	C₂×C₄	C₆₇⋊#95	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
234	C₂×C₄	#14524049	1	4	yes	—
235	C₂×C₄	#14524063	1	4	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	268	4	no	—
238	C₈	—	268	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
244	C₂²	#16863422	1	2	yes	—
245	C₂²	#16863432	1	2	yes	—
246	C₂²	#16863444	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	#16863351	1	2	yes	—
253	C₄	#16863362	1	2	yes	—
254	C₄	#16863390	1	2	yes	—
255	C₄	#16863392	1	2	yes	—
256	C₂	—	4	2	no	—
257	C₂	#22558987	1	4	yes	—
258	C₂	#22559905	1	1	yes	characteristic
259	C₂	#22559925	1	1	yes	characteristic
260	C₂	#22559935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#7878
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22558043	C₂	semidirect
135. C₆₇	#7878	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863453	2
C₂×#16863449	2
#22558043	2
C₂×#431×C₆₇	2
#7878	67