Subgroups — #30747102

Order 17152.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523975	C₂	1	1	yes	maximal, characteristic
2	#22560076	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
5	#22558049	C₂	1	1	yes	maximal, characteristic
6	#22558049	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523975	C₄	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×#14523953	—	2	1	no	—
12	C₂×C₈×Dic₆₇	—	2	1	no	—
13	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	#16863159	C₄	1	2	yes	—
16	#16863150	—	2	1	no	—
17	#16863150	—	2	1	no	—
18	#16863149	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#322×C₆₇	C₄	1	4	yes	complemented
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₆₇	—	2	1	no	—
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	#14523975	—	2	4	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	#14523953	—	4	2	no	—
35	C₈×Dic₆₇	—	4	2	no	—
36	C₆₇⋊#95	—	2	1	no	—
37	C₆₇⋊#95	—	2	1	no	—
38	C₂²×C₈×C₆₇	—	2	1	no	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₇	—	2	1	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	Q₈	1	2	yes	—
50	C₂²×Dic₆₇	—	2	2	no	—
51	C₂²×Dic₆₇	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	4	1	no	—
55	C₂×C₆₇⋊C₈	—	4	1	no	—
56	C₂×C₆₇⋊C₈	—	4	1	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₄⋊Dic₆₇	—	2	2	no	—
59	C₄⋊Dic₆₇	—	2	2	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	M₁₆	1	4	yes	—
62	C₄×Dic₆₇	—	4	1	no	—
63	C₄×Dic₆₇	—	4	1	no	—
64	C₄×Dic₆₇	—	4	1	no	—
65	C₄×Dic₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	4	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	—	2	1	no	—
74	C₂×C₈×C₆₇	—	2	1	no	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	C₄²	1	2	yes	—
78	C₂×C₈×C₆₇	C₄²	1	2	yes	—
79	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₆₇	#131	1	2	yes	—
95	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
99	C₂×Dic₆₇	—	2	4	no	—
100	C₂×Dic₆₇	—	4	2	no	—
101	C₂×Dic₆₇	—	4	2	no	—
102	C₂×Dic₆₇	—	4	2	no	—
103	C₂×Dic₆₇	—	4	2	no	—
104	C₂×Dic₆₇	—	4	2	no	—
105	C₈×C₆₇	—	2	4	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	—	2	2	no	—
110	C₆₇⋊C₈	—	4	2	no	—
111	C₆₇⋊C₈	—	4	2	no	—
112	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₆₇	#435	1	2	yes	—
114	C₂²×C₆₇	#440	1	1	yes	characteristic
115	C₂²×C₆₇	#440	1	1	yes	characteristic
116	C₂²×C₆₇	#443	1	2	yes	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
120	C₄×C₆₇	#405	1	1	yes	characteristic
121	C₄×C₆₇	#428	1	1	yes	characteristic
122	C₄×C₆₇	#428	1	1	yes	characteristic
123	C₄×C₆₇	#441	1	1	yes	characteristic
124	C₄×C₆₇	#442	1	1	yes	characteristic
125	C₄×C₆₇	#446	1	2	yes	—
126	Dic₆₇	—	4	4	no	—
127	Dic₆₇	—	8	2	no	—
128	#8289	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	#1585	1	2	yes	—
130	C₂×C₆₇	#1588	1	2	yes	—
131	C₂×C₆₇	#1771	1	1	yes	characteristic
132	C₂×C₆₇	#1772	1	1	yes	characteristic
133	C₂×C₆₇	#1776	1	1	yes	characteristic
134	C₄×#106	—	67	1	no	—
135	#1770	—	67	1	no	—
136	C₈×C₄⋊C₄	—	67	1	no	—
137	C₂×#335	—	67	1	no	—
138	C₂×#322	D₆₇	1	1	yes	characteristic
139	#1279	—	67	1	no	—
140	#1279	—	67	1	no	—
141	C₆₇	#8289	1	1	yes	characteristic, complemented, Sylow
142	C₂×#106	—	67	2	no	—
143	C₂×C₄⋊C₈	—	67	1	no	—
144	C₂×C₄⋊C₈	—	67	1	no	—
145	C₂×#97	—	134	1	no	—
146	C₂×C₄×C₈	—	134	1	no	—
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	C₄×C₄⋊C₄	—	67	1	no	—
151	C₄×C₄⋊C₄	—	67	1	no	—
152	#340	—	67	2	no	—
153	#336	—	134	1	no	—
154	#336	—	134	1	no	—
155	#335	—	67	4	no	—
156	#322	Dic₆₇	1	4	yes	complemented
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	134	1	no	—
159	C₂²×C₈	—	134	1	no	—
160	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
162	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
163	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	67	2	no	—
165	C₂×C₄⋊C₄	—	67	2	no	—
166	C₂×C₄²	—	134	1	no	—
167	C₂×C₄²	—	134	1	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	#106	—	134	4	no	—
170	C₄⋊C₈	—	67	2	no	—
171	C₄⋊C₈	—	67	2	no	—
172	C₄⋊C₈	—	134	1	no	—
173	C₄⋊C₈	—	134	1	no	—
174	#97	—	268	2	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	268	2	no	—
177	C₄×C₈	Dic₁₃₄	1	2	yes	—
178	C₄×C₈	C₄×D₆₇	1	2	yes	—
179	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
180	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
181	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
182	#95	—	134	1	no	—
183	#95	—	134	1	no	—
184	C₂²×C₄	—	134	2	no	—
185	C₂²×C₄	—	134	2	no	—
186	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
188	C₂²×C₄	#11761628	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	268	1	no	—
198	C₂×C₈	—	268	1	no	—
199	C₂×C₈	—	268	1	no	—
200	C₂×C₈	—	268	1	no	—
201	C₂×C₈	—	134	2	no	—
202	C₂×C₈	—	134	2	no	—
203	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
204	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
205	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
206	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
207	C₄⋊C₄	—	67	4	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	2	no	—
211	C₄²	—	268	1	no	—
212	C₄²	—	268	1	no	—
213	C₄²	—	268	1	no	—
214	C₄²	—	268	1	no	—
215	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
217	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
218	C₄²	#11761628	1	1	yes	characteristic
219	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	268	2	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	268	2	no	—
225	C₂×C₄	—	268	2	no	—
226	C₂×C₄	—	134	4	no	—
227	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
229	C₂×C₄	#14524002	1	1	yes	characteristic
230	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
231	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
232	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
233	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
234	C₂×C₄	#14524051	1	2	yes	—
235	C₂×C₄	#14524071	1	1	yes	characteristic
236	C₂×C₄	#14524071	1	1	yes	characteristic
237	C₂×C₄	#14524076	1	1	yes	characteristic
238	C₂×C₄	#14524076	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	268	2	no	—
245	C₈	—	268	2	no	—
246	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
247	C₂²	#16863469	1	2	yes	—
248	C₂²	#16863498	1	1	yes	characteristic
249	C₂²	#16863498	1	1	yes	characteristic
250	C₂²	#16863514	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	268	4	no	—
254	C₄	—	536	2	no	—
255	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
256	C₄	#16863354	1	1	yes	characteristic
257	C₄	#16863438	1	1	yes	characteristic
258	C₄	#16863438	1	1	yes	characteristic
259	C₄	#16863504	1	1	yes	characteristic
260	C₄	#16863509	1	1	yes	characteristic
261	C₄	#16863524	1	2	yes	—
262	C₂	#22559068	1	2	yes	—
263	C₂	#22559089	1	2	yes	—
264	C₂	#22560081	1	1	yes	characteristic
265	C₂	#22560088	1	1	yes	characteristic
266	C₂	#22560106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8289
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
18. #16863149	C₄	semidirect
22. #322×C₆₇	C₄	semidirect
141. C₆₇	#8289	semidirect
156. #322	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523975	2
#22560076	2
C₈×C₄⋊Dic₆₇	2
C₂×#16863149	2
#22558049	2
C₂×#322×C₆₇	2
#8289	67