Subgroups — #30747112

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	#22560078	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	#22558050	C₂	1	1	yes	maximal, characteristic
6	#22558050	C₂	1	1	yes	maximal, characteristic
7	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14523954	—	2	1	no	—
11	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16863151	—	2	1	no	—
15	#16863151	—	2	1	no	—
16	#16863147	C₄	1	4	yes	complemented
17	#16863118	C₄	1	4	yes	complemented
18	C₂×#106×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	#340×C₆₇	C₄	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂²×C₆₇⋊C₈	—	2	1	no	—
26	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
30	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
31	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
32	#14523954	—	4	2	no	—
33	C₄×C₆₇⋊C₈	—	2	1	no	—
34	C₄×C₆₇⋊C₈	—	4	2	no	—
35	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	—	2	1	no	—
43	C₂×C₄²×C₆₇	—	2	1	no	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#106×C₆₇	—	2	4	no	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	#95×C₆₇	—	2	1	no	—
49	#95×C₆₇	—	2	1	no	—
50	C₂×C₆₇⋊C₈	—	2	1	no	—
51	C₂×C₆₇⋊C₈	—	2	1	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	1	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₂×C₆₇⋊C₈	—	4	1	no	—
60	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
63	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	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	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	C₄²	1	2	yes	—
72	C₂×C₈×C₆₇	C₄²	1	2	yes	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
77	C₄²×C₆₇	—	4	1	no	—
78	C₄²×C₆₇	—	4	1	no	—
79	C₄²×C₆₇	—	4	1	no	—
80	C₄²×C₆₇	—	4	1	no	—
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	—	2	4	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₆₇	#131	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₆₇⋊C₈	—	2	2	no	—
108	C₆₇⋊C₈	—	4	2	no	—
109	C₆₇⋊C₈	—	4	2	no	—
110	C₆₇⋊C₈	—	4	2	no	—
111	C₆₇⋊C₈	#106	1	4	yes	complemented
112	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₆₇	#435	1	2	yes	—
114	C₂²×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
115	C₂²×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
116	C₂²×C₆₇	#444	1	2	yes	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	4	4	no	—
120	C₄×C₆₇	—	8	2	no	—
121	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
122	C₄×C₆₇	#405	1	1	yes	characteristic
123	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
124	C₄×C₆₇	#427	1	1	yes	characteristic
125	C₄×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
126	C₄×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
127	C₄×C₆₇	#446	1	2	yes	—
128	#8290	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	#1582	1	2	yes	—
130	C₂×C₆₇	#1583	1	2	yes	—
131	C₂×C₆₇	#1771	1	1	yes	characteristic
132	C₂×C₆₇	#1773	1	1	yes	characteristic
133	C₂×C₆₇	C₄×#106	1	1	yes	characteristic
134	C₄×#106	D₆₇	1	1	yes	characteristic
135	#1770	—	67	1	no	—
136	C₈×C₄⋊C₄	—	67	1	no	—
137	C₂×#334	—	67	1	no	—
138	C₂×#322	—	67	1	no	—
139	#1279	—	67	1	no	—
140	#1279	—	67	1	no	—
141	C₆₇	#8290	1	1	yes	characteristic, complemented, Sylow
142	C₂×#106	Dic₆₇	1	2	yes	—
143	C₂×C₄⋊C₈	—	67	1	no	—
144	C₂×C₄⋊C₈	—	67	1	no	—
145	C₂×#97	—	134	1	no	—
146	C₂×C₄×C₈	—	67	1	no	—
147	C₂×C₄×C₈	—	67	1	no	—
148	C₂×C₄×C₈	—	134	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
151	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
152	#340	Dic₆₇	1	2	yes	—
153	#336	—	134	1	no	—
154	#336	—	134	1	no	—
155	#334	—	67	4	no	—
156	#322	—	67	4	no	—
157	C₂²×C₈	—	67	1	no	—
158	C₂²×C₈	—	67	1	no	—
159	C₂²×C₈	—	134	1	no	—
160	C₂²×C₈	—	134	1	no	—
161	C₂²×C₈	—	134	1	no	—
162	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	#106	—	2	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	—	268	2	no	—
176	C₄×C₈	—	67	2	no	—
177	C₄×C₈	—	67	2	no	—
178	C₄×C₈	—	67	2	no	—
179	C₄×C₈	—	134	1	no	—
180	C₄×C₈	C₄×D₆₇	1	2	yes	—
181	C₄×C₈	C₄×D₆₇	1	2	yes	—
182	#95	—	2	1	no	—
183	#95	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	#11761627	1	1	yes	characteristic
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	268	2	no	—
192	C₂×C₈	—	268	1	no	—
193	C₂×C₈	—	268	1	no	—
194	C₂×C₈	—	268	1	no	—
195	C₂×C₈	—	268	1	no	—
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	67	2	no	—
198	C₂×C₈	—	67	2	no	—
199	C₂×C₈	—	67	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₂×C₈	—	134	2	no	—
202	C₂×C₈	—	134	2	no	—
203	C₂×C₈	—	134	1	no	—
204	C₂×C₈	—	134	1	no	—
205	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
206	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	#11761612	1	4	yes	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₄²	Q₈×D₆₇	1	1	yes	characteristic
218	C₄²	#11761629	1	1	yes	characteristic
219	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
228	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
229	C₂×C₄	#14524001	1	1	yes	characteristic
230	C₂×C₄	#14524005	1	1	yes	characteristic
231	C₂×C₄	#14524008	1	1	yes	characteristic
232	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761612	1	2	yes	—
234	C₂×C₄	#14524050	1	2	yes	—
235	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
236	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
237	C₂×C₄	#14524064	1	1	yes	characteristic
238	C₂×C₄	#14524064	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	268	2	no	—
242	C₈	—	268	2	no	—
243	C₈	—	268	2	no	—
244	C₈	—	67	4	no	—
245	C₈	—	134	2	no	—
246	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
247	C₂²	#16863468	1	2	yes	—
248	C₂²	SD₁₆×Dic₆₇	1	1	yes	characteristic
249	C₂²	SD₁₆×Dic₆₇	1	1	yes	characteristic
250	C₂²	#16863518	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₄×#11761612	1	1	yes	characteristic
256	C₄	#16863352	1	1	yes	characteristic
257	C₄	#106×D₆₇	1	1	yes	characteristic
258	C₄	#16863431	1	1	yes	characteristic
259	C₄	#16863483	1	1	yes	characteristic
260	C₄	#16863483	1	1	yes	characteristic
261	C₄	#16863527	1	2	yes	—
262	C₂	#22559046	1	2	yes	—
263	C₂	#22559053	1	2	yes	—
264	C₂	#22560084	1	1	yes	characteristic
265	C₂	#22560095	1	1	yes	characteristic
266	C₂	#106×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#106×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#106	D₆₇
hypocenter	C₆₇	#8290
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
16. #16863147	C₄	semidirect
17. #16863118	C₄	semidirect
111. C₆₇⋊C₈	#106	semidirect
141. C₆₇	#8290	semidirect

Maximal subgroups

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

subgroup	index
#22560078	2
C₄⋊C₄×C₆₇⋊C₈	2
C₂×#16863147	2
C₂×#16863118	2
#22558050	2
C₄×#106×C₆₇	2
#8290	67