Subgroups — #30754423

Order 17152.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560062	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal
4	#22558043	C₂	1	2	yes	maximal
5	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#14523954	—	2	1	no	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	#16863151	—	2	1	no	—
12	#16863151	C₄	1	1	yes	characteristic
13	#16863151	C₄	1	1	yes	characteristic
14	#16863148	—	2	4	no	—
15	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
16	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	—	2	1	no	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
27	C₄⋊(C₆₇⋊C₈)	Q₈	1	2	yes	—
28	#14523954	—	4	2	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	4	2	no	—
42	C₄⋊C₈×C₆₇	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₆₇⋊C₈	—	4	1	no	—
51	C₂×C₆₇⋊C₈	—	4	1	no	—
52	C₂×C₆₇⋊C₈	—	4	1	no	—
53	C₂×C₆₇⋊C₈	—	4	1	no	—
54	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	4	2	no	—
63	C₂×C₈×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	—	4	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	2	2	no	—
68	C₄²×C₆₇	—	2	1	no	—
69	C₄²×C₆₇	—	2	1	no	—
70	C₄²×C₆₇	—	4	1	no	—
71	C₄²×C₆₇	D₈	1	2	yes	—
72	C₄²×C₆₇	Q₁₆	1	2	yes	—
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₆₇	#41	1	1	yes	characteristic
76	C₄²×C₆₇	#41	1	1	yes	characteristic
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	#107	1	2	yes	—
90	C₂×C₄×C₆₇	#107	1	2	yes	—
91	C₂×C₄×C₆₇	#108	1	2	yes	—
92	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	#131	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
104	C₈×C₆₇	—	8	2	no	—
105	C₆₇⋊C₈	—	2	4	no	—
106	C₆₇⋊C₈	—	4	2	no	—
107	C₆₇⋊C₈	—	4	2	no	—
108	C₆₇⋊C₈	—	4	2	no	—
109	C₂²×C₆₇	#389	1	1	yes	characteristic
110	C₂²×C₆₇	#423	1	1	yes	characteristic
111	C₂²×C₆₇	#424	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
113	C₂²×C₆₇	#428	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
115	C₂²×C₆₇	#430	1	1	yes	characteristic
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	—	4	1	no	—
122	C₄×C₆₇	#435	1	2	yes	—
123	C₄×C₆₇	#446	1	2	yes	—
124	C₄×C₆₇	#459	1	1	yes	characteristic
125	C₄×C₆₇	#462	1	1	yes	characteristic
126	C₄×C₆₇	#474	1	1	yes	characteristic
127	C₄×C₆₇	#477	1	1	yes	characteristic
128	#9604	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	#1567	1	1	yes	characteristic
130	C₂×C₆₇	#1569	1	1	yes	characteristic
131	C₂×C₆₇	#1571	1	1	yes	characteristic
132	C₂×C₆₇	#1573	1	1	yes	characteristic
133	C₂×C₆₇	#1940	1	1	yes	characteristic
134	C₂×C₆₇	#1947	1	1	yes	characteristic
135	C₂×C₆₇	#1950	1	1	yes	characteristic
136	#1842	D₆₇	1	1	yes	characteristic
137	#1767	—	67	1	no	—
138	C₄×C₄⋊C₈	—	67	1	no	—
139	C₂×#335	—	67	2	no	—
140	#1277	—	67	2	no	—
141	C₆₇	#9604	1	1	yes	characteristic, complemented, Sylow
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄⋊C₈	—	67	1	no	—
145	C₂×C₄⋊C₈	—	67	1	no	—
146	C₂×C₄⋊C₈	—	67	2	no	—
147	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
148	C₂×#97	—	134	1	no	—
149	C₂×C₄×C₈	—	67	2	no	—
150	C₄³	C₂×D₆₇	1	1	yes	characteristic
151	#336	—	67	1	no	—
152	#336	—	67	1	no	—
153	#336	—	134	1	no	—
154	#335	—	134	4	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	67	2	no	—
157	C₂²×C₈	—	67	2	no	—
158	C₂²×C₈	—	134	1	no	—
159	C₂²×C₈	—	134	1	no	—
160	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
161	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
162	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	4	2	no	—
170	C₄⋊C₈	—	4	2	no	—
171	C₄⋊C₈	—	67	2	no	—
172	C₄⋊C₈	—	67	2	no	—
173	C₄⋊C₈	—	134	4	no	—
174	C₄⋊C₈	—	134	1	no	—
175	C₄⋊C₈	—	134	1	no	—
176	#97	—	268	2	no	—
177	C₄×C₈	—	134	2	no	—
178	C₄×C₈	—	134	2	no	—
179	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	268	1	no	—
190	C₂×C₈	—	268	1	no	—
191	C₂×C₈	—	268	1	no	—
192	C₂×C₈	—	268	1	no	—
193	C₂×C₈	—	67	4	no	—
194	C₂×C₈	—	134	2	no	—
195	C₂×C₈	—	134	2	no	—
196	C₂×C₈	—	134	2	no	—
197	C₂×C₈	—	134	2	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	#11761613	1	2	yes	—
208	C₄²	#11761618	1	2	yes	—
209	C₄²	D₄×D₆₇	1	1	yes	characteristic
210	C₄²	Q₈×D₆₇	1	1	yes	characteristic
211	C₄²	#11761627	1	1	yes	characteristic
212	C₄²	#11761629	1	1	yes	characteristic
213	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	#14523976	1	2	yes	—
226	C₂×C₄	#14523976	1	2	yes	—
227	C₂×C₄	#14523978	1	2	yes	—
228	C₂×C₄	#14524005	1	1	yes	characteristic
229	C₂×C₄	#14524008	1	1	yes	characteristic
230	C₂×C₄	#14524014	1	1	yes	characteristic
231	C₂×C₄	#14524019	1	1	yes	characteristic
232	C₂×C₄	#14524034	1	1	yes	characteristic
233	C₂×C₄	#14524042	1	1	yes	characteristic
234	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
235	C₂×C₄	#14524049	1	2	yes	—
236	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
238	C₂×C₄	#14524069	1	1	yes	characteristic
239	C₂×C₄	#14524074	1	1	yes	characteristic
240	C₈	—	8	2	no	—
241	C₈	—	268	2	no	—
242	C₈	—	268	2	no	—
243	C₈	—	268	2	no	—
244	C₈	—	134	4	no	—
245	C₂²	#16863292	1	1	yes	characteristic
246	C₂²	#16863419	1	1	yes	characteristic
247	C₂²	#16863422	1	1	yes	characteristic
248	C₂²	C₂×#14523976	1	1	yes	characteristic
249	C₂²	#16863436	1	1	yes	characteristic
250	C₂²	C₂×#14523978	1	1	yes	characteristic
251	C₂²	#16863444	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#16863466	1	2	yes	—
259	C₄	#16863525	1	2	yes	—
260	C₄	#16863588	1	1	yes	characteristic
261	C₄	#16863603	1	1	yes	characteristic
262	C₄	#16863673	1	1	yes	characteristic
263	C₄	#16863688	1	1	yes	characteristic
264	C₂	#22558967	1	1	yes	characteristic
265	C₂	#22558974	1	1	yes	characteristic
266	C₂	#22558983	1	1	yes	characteristic
267	C₂	#22558992	1	1	yes	characteristic
268	C₂	#22560930	1	1	yes	characteristic
269	C₂	#22560965	1	1	yes	characteristic
270	C₂	#22560982	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#9604
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
141. C₆₇	#9604	semidirect

Maximal subgroups

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

subgroup	index
#22560062	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863148	2
#22558043	2
C₆₇×#1842	2
#9604	67