Subgroups — #30744467

Order 17152.

388 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560352	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	#22558032	C₂	1	4	yes	maximal
4	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
6	#16863151	—	2	1	no	—
7	#16863151	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	C₂²	1	2	yes	—
10	#16863151	C₂²	1	2	yes	—
11	C₂×#125×C₆₇	—	2	1	no	—
12	#389×C₆₇	—	2	1	no	—
13	#389×C₆₇	—	2	1	no	—
14	#389×C₆₇	C₄	1	2	yes	—
15	#389×C₆₇	C₂²	1	2	yes	—
16	#384×C₆₇	C₄	1	1	yes	characteristic
17	#384×C₆₇	C₄	1	1	yes	characteristic
18	#382×C₆₇	—	2	1	no	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
23	#125×C₆₇	—	4	4	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
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	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	#95×C₆₇	—	2	2	no	—
40	#95×C₆₇	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂²×C₄×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	4	1	no	—
68	C₄²×C₆₇	—	4	1	no	—
69	C₄²×C₆₇	—	4	1	no	—
70	C₄²×C₆₇	SD₁₆	1	2	yes	—
71	C₄²×C₆₇	Q₁₆	1	2	yes	—
72	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
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	—	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	—	4	2	no	—
94	C₂×C₄×C₆₇	#101	1	2	yes	—
95	C₂×C₄×C₆₇	#103	1	2	yes	—
96	C₂×C₄×C₆₇	#103	1	2	yes	—
97	C₂×C₄×C₆₇	#131	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
101	C₆₇⋊C₈	—	8	2	no	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	#408	1	1	yes	characteristic
104	C₂²×C₆₇	#410	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
107	C₂²×C₆₇	#418	1	1	yes	characteristic
108	C₂²×C₆₇	#421	1	1	yes	characteristic
109	C₂²×C₆₇	#421	1	1	yes	characteristic
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	4	1	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	—	8	2	no	—
120	#7720	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	#1470	1	1	yes	characteristic
122	C₂×C₆₇	#1481	1	1	yes	characteristic
123	C₂×C₆₇	#1496	1	1	yes	characteristic
124	C₂×C₆₇	#1514	1	1	yes	characteristic
125	C₂×C₆₇	#1518	1	1	yes	characteristic
126	C₂×C₆₇	#1534	1	1	yes	characteristic
127	C₂×C₆₇	#1549	1	1	yes	characteristic
128	#2859	D₆₇	1	1	yes	characteristic
129	#1839	—	67	1	no	—
130	#1831	—	67	1	no	—
131	#1271	—	67	4	no	—
132	C₆₇	#7720	1	1	yes	characteristic, complemented, Sylow
133	C₂×#125	—	2	1	no	—
134	C₂×C₄⋊C₈	—	134	1	no	—
135	#389	—	2	1	no	—
136	#389	—	2	1	no	—
137	#389	Dic₆₇	1	2	yes	—
138	#389	C₂×D₆₇	1	2	yes	—
139	#384	Dic₆₇	1	1	yes	characteristic
140	#384	Dic₆₇	1	1	yes	characteristic
141	#382	—	2	1	no	—
142	C₄³	C₂×D₆₇	1	1	yes	characteristic
143	#336	—	67	2	no	—
144	#336	—	67	2	no	—
145	#336	—	134	1	no	—
146	#336	—	134	1	no	—
147	#336	—	134	1	no	—
148	C₂²×C₈	—	134	2	no	—
149	C₂²×C₈	—	134	2	no	—
150	#125	—	4	4	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
159	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₄⋊C₈	—	268	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂×C₈	—	268	2	no	—
179	C₂×C₈	—	268	2	no	—
180	C₂×C₈	—	268	2	no	—
181	C₂×C₈	—	268	2	no	—
182	C₂×C₈	—	268	2	no	—
183	C₂×C₈	—	268	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	#11761615	1	2	yes	—
199	C₄²	#11761618	1	2	yes	—
200	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	#14523962	1	2	yes	—
223	C₂×C₄	#14523967	1	2	yes	—
224	C₂×C₄	#14523967	1	2	yes	—
225	C₂×C₄	#14524050	1	2	yes	—
226	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
227	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
228	C₂×C₄	#14524069	1	2	yes	—
229	C₈	—	536	2	no	—
230	C₈	—	536	2	no	—
231	C₂²	#16863363	1	1	yes	characteristic
232	C₂²	#16863370	1	1	yes	characteristic
233	C₂²	C₂×#14523962	1	1	yes	characteristic
234	C₂²	C₂×#14523967	1	1	yes	characteristic
235	C₂²	#16863401	1	1	yes	characteristic
236	C₂²	#16863413	1	1	yes	characteristic
237	C₂²	#16863413	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	8	2	no	—
247	C₄	—	8	2	no	—
248	C₂	#22558565	1	1	yes	characteristic
249	C₂	#22558612	1	1	yes	characteristic
250	C₂	#22558678	1	1	yes	characteristic
251	C₂	#22558756	1	1	yes	characteristic
252	C₂	#22558773	1	1	yes	characteristic
253	C₂	#22558836	1	1	yes	characteristic
254	C₂	#22558894	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2859	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2859	D₆₇
hypocenter	C₆₇	#7720
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
132. C₆₇	#7720	semidirect

Maximal subgroups

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

subgroup	index
#22560352	2
#22560326	2
#22558032	2
C₆₇×#2859	2
#7720	67