Subgroups — #30754909

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863145	C₂	1	2	yes	maximal
4	C₂×#16863142	C₂	1	2	yes	maximal
5	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14523954	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	#16863145	—	2	4	no	—
12	#16863142	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#391×C₆₇	—	2	2	no	—
16	#391×C₆₇	C₄	1	4	yes	—
17	#382×C₆₇	—	2	1	no	—
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)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
27	#14523954	—	4	2	no	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#95×C₆₇	—	2	4	no	—
44	#95×C₆₇	—	2	4	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	4	no	—
50	C₂×Q₈×C₆₇	—	2	4	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂×Q₈×C₆₇	—	4	1	no	—
53	C₂×Q₈×C₆₇	—	4	1	no	—
54	C₂×Q₈×C₆₇	—	4	2	no	—
55	C₂×Q₈×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	1	no	—
67	C₄⋊C₄×C₆₇	—	4	1	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	4	2	no	—
72	C₄²×C₆₇	SD₁₆	1	2	yes	—
73	C₄²×C₆₇	Q₁₆	1	2	yes	—
74	C₄²×C₆₇	C₂×D₄	1	2	yes	—
75	C₄²×C₆₇	#41	1	2	yes	—
76	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₆₇	—	8	4	no	—
78	Q₈×C₆₇	—	8	2	no	—
79	Q₈×C₆₇	—	8	1	no	—
80	Q₈×C₆₇	—	8	1	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
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	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	#103	1	4	yes	—
92	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	#121	1	2	yes	—
94	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₆₇	#137	1	2	yes	—
100	C₂×C₄×C₆₇	#137	1	2	yes	—
101	C₆₇⋊C₈	—	4	4	no	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	#390	1	1	yes	characteristic
104	C₂²×C₆₇	#413	1	2	yes	—
105	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
106	C₂²×C₆₇	#417	1	2	yes	—
107	C₂²×C₆₇	#419	1	1	yes	characteristic
108	C₄×C₆₇	—	2	4	no	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	4	2	no	—
111	C₄×C₆₇	—	8	2	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	#461	1	2	yes	—
114	C₄×C₆₇	#462	1	2	yes	—
115	C₄×C₆₇	#489	1	2	yes	—
116	C₄×C₆₇	#496	1	1	yes	characteristic
117	C₄×C₆₇	#497	1	1	yes	characteristic
118	#9701	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	#1550	1	2	yes	—
120	C₂×C₆₇	#1559	1	2	yes	—
121	C₂×C₆₇	#1959	1	1	yes	characteristic
122	C₂×C₆₇	#1967	1	1	yes	characteristic
123	C₂×C₆₇	#1981	1	1	yes	characteristic
124	#2862	D₆₇	1	1	yes	characteristic
125	#1842	—	67	1	no	—
126	#1767	—	67	1	no	—
127	C₂×#333	—	67	2	no	—
128	C₂×#332	—	67	2	no	—
129	C₆₇	#9701	1	1	yes	characteristic, complemented, Sylow
130	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	67	2	no	—
133	C₂×C₄⋊C₈	—	67	2	no	—
134	C₂×C₄⋊C₈	—	134	2	no	—
135	C₂×#97	—	134	1	no	—
136	#391	—	2	2	no	—
137	#391	Dic₆₇	1	4	yes	—
138	#382	—	2	1	no	—
139	#382	—	2	1	no	—
140	C₄³	C₂×D₆₇	1	1	yes	characteristic
141	#336	—	134	1	no	—
142	#333	—	134	4	no	—
143	#332	—	134	4	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
146	C₂²×C₈	—	134	2	no	—
147	C₂²×C₈	—	134	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
156	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
159	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
161	C₄⋊C₈	—	268	4	no	—
162	C₄⋊C₈	—	67	4	no	—
163	C₄⋊C₈	—	134	2	no	—
164	C₄⋊C₈	—	134	2	no	—
165	C₄⋊C₈	—	134	2	no	—
166	#97	—	268	2	no	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×C₈	—	268	2	no	—
183	C₂×C₈	—	268	2	no	—
184	C₂×C₈	—	268	2	no	—
185	C₂×C₈	—	268	4	no	—
186	C₂×C₈	—	134	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	#11761615	1	2	yes	—
197	C₄²	#11761618	1	2	yes	—
198	C₄²	D₄×D₆₇	1	2	yes	—
199	C₄²	#11761627	1	2	yes	—
200	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	Q₈	—	8	2	no	—
203	Q₈	—	8	1	no	—
204	Q₈	—	8	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	#14523967	1	4	yes	—
216	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
217	C₂×C₄	#14524014	1	2	yes	—
218	C₂×C₄	#14524030	1	1	yes	characteristic
219	C₂×C₄	#14524039	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
221	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
222	C₂×C₄	#14524068	1	2	yes	—
223	C₂×C₄	#14524073	1	2	yes	—
224	C₂×C₄	#14524074	1	2	yes	—
225	C₈	—	268	4	no	—
226	C₈	—	536	2	no	—
227	C₂²	#16863299	1	1	yes	characteristic
228	C₂²	#16863380	1	2	yes	—
229	C₂²	C₂×#14523967	1	1	yes	characteristic
230	C₂²	#16863397	1	2	yes	—
231	C₂²	#16863405	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	#16863598	1	2	yes	—
238	C₄	#16863603	1	2	yes	—
239	C₄	#16863755	1	2	yes	—
240	C₄	#16863786	1	1	yes	characteristic
241	C₄	#16863791	1	1	yes	characteristic
242	C₂	#22558899	1	2	yes	—
243	C₂	#22558939	1	2	yes	—
244	C₂	#22561026	1	1	yes	characteristic
245	C₂	#22561059	1	1	yes	characteristic
246	C₂	#22561133	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×#2862	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2862	D₆₇
hypocenter	C₆₇	#9701
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
129. C₆₇	#9701	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
#22560062	2
C₂×#16863145	2
C₂×#16863142	2
C₆₇×#2862	2
#9701	67