Subgroups — #30754940

Order 17152.

486 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	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	#16863142	—	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	1	no	—
16	#391×C₆₇	—	2	1	no	—
17	#391×C₆₇	—	2	2	no	—
18	#382×C₆₇	C₄	1	4	yes	—
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	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
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	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	1	no	—
52	C₂×Q₈×C₆₇	—	4	1	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
54	C₂×Q₈×C₆₇	—	4	1	no	—
55	C₂×Q₈×C₆₇	—	4	2	no	—
56	C₂×Q₈×C₆₇	—	4	1	no	—
57	C₂×Q₈×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	1	no	—
69	C₄⋊C₄×C₆₇	—	4	1	no	—
70	C₄²×C₆₇	—	2	4	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	2	1	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	1	no	—
76	C₄²×C₆₇	C₂×D₄	1	2	yes	—
77	C₄²×C₆₇	#41	1	2	yes	—
78	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	Q₈×C₆₇	—	8	2	no	—
80	Q₈×C₆₇	—	8	2	no	—
81	Q₈×C₆₇	—	8	2	no	—
82	Q₈×C₆₇	—	8	1	no	—
83	Q₈×C₆₇	—	8	1	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	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	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	4	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	#101	1	4	yes	—
96	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₆₇	#121	1	2	yes	—
98	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#135	1	2	yes	—
101	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#137	1	2	yes	—
105	C₆₇⋊C₈	—	4	4	no	—
106	C₆₇⋊C₈	—	8	2	no	—
107	C₂²×C₆₇	#390	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×#101	1	2	yes	—
109	C₂²×C₆₇	#416	1	1	yes	characteristic
110	C₂²×C₆₇	#417	1	2	yes	—
111	C₂²×C₆₇	#417	1	1	yes	characteristic
112	C₄×C₆₇	—	2	4	no	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	4	2	no	—
115	C₄×C₆₇	—	8	2	no	—
116	C₄×C₆₇	—	8	2	no	—
117	C₄×C₆₇	#463	1	2	yes	—
118	C₄×C₆₇	#464	1	2	yes	—
119	C₄×C₆₇	#489	1	2	yes	—
120	C₄×C₆₇	#496	1	1	yes	characteristic
121	C₄×C₆₇	#497	1	1	yes	characteristic
122	#9708	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	#1552	1	2	yes	—
124	C₂×C₆₇	#1561	1	2	yes	—
125	C₂×C₆₇	#1959	1	1	yes	characteristic
126	C₂×C₆₇	#1966	1	1	yes	characteristic
127	C₂×C₆₇	#1983	1	1	yes	characteristic
128	#2862	D₆₇	1	1	yes	characteristic
129	#1842	—	67	1	no	—
130	#1767	—	67	1	no	—
131	C₂×#332	—	67	2	no	—
132	C₂×#332	—	67	2	no	—
133	C₆₇	#9708	1	1	yes	characteristic, complemented, Sylow
134	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	67	2	no	—
137	C₂×C₄⋊C₈	—	67	2	no	—
138	C₂×C₄⋊C₈	—	134	2	no	—
139	C₂×#97	—	134	1	no	—
140	#391	—	2	1	no	—
141	#391	—	2	1	no	—
142	#391	—	2	2	no	—
143	#382	Dic₆₇	1	4	yes	—
144	C₄³	C₂×D₆₇	1	1	yes	characteristic
145	#336	—	134	1	no	—
146	#332	—	134	4	no	—
147	#332	—	134	4	no	—
148	C₂²×Q₈	—	2	1	no	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂²×C₈	—	134	2	no	—
151	C₂²×C₈	—	134	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
160	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
161	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	C₄⋊C₈	—	268	4	no	—
166	C₄⋊C₈	—	67	4	no	—
167	C₄⋊C₈	—	134	2	no	—
168	C₄⋊C₈	—	134	2	no	—
169	C₄⋊C₈	—	134	2	no	—
170	#97	—	268	2	no	—
171	#95	—	2	4	no	—
172	#95	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
184	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	—	268	2	no	—
190	C₂×C₈	—	268	2	no	—
191	C₂×C₈	—	268	4	no	—
192	C₂×C₈	—	134	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	1	no	—
204	C₄²	D₄×D₆₇	1	2	yes	—
205	C₄²	#11761627	1	2	yes	—
206	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
207	Q₈	—	8	2	no	—
208	Q₈	—	8	1	no	—
209	Q₈	—	8	1	no	—
210	Q₈	—	8	2	no	—
211	Q₈	—	8	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#14523962	1	4	yes	—
224	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
225	C₂×C₄	#14524014	1	2	yes	—
226	C₂×C₄	#14524030	1	1	yes	characteristic
227	C₂×C₄	#14524039	1	1	yes	characteristic
228	C₂×C₄	#14524063	1	2	yes	—
229	C₂×C₄	#14524068	1	2	yes	—
230	C₂×C₄	#14524068	1	1	yes	characteristic
231	C₂×C₄	#14524073	1	2	yes	—
232	C₂×C₄	#14524073	1	1	yes	characteristic
233	C₈	—	268	4	no	—
234	C₈	—	536	2	no	—
235	C₂²	#16863299	1	1	yes	characteristic
236	C₂²	C₂×#14523962	1	2	yes	—
237	C₂²	#16863392	1	1	yes	characteristic
238	C₂²	#16863397	1	2	yes	—
239	C₂²	#16863397	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	#16863608	1	2	yes	—
246	C₄	#16863615	1	2	yes	—
247	C₄	#16863755	1	2	yes	—
248	C₄	#16863786	1	1	yes	characteristic
249	C₄	#16863791	1	1	yes	characteristic
250	C₂	#22558909	1	2	yes	—
251	C₂	#22558947	1	2	yes	—
252	C₂	#22561026	1	1	yes	characteristic
253	C₂	#22561054	1	1	yes	characteristic
254	C₂	#22561142	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	#9708
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
133. C₆₇	#9708	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×#16863142	2
C₆₇×#2862	2
#9708	67