Subgroups — #30748683

Order 17152.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560189	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	2	yes	maximal, complemented
3	#22558044	C₂	1	2	yes	maximal, complemented
4	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
7	#16863151	—	2	2	no	—
8	#16863149	—	2	4	no	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#431×C₆₇	—	2	4	no	—
11	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
12	C₂×C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	—	2	4	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
18	C₂²⋊(C₆₇⋊C₈)	—	4	4	no	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₆₇	—	2	2	no	—
25	C₂×C₄²×C₆₇	—	2	2	no	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	—	2	4	no	—
31	C₂²⋊C₈×C₆₇	—	2	4	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	D₄	1	4	yes	—
34	C₂×C₆₇⋊C₈	—	4	4	no	—
35	C₂×C₆₇⋊C₈	—	4	4	no	—
36	C₂×C₆₇⋊C₈	—	4	4	no	—
37	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	—	2	1	no	—
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	—	2	2	no	—
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	D₈	1	2	yes	—
49	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₆₇	—	2	1	no	—
58	C₄²×C₆₇	—	2	1	no	—
59	C₄²×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	C₂×D₄	1	2	yes	—
62	C₄²×C₆₇	#41	1	2	yes	—
63	C₂³×C₆₇	—	2	1	no	—
64	C₂³×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	—	2	1	no	—
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	#107	1	2	yes	—
68	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	#107	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#121	1	2	yes	—
85	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	#131	1	4	yes	—
87	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#135	1	2	yes	—
90	C₈×C₆₇	—	2	4	no	—
91	C₈×C₆₇	—	4	2	no	—
92	C₆₇⋊C₈	—	8	4	no	—
93	C₂²×C₆₇	—	2	2	no	—
94	C₂²×C₆₇	—	2	4	no	—
95	C₂²×C₆₇	—	4	1	no	—
96	C₂²×C₆₇	—	4	1	no	—
97	C₂²×C₆₇	—	4	2	no	—
98	C₂²×C₆₇	#380	1	1	yes	characteristic
99	C₂²×C₆₇	#424	1	2	yes	—
100	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
101	C₂²×C₆₇	#427	1	1	yes	characteristic
102	C₂²×C₆₇	#430	1	2	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	—	4	1	no	—
106	C₄×C₆₇	—	4	1	no	—
107	C₄×C₆₇	#408	1	2	yes	—
108	C₄×C₆₇	#433	1	2	yes	—
109	C₄×C₆₇	#466	1	2	yes	—
110	C₄×C₆₇	#467	1	2	yes	—
111	#8599	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	—	4	2	no	—
113	C₂×C₆₇	#1571	1	2	yes	—
114	C₂×C₆₇	#1572	1	2	yes	—
115	C₂×C₆₇	#1798	1	1	yes	characteristic
116	C₂×C₆₇	#1808	1	1	yes	characteristic
117	C₂×C₆₇	#1811	1	1	yes	characteristic
118	C₂×#431	D₆₇	1	1	yes	characteristic
119	#1797	—	67	2	no	—
120	C₂×#335	—	67	2	no	—
121	#1277	—	67	2	no	—
122	C₆₇	#8599	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
124	#431	—	2	4	no	—
125	C₂×C₄⋊C₈	—	67	4	no	—
126	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
127	C₂×C₂²⋊C₈	—	134	2	no	—
128	C₂×C₂²⋊C₈	Dic₆₇	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	#336	—	134	2	no	—
132	#335	—	134	4	no	—
133	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
134	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
135	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
136	C₂²×C₈	—	134	4	no	—
137	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
138	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
142	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	134	4	no	—
147	C₄⋊C₈	—	134	4	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	268	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	C₂²⋊D₆₇	1	4	yes	—
152	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
163	C₂²×C₄	D₅₃₆	1	2	yes	—
164	C₂²×C₄	Dic₂₆₈	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	268	4	no	—
172	C₂×C₈	—	268	4	no	—
173	C₂×C₈	—	268	4	no	—
174	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
180	C₄²	#11761628	1	2	yes	—
181	C₂³	—	2	1	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	1	no	—
184	C₂³	—	2	2	no	—
185	C₂³	#14523977	1	2	yes	—
186	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#14523977	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
201	C₂×C₄	#14524002	1	1	yes	characteristic
202	C₂×C₄	#14524017	1	2	yes	—
203	C₂×C₄	#14524022	1	2	yes	—
204	C₂×C₄	#14524050	1	4	yes	—
205	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
207	C₂×C₄	#14524065	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	4	2	no	—
210	C₈	—	536	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	#16863256	1	1	yes	characteristic
217	C₂²	#16863423	1	2	yes	—
218	C₂²	C₂×#14523977	1	1	yes	characteristic
219	C₂²	#16863434	1	1	yes	characteristic
220	C₂²	#16863445	1	2	yes	—
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	#16863363	1	2	yes	—
226	C₄	#16863457	1	2	yes	—
227	C₄	#16863632	1	2	yes	—
228	C₄	#16863637	1	2	yes	—
229	C₂	—	4	2	no	—
230	C₂	#22558984	1	2	yes	—
231	C₂	#22558989	1	2	yes	—
232	C₂	#22560193	1	1	yes	characteristic
233	C₂	#22560238	1	1	yes	characteristic
234	C₂	#22560251	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#8599
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. C₂×#16863149	C₂	semidirect
3. #22558044	C₂	semidirect
122. C₆₇	#8599	semidirect

Maximal subgroups

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

subgroup	index
#22560189	2
C₂×#16863149	2
#22558044	2
C₂×#431×C₆₇	2
#8599	67