Subgroups — #30773196

Order 17152.

526 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	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863637	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863199	C₂	1	2	yes	maximal, complemented
4	#22558300	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#14524022	—	2	1	no	—
8	C₂²×C₆₇⋊C₁₆	—	2	4	no	—
9	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#16863637	—	2	2	no	—
12	#16863637	C₄	1	4	yes	complemented
13	C₂×#14523977	C₂²	1	1	yes	characteristic
14	C₂×#14523968	—	2	1	no	—
15	#16863199	—	2	4	no	—
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	#380×C₆₇	—	2	1	no	—
19	#340×C₆₇	—	2	1	no	—
20	C₂²×Dic₁₃₄	—	2	1	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	1	no	—
23	C₂×Dic₂₆₈	—	2	2	no	—
24	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
25	#14524022	—	2	4	no	—
26	#14524022	—	4	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	1	no	—
28	C₂×Dic₆₇⋊C₄	—	4	1	no	—
29	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
30	C₂×C₆₇⋊C₁₆	—	2	8	no	—
31	C₂×C₆₇⋊C₁₆	—	2	2	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
34	#14523977	—	2	1	no	—
35	#14523977	C₂×C₄	1	2	yes	—
36	#14523968	—	2	4	no	—
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
46	#107×C₆₇	—	2	2	no	—
47	#107×C₆₇	—	2	2	no	—
48	#95×C₆₇	—	4	1	no	—
49	C₂×Dic₁₃₄	—	2	2	no	—
50	C₂×Dic₁₃₄	—	4	2	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	1	no	—
53	C₂²×Dic₆₇	—	4	1	no	—
54	C₂²×Dic₆₇	—	4	1	no	—
55	Dic₂₆₈	—	2	4	no	—
56	Dic₂₆₈	—	4	1	no	—
57	Dic₂₆₈	—	4	1	no	—
58	C₄⋊Dic₆₇	—	2	2	no	—
59	C₄⋊Dic₆₇	—	4	1	no	—
60	Dic₆₇⋊C₄	—	4	4	no	—
61	C₂²⋊Dic₆₇	—	4	4	no	—
62	C₆₇⋊C₁₆	—	2	4	no	—
63	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	#41	1	2	yes	—
80	C₄⋊C₄×C₆₇	—	4	2	no	—
81	C₄⋊C₄×C₆₇	—	4	2	no	—
82	C₂²⋊C₄×C₆₇	—	8	2	no	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	#102	1	2	yes	—
86	C₂³×C₆₇	SD₃₂	1	2	yes	—
87	C₂³×C₆₇	Q₃₂	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	8	2	no	—
94	C₂×C₄×C₆₇	—	8	2	no	—
95	C₂×C₄×C₆₇	—	8	2	no	—
96	C₂×C₄×C₆₇	#102	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#121	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#135	1	2	yes	—
101	C₂×Dic₆₇	—	4	2	no	—
102	C₂×Dic₆₇	—	4	2	no	—
103	C₂×Dic₆₇	—	8	2	no	—
104	C₂×Dic₆₇	—	8	2	no	—
105	Dic₁₃₄	—	4	2	no	—
106	Dic₁₃₄	—	4	2	no	—
107	Dic₁₃₄	—	8	2	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	C₄×D₄	1	2	yes	—
110	C₈×C₆₇	#120	1	1	yes	characteristic
111	C₈×C₆₇	#123	1	1	yes	characteristic
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	2	4	no	—
115	C₂²×C₆₇	—	2	4	no	—
116	C₂²×C₆₇	#358	1	4	yes	complemented
117	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
118	C₂²×C₆₇	#466	1	2	yes	—
119	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₆₇	#508	1	2	yes	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	16	2	no	—
124	C₄×C₆₇	#386	1	1	yes	characteristic
125	C₄×C₆₇	#416	1	1	yes	characteristic
126	C₄×C₆₇	#465	1	2	yes	—
127	Dic₆₇	—	8	2	no	—
128	Dic₆₇	—	8	2	no	—
129	#13060	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	4	no	—
131	C₂×C₆₇	#1893	1	1	yes	characteristic
132	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
133	C₂×C₆₇	#2138	1	1	yes	characteristic
134	C₂×C₆₇	#2216	1	2	yes	—
135	C₂×C₆₇	#2217	1	2	yes	—
136	C₂³×C₁₆	—	67	1	no	—
137	C₂×#467	—	67	1	no	—
138	C₂×#358	—	67	2	no	—
139	#1808	D₆₇	1	1	yes	characteristic
140	#1380	—	67	2	no	—
141	C₆₇	#13060	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₁₆	—	67	1	no	—
143	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	—	134	1	no	—
145	C₂²×C₁₆	—	67	2	no	—
146	C₂²×C₁₆	—	134	4	no	—
147	C₂²×C₁₆	—	134	2	no	—
148	#467	—	67	4	no	—
149	#467	—	134	2	no	—
150	C₂×#107	—	67	1	no	—
151	C₂×#107	C₂×D₆₇	1	2	yes	—
152	C₂×#103	—	134	1	no	—
153	#380	—	2	1	no	—
154	#358	—	134	4	no	—
155	#340	—	2	1	no	—
156	C₂²×Q₈	—	134	1	no	—
157	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	67	2	no	—
159	C₂×Q₁₆	—	134	2	no	—
160	C₂×Q₁₆	—	134	1	no	—
161	C₂×Q₁₆	—	134	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₄×D₆₇	1	2	yes	—
165	C₂²×C₈	D₂₆₈	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
168	C₂²⋊Q₈	—	268	2	no	—
169	C₂²⋊Q₈	—	134	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	268	1	no	—
172	C₂×C₄⋊C₄	—	134	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	268	1	no	—
175	C₂×C₁₆	—	67	4	no	—
176	C₂×C₁₆	—	134	8	no	—
177	C₂×C₁₆	—	134	2	no	—
178	C₂×C₁₆	—	134	2	no	—
179	#107	—	2	2	no	—
180	#107	—	2	2	no	—
181	#107	—	67	2	no	—
182	#107	—	134	1	no	—
183	#103	—	134	4	no	—
184	#95	—	4	1	no	—
185	C₂⁴	#11761613	1	1	yes	characteristic
186	C₂×Q₈	—	268	1	no	—
187	C₂×Q₈	—	268	2	no	—
188	C₂×Q₈	—	268	2	no	—
189	C₂×Q₈	—	134	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	268	1	no	—
194	C₂²×C₄	—	268	1	no	—
195	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
196	C₂²×C₄	#11761613	1	1	yes	characteristic
197	C₂²×C₄	#11761615	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
199	Q₁₆	—	268	1	no	—
200	Q₁₆	—	268	1	no	—
201	Q₁₆	—	134	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
207	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
208	C₂×C₈	D₄×D₆₇	1	2	yes	—
209	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
210	C₂×C₈	#11761627	1	2	yes	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	268	4	no	—
214	C₄⋊C₄	—	268	1	no	—
215	C₄⋊C₄	—	134	2	no	—
216	C₂²⋊C₄	—	8	2	no	—
217	C₂²⋊C₄	—	268	4	no	—
218	C₁₆	—	134	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#14523963	1	2	yes	—
222	C₂³	#14523987	1	2	yes	—
223	C₂³	#14523989	1	2	yes	—
224	C₂³	C₂×#11761613	1	1	yes	characteristic
225	Q₈	—	268	2	no	—
226	Q₈	—	268	2	no	—
227	Q₈	—	536	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	268	2	no	—
236	C₂×C₄	—	268	2	no	—
237	C₂×C₄	—	536	2	no	—
238	C₂×C₄	—	536	2	no	—
239	C₂×C₄	#14523963	1	2	yes	—
240	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	#14524015	1	2	yes	—
242	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
243	C₂×C₄	#14524063	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
246	C₈	#14524013	1	1	yes	characteristic
247	C₈	#14524027	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#16863199	1	4	yes	complemented
253	C₂²	C₂×#14523963	1	1	yes	characteristic
254	C₂²	#16863630	1	2	yes	—
255	C₂²	C₂×#14523987	1	1	yes	characteristic
256	C₂²	C₂×#14523989	1	1	yes	characteristic
257	C₂²	#16863826	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	16	2	no	—
260	C₄	—	536	2	no	—
261	C₄	—	536	2	no	—
262	C₄	#16863273	1	1	yes	characteristic
263	C₄	#16863390	1	1	yes	characteristic
264	C₄	#16863627	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	#22560635	1	1	yes	characteristic
267	C₂	C₂×#16863199	1	1	yes	characteristic
268	C₂	#22561914	1	1	yes	characteristic
269	C₂	#22562298	1	2	yes	—
270	C₂	#22562303	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1808	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1808	D₆₇
hypocenter	C₆₇	#13060
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
3. C₂×#16863199	C₂	semidirect
4. #22558300	C₂	semidirect
12. #16863637	C₄	semidirect
116. C₂²×C₆₇	#358	semidirect
141. C₆₇	#13060	semidirect
252. C₂²	#16863199	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂×#16863637	2
C₂×#16863199	2
#22558300	2
C₆₇×#1808	2
#13060	67