Subgroups — #30755125

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22561160	C₂	1	2	yes	maximal
3	#22560076	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863141	C₂	1	2	yes	maximal
5	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	—	2	1	no	—
8	C₂×#14524043	C₂²	1	2	yes	—
9	#16863776	—	2	2	no	—
10	#16863776	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14523953	—	2	1	no	—
13	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	#16863159	—	2	4	no	—
15	#16863150	—	2	1	no	—
16	#16863141	—	2	4	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#334×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	2	4	no	—
23	C₂×Dic₂₆₈	—	4	2	no	—
24	C₂×Dic₂₆₈	—	4	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	—	4	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	#14524043	—	2	2	no	—
29	#14524043	—	2	2	no	—
30	#14524043	—	2	2	no	—
31	#14524043	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
38	#14523953	—	4	2	no	—
39	C₆₇⋊#95	—	2	1	no	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	—	2	2	no	—
45	C₄⋊C₈×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	D₄	1	2	yes	—
48	C₂×Dic₁₃₄	—	2	4	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂×Dic₁₃₄	—	4	2	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	2	no	—
53	C₂×Dic₁₃₄	—	4	2	no	—
54	C₂²×Dic₆₇	—	2	2	no	—
55	C₂²×Dic₆₇	—	4	2	no	—
56	Dic₂₆₈	—	4	8	no	—
57	Dic₂₆₈	—	4	8	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	4	1	no	—
60	C₂×C₆₇⋊C₈	—	4	1	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	2	4	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₄⋊Dic₆₇	—	4	2	no	—
66	C₄×Dic₆₇	—	4	1	no	—
67	C₄×Dic₆₇	—	4	1	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	4	4	no	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₆₇	D₈	1	2	yes	—
78	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	#41	1	1	yes	characteristic
83	C₂³×C₆₇	#121	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#137	1	4	yes	—
96	C₂×Dic₆₇	—	4	4	no	—
97	C₂×Dic₆₇	—	4	2	no	—
98	C₂×Dic₆₇	—	4	2	no	—
99	C₂×Dic₆₇	—	4	2	no	—
100	C₂×Dic₆₇	—	8	2	no	—
101	C₂×Dic₆₇	—	8	2	no	—
102	Dic₁₃₄	—	4	4	no	—
103	Dic₁₃₄	—	4	4	no	—
104	Dic₁₃₄	—	8	2	no	—
105	Dic₁₃₄	—	8	2	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	—	8	2	no	—
108	C₈×C₆₇	#102	1	4	yes	—
109	C₆₇⋊C₈	—	4	2	no	—
110	C₂²×C₆₇	#390	1	1	yes	characteristic
111	C₂²×C₆₇	#459	1	1	yes	characteristic
112	C₂²×C₆₇	#460	1	1	yes	characteristic
113	C₂²×C₆₇	#470	1	2	yes	—
114	C₂²×C₆₇	#473	1	2	yes	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
118	C₄×C₆₇	#418	1	1	yes	characteristic
119	C₄×C₆₇	#441	1	2	yes	—
120	C₄×C₆₇	#459	1	1	yes	characteristic
121	C₄×C₆₇	#460	1	1	yes	characteristic
122	C₄×C₆₇	#497	1	2	yes	—
123	Dic₆₇	—	8	2	no	—
124	Dic₆₇	—	8	4	no	—
125	#9742	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	#1689	1	2	yes	—
127	C₂×C₆₇	#1690	1	2	yes	—
128	C₂×C₆₇	#1971	1	1	yes	characteristic
129	C₂×C₆₇	#1979	1	1	yes	characteristic
130	C₂×C₆₇	#1982	1	1	yes	characteristic
131	C₂×#494	—	67	1	no	—
132	#1988	—	67	2	no	—
133	#1770	—	67	1	no	—
134	C₂×#334	D₆₇	1	1	yes	characteristic
135	C₂×#332	—	67	2	no	—
136	C₆₇	#9742	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₁₆	—	134	1	no	—
138	C₂²×Q₁₆	—	134	1	no	—
139	C₂×C₄⋊Q₈	—	67	2	no	—
140	#494	—	67	4	no	—
141	#494	—	134	2	no	—
142	C₂×C₄⋊C₈	—	67	1	no	—
143	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
144	C₂×#97	—	134	1	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	67	1	no	—
147	#340	—	134	4	no	—
148	#336	—	134	1	no	—
149	#334	—	2	2	no	—
150	#332	—	134	4	no	—
151	C₂²×Q₈	—	134	2	no	—
152	C₂×Q₁₆	—	268	2	no	—
153	C₂×Q₁₆	—	268	2	no	—
154	C₂×Q₁₆	—	268	2	no	—
155	C₂×Q₁₆	—	268	2	no	—
156	C₂×Q₁₆	—	134	4	no	—
157	C₂×Q₁₆	—	134	4	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	134	1	no	—
160	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
161	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
162	C₄⋊Q₈	—	67	4	no	—
163	C₄⋊Q₈	—	134	2	no	—
164	C₄⋊Q₈	—	134	2	no	—
165	C₄⋊Q₈	—	134	2	no	—
166	C₂×C₄⋊C₄	—	67	2	no	—
167	C₂×C₄⋊C₄	—	134	2	no	—
168	C₂×C₄⋊C₄	—	134	2	no	—
169	C₂×C₄²	—	134	1	no	—
170	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	67	2	no	—
174	C₄⋊C₈	—	134	1	no	—
175	#97	—	268	2	no	—
176	C₄×C₈	C₄×D₆₇	1	2	yes	—
177	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
178	#95	—	134	1	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	2	no	—
181	C₂×Q₈	—	268	2	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	2	no	—
184	C₂×Q₈	—	134	4	no	—
185	C₂²×C₄	—	268	2	no	—
186	C₂²×C₄	—	134	2	no	—
187	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
189	C₂²×C₄	#11761629	1	1	yes	characteristic
190	Q₁₆	—	268	8	no	—
191	Q₁₆	—	268	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	268	1	no	—
197	C₂×C₈	—	268	1	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂×C₈	D₅₃₆	1	2	yes	—
202	C₂×C₈	#11761617	1	2	yes	—
203	C₄⋊C₄	—	268	2	no	—
204	C₄⋊C₄	—	268	2	no	—
205	C₄⋊C₄	—	134	4	no	—
206	C₄⋊C₄	—	134	4	no	—
207	C₄⋊C₄	—	134	4	no	—
208	C₄²	—	268	1	no	—
209	C₄²	—	268	1	no	—
210	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₄²	D₄×D₆₇	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₄²	#11761629	1	1	yes	characteristic
214	C₂³	#14524018	1	1	yes	characteristic
215	Q₈	—	268	4	no	—
216	Q₈	—	268	4	no	—
217	Q₈	—	536	2	no	—
218	Q₈	—	536	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	268	4	no	—
221	C₂×C₄	—	268	2	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	536	2	no	—
225	C₂×C₄	—	536	2	no	—
226	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	#14524005	1	1	yes	characteristic
228	C₂×C₄	#14524005	1	1	yes	characteristic
229	C₂×C₄	#14524018	1	1	yes	characteristic
230	C₂×C₄	#14524032	1	1	yes	characteristic
231	C₂×C₄	#14524039	1	1	yes	characteristic
232	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
234	C₂×C₄	#14524071	1	1	yes	characteristic
235	C₂×C₄	#14524071	1	1	yes	characteristic
236	C₂×C₄	#14524072	1	4	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	8	2	no	—
239	C₈	—	268	2	no	—
240	C₈	#14523965	1	4	yes	—
241	C₂²	#16863297	1	1	yes	characteristic
242	C₂²	#16863592	1	1	yes	characteristic
243	C₂²	#16863597	1	1	yes	characteristic
244	C₂²	#16863655	1	2	yes	—
245	C₂²	#16863672	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	536	4	no	—
249	C₄	—	536	2	no	—
250	C₄	C₂×#14523965	1	1	yes	characteristic
251	C₄	#16863402	1	1	yes	characteristic
252	C₄	#16863502	1	2	yes	—
253	C₄	#16863592	1	1	yes	characteristic
254	C₄	#16863597	1	1	yes	characteristic
255	C₄	#16863787	1	2	yes	—
256	C₂	#22559699	1	2	yes	—
257	C₂	#22559704	1	2	yes	—
258	C₂	#22561078	1	1	yes	characteristic
259	C₂	#22561122	1	1	yes	characteristic
260	C₂	#22561135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#9742
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
10. #16863776	C₄	semidirect
136. C₆₇	#9742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
#22561160	2
#22560076	2
C₂×#16863141	2
C₂×#334×C₆₇	2
#9742	67