Subgroups — #30745250

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	2	yes	maximal
2	#22558040	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863449	—	2	2	no	—
5	#16863449	—	2	2	no	—
6	#16863449	C₄	1	4	yes	—
7	#16863449	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
11	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₆₇	—	2	2	no	—
14	C₄×M₁₆×C₆₇	—	2	2	no	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
17	C₂²×C₆₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
22	C₄×C₆₇⋊C₈	—	2	4	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	—	2	4	no	—
28	C₂×M₁₆×C₆₇	—	2	4	no	—
29	C₂×M₁₆×C₆₇	C₈	1	8	yes	complemented
30	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	#97×C₆₇	—	2	4	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	8	no	—
43	C₂×C₆₇⋊C₈	—	4	8	no	—
44	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
57	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	M₁₆×C₆₇	—	2	16	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	C₂×C₈	1	8	yes	complemented
65	C₄²×C₆₇	C₄²	1	1	yes	characteristic
66	C₄²×C₆₇	C₄²	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₆₇	C₂×D₄	1	2	yes	—
75	C₂³×C₆₇	—	4	1	no	—
76	C₂³×C₆₇	—	4	1	no	—
77	C₂³×C₆₇	#95	1	2	yes	—
78	C₂³×C₆₇	C₄⋊C₈	1	4	yes	—
79	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	—	4	1	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#95	1	2	yes	—
91	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
94	C₂×C₄×C₆₇	#97	1	4	yes	—
95	C₂×C₄×C₆₇	C₂²⋊C₈	1	8	yes	—
96	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
103	C₈×C₆₇	—	2	8	no	—
104	C₆₇⋊C₈	—	4	8	no	—
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	4	4	no	—
110	C₂²×C₆₇	#336	1	4	yes	—
111	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	#336	1	4	yes	—
118	C₄×C₆₇	H(3,ℤ/4)	1	2	yes	—
119	C₄×C₆₇	#338	1	2	yes	—
120	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
121	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
122	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
123	#7860	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	4	no	—
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
127	C₂×C₆₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#338	1	1	yes	characteristic
129	C₂×#431	—	67	2	no	—
130	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
131	#1275	—	67	4	no	—
132	C₆₇	#7860	1	1	yes	characteristic, complemented, Sylow
133	C₂²×M₁₆	Dic₆₇	1	2	yes	—
134	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
135	#431	—	67	4	no	—
136	#431	—	67	4	no	—
137	#431	—	134	2	no	—
138	#431	—	134	2	no	—
139	C₂×C₄⋊C₈	—	134	2	no	—
140	C₂×C₂²⋊C₈	—	134	2	no	—
141	C₄×M₁₆	—	2	2	no	—
142	C₄×M₁₆	—	2	2	no	—
143	C₂×#97	C₂×D₆₇	1	2	yes	—
144	C₂×C₄×C₈	—	67	4	no	—
145	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
146	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
147	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
148	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	C₆₇⋊C₈	1	8	yes	complemented
152	C₂²×C₈	—	134	4	no	—
153	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
154	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
155	C₂×C₄²	C₄×D₆₇	1	2	yes	—
156	C₂×C₄²	C₄×D₆₇	1	2	yes	—
157	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
161	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
162	C₄⋊C₈	—	134	4	no	—
163	C₄⋊C₈	—	134	4	no	—
164	C₂²⋊C₈	—	134	4	no	—
165	C₂²⋊C₈	—	134	4	no	—
166	#97	—	2	4	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	67	8	no	—
170	C₄×C₈	—	134	4	no	—
171	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
175	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
176	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
177	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
178	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
179	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
180	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
184	C₂²×C₄	#11761612	1	4	yes	—
185	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
188	M₁₆	—	2	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	268	8	no	—
192	C₂×C₈	—	134	8	no	—
193	C₂×C₈	C₂×C₆₇⋊C₈	1	8	yes	complemented
194	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
195	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
196	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
197	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
198	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₄²	Dic₆₇⋊C₄	1	2	yes	—
201	C₄²	Dic₆₇⋊C₄	1	2	yes	—
202	C₄²	C₂×C₄×D₆₇	1	2	yes	—
203	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
204	C₂³	—	4	1	no	—
205	C₂³	—	4	1	no	—
206	C₂³	C₆₇⋊#95	1	2	yes	—
207	C₂³	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
208	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
218	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
219	C₂×C₄	C₆₇⋊#95	1	2	yes	—
220	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
221	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
222	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
223	C₂×C₄	#14523954	1	4	yes	—
224	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	8	yes	—
225	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
226	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
227	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
230	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
231	C₂×C₄	C₂×#11761612	1	2	yes	—
232	C₈	—	2	8	no	—
233	C₈	—	268	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	4	no	—
239	C₂²	#16863151	1	4	yes	—
240	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	#16863151	1	4	yes	—
247	C₄	#16863153	1	2	yes	—
248	C₄	C₆₇⋊#338	1	2	yes	—
249	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
250	C₄	C₂×#14523954	1	1	yes	characteristic
251	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	—	2	2	no	—
254	C₂	C₂×#16863151	1	1	yes	characteristic
255	C₂	C₂×#16863153	1	1	yes	characteristic
256	C₂	C₂×C₆₇⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#7860
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22558040	C₂	semidirect
29. C₂×M₁₆×C₆₇	C₈	semidirect
64. C₂×C₈×C₆₇	C₂×C₈	semidirect
132. C₆₇	#7860	semidirect
151. C₂×M₁₆	C₆₇⋊C₈	semidirect
193. C₂×C₈	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
#22558040	2
C₂×C₄×M₁₆×C₆₇	2
#7860	67