Subgroups — C₂×#22559119 #30825596

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863526	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559119	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863128	C₂	1	2	yes	maximal
6	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
9	#16863526	—	2	4	no	—
10	#16863526	C₂²	1	8	yes	—
11	C₄×Dic₂₆₈	C₄	1	4	yes	—
12	C₄×Dic₂₆₈	C₂²	1	4	yes	—
13	C₂×#14523977	C₄	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₆₇⋊C₈	—	2	2	no	—
16	C₂×#14523968	C₄	1	2	yes	—
17	C₂×#14523953	—	2	2	no	—
18	#16863147	C₂²	1	4	yes	—
19	#16863128	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#322×C₆₇	C₂²	1	4	yes	—
24	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₆₈	—	2	4	no	—
26	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₄×Dic₁₃₄	—	2	2	no	—
29	C₄×Dic₁₃₄	—	2	2	no	—
30	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₃₄	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
33	C₂×Dic₆₇⋊C₄	—	2	2	no	—
34	C₂×C₄×Dic₆₇	—	2	2	no	—
35	#14523977	C₂×C₄	1	4	yes	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
38	Dic₆₇⋊C₈	—	2	8	no	—
39	#14523968	C₂×C₄	1	8	yes	—
40	#14523953	—	2	8	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₆₇	—	2	1	no	—
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₄×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂×Dic₁₃₄	—	2	4	no	—
53	C₂×Dic₁₃₄	—	2	4	no	—
54	C₂×Dic₁₃₄	—	2	2	no	—
55	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₆₇	—	2	2	no	—
57	C₂²×Dic₆₇	—	2	2	no	—
58	Dic₂₆₈	—	2	8	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	4	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
63	Dic₆₇⋊C₄	—	2	4	no	—
64	Dic₆₇⋊C₄	—	2	4	no	—
65	C₄×Dic₆₇	—	2	4	no	—
66	C₄×Dic₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	#41	1	1	yes	characteristic
81	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₆₇	#131	1	2	yes	—
89	C₂×C₄×C₆₇	#137	1	4	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₆₇	—	2	4	no	—
94	C₂×Dic₆₇	—	2	4	no	—
95	C₂×Dic₆₇	—	4	4	no	—
96	C₂×Dic₆₇	—	4	4	no	—
97	Dic₁₃₄	—	2	4	no	—
98	Dic₁₃₄	—	2	4	no	—
99	Dic₁₃₄	—	4	4	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	4	no	—
102	C₈×C₆₇	—	4	2	no	—
103	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
104	C₆₇⋊C₈	—	4	4	no	—
105	C₂²×C₆₇	#436	1	2	yes	—
106	C₂²×C₆₇	#441	1	2	yes	—
107	C₂²×C₆₇	#444	1	2	yes	—
108	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	#436	1	2	yes	—
112	C₄×C₆₇	#441	1	2	yes	—
113	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
114	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
115	C₄×C₆₇	C₂×#137	1	2	yes	—
116	Dic₆₇	—	4	4	no	—
117	Dic₆₇	—	4	4	no	—
118	C₂×#1593	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	#1593	1	4	yes	complemented
120	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
123	C₂×#446	—	67	2	no	—
124	C₂×C₄×Q₁₆	—	67	1	no	—
125	#1593	—	67	8	no	—
126	C₂×#334	—	67	1	no	—
127	C₂×Q₈⋊C₈	—	67	2	no	—
128	C₂×#322	D₆₇	1	1	yes	characteristic
129	C₆₇	C₂×#1593	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	67	1	no	—
131	C₂×C₄×Q₈	—	67	2	no	—
132	#446	—	67	8	no	—
133	#446	—	134	4	no	—
134	C₄×Q₁₆	—	67	4	no	—
135	C₄×Q₁₆	—	67	4	no	—
136	C₂×#107	—	67	1	no	—
137	C₂×C₄⋊C₈	—	67	2	no	—
138	C₂×C₄⋊C₈	—	134	2	no	—
139	C₂×#103	—	67	2	no	—
140	C₂×#97	—	134	2	no	—
141	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
144	#334	—	67	4	no	—
145	Q₈⋊C₈	—	67	8	no	—
146	#322	C₂×D₆₇	1	4	yes	—
147	C₂²×Q₈	—	67	2	no	—
148	C₂×Q₁₆	—	67	4	no	—
149	C₂×Q₁₆	—	134	4	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	134	2	no	—
152	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
154	C₂²×C₈	D₂₆₈	1	2	yes	—
155	C₄×Q₈	—	67	4	no	—
156	C₄×Q₈	—	67	4	no	—
157	C₄×Q₈	—	134	2	no	—
158	C₄×Q₈	—	134	2	no	—
159	C₂×C₄⋊C₄	—	67	2	no	—
160	C₂×C₄⋊C₄	—	134	2	no	—
161	C₂×C₄²	—	134	2	no	—
162	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
163	#107	—	67	4	no	—
164	C₄⋊C₈	—	67	4	no	—
165	C₄⋊C₈	—	134	8	no	—
166	C₄⋊C₈	—	134	2	no	—
167	#103	—	67	8	no	—
168	#97	—	134	8	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₄×D₆₇	1	2	yes	—
171	C₄×C₈	D₂₆₈	1	2	yes	—
172	C₄×C₈	C₂²×D₆₇	1	2	yes	—
173	C₄×C₈	C₂²×D₆₇	1	2	yes	—
174	C₄×C₈	C₂²×D₆₇	1	2	yes	—
175	C₂×Q₈	—	67	4	no	—
176	C₂×Q₈	—	134	4	no	—
177	C₂×Q₈	—	134	4	no	—
178	C₂×Q₈	—	134	2	no	—
179	C₂²×C₄	—	134	2	no	—
180	C₂²×C₄	—	134	2	no	—
181	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
183	C₂²×C₄	#11761628	1	1	yes	characteristic
184	Q₁₆	—	134	8	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	268	4	no	—
190	C₂×C₈	—	134	4	no	—
191	C₂×C₈	#11761611	1	4	yes	—
192	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
194	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
195	C₄⋊C₄	—	67	4	no	—
196	C₄⋊C₄	—	134	2	no	—
197	C₄⋊C₄	—	134	4	no	—
198	C₄⋊C₄	—	134	4	no	—
199	C₄²	—	134	4	no	—
200	C₄²	—	134	4	no	—
201	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
202	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
203	C₄²	#11761628	1	1	yes	characteristic
204	C₄²	C₂³×D₆₇	1	1	yes	characteristic
205	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
206	Q₈	—	268	4	no	—
207	Q₈	—	134	4	no	—
208	Q₈	—	134	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	268	4	no	—
211	C₂×C₄	—	268	4	no	—
212	C₂×C₄	—	134	4	no	—
213	C₂×C₄	—	134	4	no	—
214	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
216	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
217	C₂×C₄	C₂×#11761611	1	2	yes	—
218	C₂×C₄	#14524049	1	2	yes	—
219	C₂×C₄	#14524076	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
222	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	4	no	—
225	C₈	—	4	2	no	—
226	C₈	—	268	4	no	—
227	C₈	C₂×#11761611	1	4	yes	—
228	C₂²	#16863475	1	2	yes	—
229	C₂²	#16863504	1	2	yes	—
230	C₂²	#16863519	1	2	yes	—
231	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	268	4	no	—
235	C₄	—	268	4	no	—
236	C₄	#16863475	1	2	yes	—
237	C₄	#16863504	1	2	yes	—
238	C₄	C₂²×#11761611	1	1	yes	characteristic
239	C₄	C₂×#14524049	1	1	yes	characteristic
240	C₄	C₂×#14524076	1	2	yes	—
241	C₂	#22559119	1	4	yes	direct product factor
242	C₂	C₂×#16863475	1	1	yes	characteristic
243	C₂	C₂×#16863504	1	1	yes	characteristic
244	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	C₂×#1593
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #22559119	C₂	direct (N × H)
119. C₂×C₆₇	#1593	semidirect
129. C₆₇	C₂×#1593	semidirect
241. C₂	#22559119	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863526	2
C₂×C₄×Dic₂₆₈	2
#22559119	2
C₂×#16863147	2
C₂×#16863128	2
C₂×#322×C₆₇	2
C₂×#1593	67

Subgroups — C2×#22559119 #30825596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559119 #30825596