Subgroups — C₂×#22558042 #30745267

Order 17152.

974 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#22558042	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863449	—	2	2	no	—
5	#16863449	—	2	2	no	—
6	#16863449	C₄	1	4	yes	complemented
7	#16863449	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	complemented
9	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
11	C₂²×C₄²×C₆₇	—	2	2	no	—
12	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₄³×C₆₇	—	2	2	no	—
15	C₄³×C₆₇	C₄	1	2	yes	—
16	C₄³×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₆₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	complemented
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	—	2	4	no	—
24	C₂³×C₄×C₆₇	—	2	2	no	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	D₄	1	2	yes	—
27	C₂×C₄²×C₆₇	—	2	16	no	—
28	C₂×C₄²×C₆₇	—	2	4	no	—
29	C₂×C₄²×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	—	2	4	no	—
31	C₂×C₄²×C₆₇	—	2	4	no	—
32	C₂×C₄²×C₆₇	C₈	1	8	yes	complemented
33	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₆₇⋊C₈	—	2	8	no	—
44	C₂×C₆₇⋊C₈	—	4	8	no	—
45	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	—	2	16	no	—
47	C₂²×C₄×C₆₇	—	2	8	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₄²×C₆₇	—	2	16	no	—
65	C₄²×C₆₇	—	2	16	no	—
66	C₄²×C₆₇	—	2	8	no	—
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	C₄²	1	2	yes	—
69	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×C₈	1	8	yes	complemented
76	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
79	C₂³×C₆₇	—	2	4	no	—
80	C₂³×C₆₇	#95	1	1	yes	characteristic
81	C₂³×C₆₇	#95	1	1	yes	characteristic
82	C₂³×C₆₇	C₂²⋊C₈	1	4	yes	—
83	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	16	no	—
85	C₂×C₄×C₆₇	—	2	32	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	#95	1	2	yes	—
93	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#95	1	2	yes	—
96	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₆₇	#97	1	2	yes	—
99	C₂×C₄×C₆₇	#97	1	2	yes	—
100	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₆₇	#104	1	8	yes	—
102	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
103	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
110	C₆₇⋊C₈	—	4	8	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	#336	1	2	yes	—
115	C₂²×C₆₇	#336	1	2	yes	—
116	C₂²×C₆₇	#339	1	4	yes	—
117	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
118	C₂²×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
119	C₄×C₆₇	—	2	16	no	—
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	#336	1	2	yes	—
123	C₄×C₆₇	#336	1	2	yes	—
124	C₄×C₆₇	#339	1	4	yes	—
125	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
126	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
127	C₄×C₆₇	C₂×#104	1	4	yes	—
128	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
129	C₂×#1276	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	2	no	—
131	C₂×C₆₇	—	2	2	no	—
132	C₂×C₆₇	#1276	1	4	yes	complemented
133	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#339	1	2	yes	—
135	C₂×#431	—	67	2	no	—
136	C₂×C₄³	D₆₇	1	1	yes	characteristic
137	#1276	—	67	4	no	—
138	C₆₇	C₂×#1276	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄²	—	2	2	no	—
140	C₂²×C₄²	Dic₆₇	1	2	yes	—
141	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
142	#431	—	67	4	no	—
143	#431	—	67	4	no	—
144	#431	—	134	2	no	—
145	#431	—	134	2	no	—
146	C₂×C₄⋊C₈	—	67	4	no	—
147	C₂×C₂²⋊C₈	—	134	2	no	—
148	C₂×C₄×C₈	—	134	2	no	—
149	C₄³	—	2	2	no	—
150	C₄³	Dic₆₇	1	2	yes	—
151	C₄³	C₂×D₆₇	1	2	yes	—
152	C₂³×C₄	—	2	2	no	—
153	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
154	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
155	C₂²×C₈	—	134	4	no	—
156	C₂×C₄²	—	2	16	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	—	2	4	no	—
160	C₂×C₄²	—	2	4	no	—
161	C₂×C₄²	C₆₇⋊C₈	1	8	yes	complemented
162	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
163	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₆₇	1	2	yes	—
165	C₂×C₄²	C₄×D₆₇	1	2	yes	—
166	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
167	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
168	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
169	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
171	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
172	C₄⋊C₈	—	67	8	no	—
173	C₄⋊C₈	—	134	4	no	—
174	C₂²⋊C₈	—	134	4	no	—
175	C₂²⋊C₈	—	134	4	no	—
176	C₄×C₈	—	134	4	no	—
177	C₄×C₈	—	134	4	no	—
178	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
185	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
186	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
187	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
188	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
190	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
191	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
194	C₂²×C₄	#11761612	1	4	yes	—
195	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
197	C₂×C₈	—	268	8	no	—
198	C₂×C₈	—	134	8	no	—
199	C₄²	—	2	16	no	—
200	C₄²	—	2	16	no	—
201	C₄²	—	2	8	no	—
202	C₄²	—	2	4	no	—
203	C₄²	C₄×Dic₆₇	1	2	yes	—
204	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
205	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
206	C₄²	Dic₆₇⋊C₄	1	2	yes	—
207	C₄²	Dic₆₇⋊C₄	1	2	yes	—
208	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
209	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
210	C₄²	C₂×C₆₇⋊C₈	1	8	yes	complemented
211	C₄²	C₂×C₄×D₆₇	1	2	yes	—
212	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
213	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
214	C₂³	—	2	4	no	—
215	C₂³	C₆₇⋊#95	1	1	yes	characteristic
216	C₂³	C₆₇⋊#95	1	1	yes	characteristic
217	C₂³	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
218	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	2	32	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	C₆₇⋊#95	1	2	yes	—
228	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
229	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
230	C₂×C₄	C₆₇⋊#95	1	2	yes	—
231	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
232	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
233	C₂×C₄	#14523954	1	2	yes	—
234	C₂×C₄	#14523954	1	2	yes	—
235	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
236	C₂×C₄	#14523971	1	8	yes	—
237	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
238	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
239	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
240	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
244	C₂×C₄	C₂×#11761612	1	2	yes	—
245	C₈	—	268	8	no	—
246	C₂²	—	2	8	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	#16863151	1	2	yes	—
250	C₂²	#16863151	1	2	yes	—
251	C₂²	#16863156	1	4	yes	—
252	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
253	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
254	C₄	—	2	16	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	#16863151	1	2	yes	—
258	C₄	#16863151	1	2	yes	—
259	C₄	#16863156	1	4	yes	—
260	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
261	C₄	C₂×#14523954	1	1	yes	characteristic
262	C₄	C₂×#14523971	1	4	yes	—
263	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
264	C₂	—	2	2	no	—
265	C₂	—	2	2	no	—
266	C₂	#22558042	1	4	yes	direct product factor
267	C₂	C₂×#16863151	1	1	yes	characteristic
268	C₂	C₂×#16863156	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₆₈
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄³×C₆₇	C₂
socle	C₂³×C₆₇	#95
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	C₂×#1276
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22558042	C₂	direct (N × H)
6. #16863449	C₄	semidirect
8. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
19. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
32. C₂×C₄²×C₆₇	C₈	semidirect
75. C₄²×C₆₇	C₂×C₈	semidirect
132. C₂×C₆₇	#1276	semidirect
138. C₆₇	C₂×#1276	semidirect
161. C₂×C₄²	C₆₇⋊C₈	semidirect
210. C₄²	C₂×C₆₇⋊C₈	semidirect
266. C₂	#22558042	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×#16863449	2
#22558042	2
C₂×C₄³×C₆₇	2
C₂×#1276	67

Subgroups — C2×#22558042 #30745267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558042 #30745267