Subgroups — C₂×#22568804 #31719498

Order 17152.

2766 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863879	C₂	1	2	yes	maximal
2	#22568804	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863438	C₂	1	6	yes	maximal, complemented
4	C₂×#16863434	C₂	1	3	yes	maximal, complemented
5	C₂²×#14523975	C₂	1	3	yes	maximal, complemented
6	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863879	—	2	8	no	—
8	#16863879	C₂²	1	16	yes	—
9	C₂×Q₈×Dic₆₇	—	2	2	no	—
10	C₂×D₄×Dic₆₇	—	2	6	no	—
11	C₂×#14524002	C₂²	1	6	yes	—
12	C₂²×C₄⋊Dic₆₇	C₂²	1	6	yes	—
13	#16863438	C₂²	1	48	yes	complemented
14	#16863434	C₂²	1	24	yes	complemented
15	C₂×#14523977	C₂²	1	6	yes	—
16	C₂×#14523975	C₂²	1	3	yes	—
17	C₂×#14523975	C₂²	1	6	yes	—
18	C₂×#14523975	C₂²	1	1	yes	characteristic
19	C₂×#14523975	C₂²	1	24	yes	complemented
20	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₄	1	16	yes	complemented
22	C₂×#131×C₆₇	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₆₇	C₂²	1	3	yes	—
24	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
25	C₂³×Dic₆₇	—	2	6	no	—
26	Q₈×Dic₆₇	—	2	16	no	—
27	D₄×Dic₆₇	—	2	48	no	—
28	#14524002	—	2	12	no	—
29	#14524002	C₂³	1	24	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	6	no	—
31	C₂×C₄⋊Dic₆₇	—	2	12	no	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
34	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
35	C₂×C₂²⋊Dic₆₇	—	2	6	no	—
36	C₂×C₄×Dic₆₇	—	2	6	no	—
37	#14523977	C₂³	1	24	yes	—
38	#14523975	C₂³	1	12	yes	—
39	#14523975	C₂³	1	24	yes	—
40	#14523975	C₂³	1	4	yes	—
41	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
42	C₂×#41×C₆₇	D₄	1	4	yes	—
43	C₂×#41×C₆₇	Q₈	1	8	yes	—
44	C₂×#41×C₆₇	C₂³	1	4	yes	—
45	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	D₄	1	3	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
48	#131×C₆₇	C₂×C₄	1	64	yes	complemented
49	C₂×M₁₆×C₆₇	C₂×C₄	1	24	yes	complemented
50	C₂×M₁₆×C₆₇	C₂³	1	12	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	24	yes	complemented
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
55	C₂²×Dic₆₇	—	2	2	no	—
56	C₂²×Dic₆₇	—	2	6	no	—
57	C₂²×Dic₆₇	—	2	24	no	—
58	C₂²×Dic₆₇	—	8	12	no	—
59	C₄⋊Dic₆₇	—	2	6	no	—
60	C₄⋊Dic₆₇	—	2	24	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
63	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
64	C₂²⋊Dic₆₇	—	2	24	no	—
65	C₄×Dic₆₇	—	2	24	no	—
66	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
67	#41×C₆₇	C₄⋊C₄	1	32	yes	—
68	#41×C₆₇	C₂²×C₄	1	16	yes	—
69	#41×C₆₇	C₂×Q₈	1	16	yes	—
70	C₂×Q₈×C₆₇	C₄⋊C₄	1	8	yes	—
71	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
72	C₂×D₄×C₆₇	C₄⋊C₄	1	24	yes	—
73	C₂×D₄×C₆₇	C₂×D₄	1	12	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₆₇	C₂×Q₈	1	12	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
80	M₁₆×C₆₇	C₂²×C₄	1	48	yes	complemented
81	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	48	yes	complemented
84	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
85	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
86	C₂³×C₆₇	—	2	12	no	—
87	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
88	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
89	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
91	D₄×C₆₇	C₂×C₄⋊C₄	1	48	yes	—
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
93	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
98	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
100	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
101	C₂×Dic₆₇	—	2	4	no	—
102	C₂×Dic₆₇	—	2	12	no	—
103	C₂×Dic₆₇	—	4	4	no	—
104	C₂×Dic₆₇	—	4	12	no	—
105	C₂×Dic₆₇	—	8	24	no	—
106	C₈×C₆₇	C₂³×C₄	1	4	yes	—
107	C₈×C₆₇	C₂³×C₄	1	12	yes	—
108	C₂²×C₆₇	—	2	24	no	—
109	C₂²×C₆₇	—	2	12	no	—
110	C₂²×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
111	C₂²×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
112	C₂²×C₆₇	#577	1	4	yes	—
113	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
114	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
115	C₄×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
116	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
117	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
118	Dic₆₇	—	4	4	no	—
119	Dic₆₇	—	4	12	no	—
120	C₂×#2912	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	12	no	—
122	C₂×C₆₇	#2912	1	4	yes	complemented
123	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₆₇	C₂×#577	1	2	yes	—
125	C₂²×#131	D₆₇	1	1	yes	characteristic
126	C₂×#520	—	67	2	no	—
127	#2912	—	67	16	no	—
128	C₂×#428	—	67	6	no	—
129	C₂×#427	—	67	3	no	—
130	C₂²×#106	—	67	3	no	—
131	C₆₇	C₂×#2912	1	1	yes	characteristic, complemented, Sylow
132	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
133	C₂×#131	Dic₆₇	1	16	yes	complemented
134	C₂×#131	C₂×D₆₇	1	8	yes	—
135	C₂²×M₁₆	C₂×D₆₇	1	3	yes	—
136	C₂³×C₈	C₂×D₆₇	1	3	yes	—
137	#520	—	67	16	no	—
138	#520	—	134	8	no	—
139	C₂×C₄×Q₈	—	134	2	no	—
140	C₂×C₄×D₄	—	134	6	no	—
141	C₂×#117	—	67	6	no	—
142	C₂²×C₄⋊C₄	—	67	6	no	—
143	#428	—	67	48	no	—
144	#427	—	67	24	no	—
145	C₂×#107	—	67	6	no	—
146	C₂×#106	—	67	3	no	—
147	C₂×#106	—	67	6	no	—
148	C₂×#106	—	67	24	no	—
149	C₂×#106	—	67	1	no	—
150	C₂×#41	Dic₁₃₄	1	8	yes	—
151	C₂×#41	D₂₆₈	1	4	yes	—
152	C₂×#41	C₂×Dic₆₇	1	8	yes	—
153	C₂×#41	C₂²×D₆₇	1	4	yes	—
154	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
155	C₂²×D₄	D₂₆₈	1	3	yes	—
156	C₂³×C₄	—	134	6	no	—
157	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
158	#131	C₂×Dic₆₇	1	64	yes	complemented
159	C₂×M₁₆	C₂×Dic₆₇	1	24	yes	complemented
160	C₂×M₁₆	C₂²×D₆₇	1	12	yes	—
161	C₂²×C₈	C₂×Dic₆₇	1	24	yes	complemented
162	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
164	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
165	C₄×Q₈	—	134	16	no	—
166	C₄×D₄	—	134	48	no	—
167	#117	—	67	24	no	—
168	#117	—	134	12	no	—
169	C₂×C₄⋊C₄	—	67	6	no	—
170	C₂×C₄⋊C₄	—	67	24	no	—
171	C₂×C₄⋊C₄	—	67	2	no	—
172	C₂×C₄⋊C₄	—	134	6	no	—
173	C₂×C₄⋊C₄	—	134	12	no	—
174	C₂×C₂²⋊C₄	—	134	6	no	—
175	C₂×C₄²	—	134	6	no	—
176	#107	—	67	24	no	—
177	#106	—	67	12	no	—
178	#106	—	67	24	no	—
179	#106	—	67	4	no	—
180	C₂⁴	C₂×D₂₆₈	1	3	yes	—
181	#41	C₄⋊Dic₆₇	1	32	yes	—
182	#41	C₂²×Dic₆₇	1	16	yes	—
183	#41	C₂×Dic₁₃₄	1	16	yes	—
184	C₂×Q₈	C₄⋊Dic₆₇	1	8	yes	—
185	C₂×Q₈	C₂×D₂₆₈	1	4	yes	—
186	C₂×D₄	C₄⋊Dic₆₇	1	24	yes	—
187	C₂×D₄	C₂×D₂₆₈	1	12	yes	—
188	C₂²×C₄	—	536	12	no	—
189	C₂²×C₄	—	134	2	no	—
190	C₂²×C₄	—	134	6	no	—
191	C₂²×C₄	—	134	24	no	—
192	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
193	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
194	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
195	C₂²×C₄	C₂×Dic₁₃₄	1	12	yes	—
196	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
197	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
198	M₁₆	C₂²×Dic₆₇	1	48	yes	complemented
199	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
200	C₂×C₈	C₂²×Dic₆₇	1	12	yes	—
201	C₂×C₈	C₂²×Dic₆₇	1	48	yes	complemented
202	C₂×C₈	C₂³×D₆₇	1	2	yes	—
203	C₂×C₈	C₂³×D₆₇	1	6	yes	—
204	C₄⋊C₄	—	67	12	no	—
205	C₄⋊C₄	—	67	4	no	—
206	C₄⋊C₄	—	134	6	no	—
207	C₄⋊C₄	—	134	24	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₂²⋊C₄	—	134	24	no	—
210	C₄²	—	134	24	no	—
211	C₂³	—	2	12	no	—
212	C₂³	C₂×C₄⋊Dic₆₇	1	12	yes	—
213	C₂³	C₂²×D₂₆₈	1	6	yes	—
214	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
215	Q₈	C₂×C₄⋊Dic₆₇	1	16	yes	—
216	D₄	C₂×C₄⋊Dic₆₇	1	48	yes	—
217	C₂×C₄	—	268	12	no	—
218	C₂×C₄	—	268	4	no	—
219	C₂×C₄	—	536	24	no	—
220	C₂×C₄	—	134	4	no	—
221	C₂×C₄	—	134	12	no	—
222	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
223	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
224	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
225	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
226	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
227	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
228	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
229	C₂×C₄	C₂²×Dic₁₃₄	1	12	yes	—
230	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
231	C₈	C₂³×Dic₆₇	1	4	yes	—
232	C₈	C₂³×Dic₆₇	1	12	yes	—
233	C₂²	—	2	24	no	—
234	C₂²	—	2	12	no	—
235	C₂²	C₂²×C₄⋊Dic₆₇	1	12	yes	—
236	C₂²	C₂²×C₄⋊Dic₆₇	1	2	yes	—
237	C₂²	#16864208	1	4	yes	—
238	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
239	C₄	—	268	12	no	—
240	C₄	—	268	4	no	—
241	C₄	C₂²×C₄⋊Dic₆₇	1	2	yes	—
242	C₄	C₂²×C₄⋊Dic₆₇	1	12	yes	—
243	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
244	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
245	C₂	—	2	12	no	—
246	C₂	#22568804	1	4	yes	direct product factor
247	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
248	C₂	C₂×#16864208	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#131×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂²×#41×C₆₇	C₂²
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×#131	D₆₇
hypocenter	C₆₇	C₂×#2912
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #22568804	C₂	direct (N × H)
3. C₂×#16863438	C₂	semidirect
4. C₂×#16863434	C₂	semidirect
5. C₂²×#14523975	C₂	semidirect
13. #16863438	C₂²	semidirect
14. #16863434	C₂²	semidirect
19. C₂×#14523975	C₂²	semidirect
21. C₂×#131×C₆₇	C₄	semidirect
48. #131×C₆₇	C₂×C₄	semidirect
49. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
51. C₂²×C₈×C₆₇	C₂×C₄	semidirect
80. M₁₆×C₆₇	C₂²×C₄	semidirect
83. C₂×C₈×C₆₇	C₂²×C₄	semidirect
122. C₂×C₆₇	#2912	semidirect
131. C₆₇	C₂×#2912	semidirect
133. C₂×#131	Dic₆₇	semidirect
158. #131	C₂×Dic₆₇	semidirect
159. C₂×M₁₆	C₂×Dic₆₇	semidirect
161. C₂²×C₈	C₂×Dic₆₇	semidirect
198. M₁₆	C₂²×Dic₆₇	semidirect
201. C₂×C₈	C₂²×Dic₆₇	semidirect
246. C₂	#22568804	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×#16863879	2
#22568804	2
C₂×#16863438	2
C₂×#16863434	2
C₂²×#14523975	2
C₂²×#131×C₆₇	2
C₂×#2912	67

Subgroups — C2×#22568804 #31719498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22568804 #31719498