Subgroups — C₂×#22558512 #30821537

Order 17152.

994 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11761612	C₂	1	2	yes	maximal
2	#22558512	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863118	C₂	1	2	yes	maximal, complemented
4	C₂×#16863117	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₄	1	4	yes	—
8	C₄×#11761612	C₄	1	8	yes	—
9	C₄×#11761612	C₂²	1	8	yes	—
10	C₂×#14523954	C₄	1	4	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
13	#16863118	C₂²	1	8	yes	complemented
14	#16863117	C₂²	1	4	yes	complemented
15	C₈×C₆₇⋊C₈	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
19	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
20	C₂×#97×C₆₇	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	16	no	—
24	C₂×#11761612	C₂×C₄	1	16	yes	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	#14523954	C₂×C₄	1	16	yes	—
28	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
31	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₆₇	—	2	8	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	#97×C₆₇	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄×C₈×C₆₇	C₂³	1	2	yes	—
49	#11761612	—	2	32	no	—
50	C₂×C₆₇⋊C₈	—	2	8	no	—
51	C₂×C₆₇⋊C₈	M₁₆	1	16	yes	complemented
52	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
53	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	8	no	—
57	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₆₇	—	2	16	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	M₁₆	1	8	yes	complemented
68	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₆₇	C₄²	1	4	yes	—
71	C₄²×C₆₇	C₄²	1	2	yes	—
72	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂³×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
80	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	8	no	—
85	C₂×C₄×C₆₇	—	2	8	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
92	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₆₇	—	2	4	no	—
96	C₈×C₆₇	C₂×M₁₆	1	8	yes	complemented
97	C₆₇⋊C₈	—	2	8	no	—
98	C₆₇⋊C₈	C₂×M₁₆	1	16	yes	complemented
99	C₂²×C₆₇	—	2	4	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	4	4	no	—
102	C₂²×C₆₇	C₄×M₁₆	1	4	yes	—
103	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
104	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
105	C₄×C₆₇	—	2	4	no	—
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	C₄×M₁₆	1	4	yes	—
108	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
109	C₄×C₆₇	C₂²×M₁₆	1	4	yes	—
110	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
111	C₂×#1455	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	—	4	2	no	—
113	C₂×C₆₇	#1455	1	4	yes	complemented
114	C₂×C₆₇	C₂×C₄×M₁₆	1	2	yes	—
115	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
116	C₂×C₄×M₁₆	—	67	2	no	—
117	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
118	#1455	—	67	8	no	—
119	C₂×#322	—	67	1	no	—
120	C₂×#322	—	67	2	no	—
121	C₂×C₈²	—	67	1	no	—
122	C₆₇	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	—	67	4	no	—
124	C₂²×M₁₆	Dic₆₇	1	2	yes	—
125	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
126	C₄×M₁₆	—	67	8	no	—
127	C₄×M₁₆	—	67	8	no	—
128	C₄×M₁₆	Dic₆₇	1	4	yes	—
129	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
130	C₂×#97	—	67	4	no	—
131	C₂×#97	Dic₆₇	1	2	yes	—
132	C₂×C₄×C₈	—	67	2	no	—
133	C₂×C₄×C₈	—	67	2	no	—
134	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
136	#322	—	67	4	no	—
137	#322	—	67	8	no	—
138	C₈²	—	67	4	no	—
139	C₂³×C₄	C₄×D₆₇	1	2	yes	—
140	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	8	no	—
142	C₂×M₁₆	—	67	16	no	—
143	C₂×M₁₆	—	134	16	no	—
144	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
145	C₂²×C₈	—	67	4	no	—
146	C₂²×C₈	—	67	4	no	—
147	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
148	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
149	C₂×C₄²	C₄×D₆₇	1	2	yes	—
150	C₂×C₄²	C₄×D₆₇	1	4	yes	—
151	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
153	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
154	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
155	#97	—	67	16	no	—
156	#97	C₂×Dic₆₇	1	8	yes	—
157	C₄×C₈	—	67	4	no	—
158	C₄×C₈	—	67	4	no	—
159	C₄×C₈	—	67	4	no	—
160	C₄×C₈	—	67	4	no	—
161	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
162	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
163	C₄×C₈	C₂²×D₆₇	1	2	yes	—
164	C₄×C₈	C₂²×D₆₇	1	2	yes	—
165	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
170	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
171	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
172	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
176	M₁₆	—	2	16	no	—
177	M₁₆	—	134	32	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	67	16	no	—
180	C₂×C₈	—	67	8	no	—
181	C₂×C₈	—	67	8	no	—
182	C₂×C₈	—	134	8	no	—
183	C₂×C₈	#11761612	1	8	yes	complemented
184	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
185	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
186	C₄²	C₄×Dic₆₇	1	4	yes	—
187	C₄²	C₄×Dic₆₇	1	2	yes	—
188	C₄²	C₂×C₄×D₆₇	1	2	yes	—
189	C₄²	C₂×C₄×D₆₇	1	4	yes	—
190	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
191	C₄²	C₂²×Dic₆₇	1	2	yes	—
192	C₄²	C₂³×D₆₇	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
196	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
205	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₆₇	1	8	yes	—
208	C₂×C₄	C₂×#11761612	1	4	yes	—
209	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
210	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
211	C₈	—	2	4	no	—
212	C₈	—	67	16	no	—
213	C₈	—	134	8	no	—
214	C₈	C₂×#11761612	1	8	yes	complemented
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	M₁₆×Dic₆₇	1	4	yes	—
219	C₂²	C₄×#11761612	1	2	yes	—
220	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	M₁₆×Dic₆₇	1	4	yes	—
224	C₄	C₄×#11761612	1	2	yes	—
225	C₄	C₂×M₁₆×D₆₇	1	4	yes	—
226	C₄	C₂²×#11761612	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#22558512	1	4	yes	direct product factor
229	C₂	C₂×M₁₆×Dic₆₇	1	2	yes	—
230	C₂	C₂×C₄×#11761612	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄²
prime radical	C₂⁴×C₆₇	C₄²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₂×#1455
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22558512	C₂	direct (N × H)
3. C₂×#16863118	C₂	semidirect
4. C₂×#16863117	C₂	semidirect
5. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
13. #16863118	C₂²	semidirect
14. #16863117	C₂²	semidirect
15. C₈×C₆₇⋊C₈	C₂²	semidirect
51. C₂×C₆₇⋊C₈	M₁₆	semidirect
67. C₂×C₈×C₆₇	M₁₆	semidirect
96. C₈×C₆₇	C₂×M₁₆	semidirect
98. C₆₇⋊C₈	C₂×M₁₆	semidirect
113. C₂×C₆₇	#1455	semidirect
122. C₆₇	C₂×#1455	semidirect
183. C₂×C₈	#11761612	semidirect
214. C₈	C₂×#11761612	semidirect
228. C₂	#22558512	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×C₄×#11761612	2
#22558512	2
C₂×#16863118	2
C₂×#16863117	2
C₂×C₈×C₆₇⋊C₈	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#1455	67

Subgroups — C2×#22558512 #30821537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558512 #30821537