Subgroups — C₂×#22562008 #31008896

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
7	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	complemented
8	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	8	yes	complemented
10	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₆₇⋊C₁₆	—	2	8	no	—
20	C₂×C₆₇⋊C₁₆	C₂³	1	8	yes	—
21	C₂³×C₄×C₆₇	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	—	2	4	no	—
25	C₂²×C₈×C₆₇	—	2	4	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₈	1	4	yes	—
35	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₆₇	C₂³	1	2	yes	—
44	C₄×C₈×C₆₇	C₂³	1	2	yes	—
45	C₆₇⋊C₁₆	—	2	8	no	—
46	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	C₁₆	1	8	yes	complemented
51	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	—	2	8	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	#41	1	4	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×C₈	1	4	yes	—
74	C₄²×C₆₇	C₂×C₈	1	4	yes	—
75	C₄²×C₆₇	C₂×C₈	1	2	yes	—
76	C₄²×C₆₇	C₂×C₈	1	2	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	C₂×C₁₆	1	4	yes	—
82	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
83	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	8	no	—
90	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
91	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×C₁₆	1	16	yes	complemented
93	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
94	C₂×C₄×C₆₇	#117	1	2	yes	—
95	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	#117	1	2	yes	—
106	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
107	C₈×C₆₇	C₂×#41	1	2	yes	—
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	8	no	—
111	C₂²×C₆₇	#431	1	2	yes	—
112	C₂²×C₆₇	C₂²×C₁₆	1	4	yes	—
113	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
114	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
115	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	#431	1	2	yes	—
119	C₄×C₆₇	C₂²×C₁₆	1	4	yes	—
120	C₄×C₆₇	C₂²×C₁₆	1	4	yes	—
121	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
122	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
123	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×#2162	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	4	no	—
126	C₂×C₆₇	#2162	1	4	yes	complemented
127	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
128	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
129	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
130	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
131	#2162	—	67	8	no	—
132	C₂×C₄⋊C₁₆	—	67	2	no	—
133	C₂×C₂²⋊C₁₆	—	67	2	no	—
134	C₂×C₄×C₁₆	—	67	2	no	—
135	C₆₇	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
137	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
139	C₂²×C₁₆	—	67	4	no	—
140	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
141	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
142	C₂×C₄×C₈	Dic₆₇	1	4	yes	—
143	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₆₇	1	4	yes	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	C₄⋊C₁₆	—	67	8	no	—
147	C₂²⋊C₁₆	—	67	8	no	—
148	C₄×C₁₆	—	67	8	no	—
149	C₂³×C₄	C₆₇⋊C₈	1	1	yes	characteristic
150	C₂³×C₄	C₆₇⋊C₈	1	1	yes	characteristic
151	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
156	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
158	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄²	C₆₇⋊C₈	1	4	yes	—
163	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
166	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂×C₁₆	—	67	8	no	—
168	C₂×C₁₆	—	134	8	no	—
169	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
170	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
171	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
172	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
173	C₄×C₈	C₂²×D₆₇	1	2	yes	—
174	C₄×C₈	C₂²×D₆₇	1	2	yes	—
175	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₆₇⋊C₁₆	1	8	yes	complemented
180	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
182	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
184	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
187	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	#11761612	1	4	yes	—
196	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
197	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
198	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
200	C₂×C₈	#11761628	1	4	yes	—
201	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
202	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
203	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
204	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
205	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
206	C₄²	C₂²×Dic₆₇	1	2	yes	—
207	C₄²	C₂²×Dic₆₇	1	2	yes	—
208	C₁₆	—	134	8	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	C₂×C₆₇⋊C₁₆	1	4	yes	—
212	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
213	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	C₂×C₆₇⋊C₁₆	1	4	yes	—
221	C₂×C₄	C₂×C₆₇⋊C₁₆	1	4	yes	—
222	C₂×C₄	C₂×C₆₇⋊C₁₆	1	16	yes	complemented
223	C₂×C₄	#14523983	1	4	yes	—
224	C₂×C₄	#14524002	1	2	yes	—
225	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
226	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
227	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
228	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
229	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
230	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
231	C₂×C₄	C₂×#11761612	1	2	yes	—
232	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	#14524002	1	2	yes	—
236	C₈	C₂×#11761612	1	4	yes	—
237	C₈	C₂×#11761628	1	2	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	#16863449	1	2	yes	—
242	C₂²	C₂²×C₆₇⋊C₁₆	1	4	yes	—
243	C₂²	C₂²×C₆₇⋊C₁₆	1	2	yes	—
244	C₂²	C₂×#14523983	1	2	yes	—
245	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	#16863449	1	2	yes	—
249	C₄	C₂²×C₆₇⋊C₁₆	1	4	yes	—
250	C₄	C₂²×C₆₇⋊C₁₆	1	4	yes	—
251	C₄	C₂×#14523983	1	4	yes	—
252	C₄	C₂×#14524002	1	1	yes	characteristic
253	C₄	C₂²×#11761612	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#22562008	1	4	yes	direct product factor
256	C₂	C₂×#16863449	1	1	yes	characteristic
257	C₂	C₂³×C₆₇⋊C₁₆	1	1	yes	characteristic
258	C₂	C₂²×#14523983	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₆₇
commutator	C₂×C₆₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#2162
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #22562008	C₂	direct (N × H)
2. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
4. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
7. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
9. C₄×C₆₇⋊C₁₆	C₂²	semidirect
50. C₂²×C₄×C₆₇	C₁₆	semidirect
92. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
126. C₂×C₆₇	#2162	semidirect
135. C₆₇	C₂×#2162	semidirect
179. C₂²×C₄	C₆₇⋊C₁₆	semidirect
222. C₂×C₄	C₂×C₆₇⋊C₁₆	semidirect
255. C₂	#22562008	direct (N × H)

Maximal subgroups

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

subgroup	index
#22562008	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂²×C₄×C₈×C₆₇	2
C₂×#2162	67

Subgroups — C2×#22562008 #31008896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562008 #31008896