Subgroups — C₂×#22562115 #31009316

Order 17152.

542 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22562115	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863171	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₃₄	C₄	1	4	yes	—
13	C₈×Dic₁₃₄	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
15	C₂×Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
16	C₂×Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
20	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
21	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
22	#16863171	C₂²	1	4	yes	—
23	C₁₆×Dic₆₇	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#346×C₆₇	C₂²	1	4	yes	—
28	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	C₈	1	4	yes	—
32	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
34	C₂×Dic₆₇⋊C₄	C₈	1	1	yes	characteristic
35	C₂×Dic₆₇⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₆₇⋊C₁₆	—	2	2	no	—
39	C₂×C₆₇⋊C₁₆	—	2	2	no	—
40	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
41	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
42	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
43	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
44	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
45	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
46	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
47	C₈×Dic₆₇	C₂³	1	2	yes	—
48	C₈×Dic₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₆₇	—	2	2	no	—
53	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
54	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
55	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
56	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂×Dic₁₃₄	—	2	4	no	—
59	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
60	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
68	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
69	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
70	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
71	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
72	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
73	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
74	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
75	C₆₇⋊C₁₆	—	2	2	no	—
76	C₆₇⋊C₁₆	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₆₇	#41	1	2	yes	—
85	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₆₇	C₂×C₈	1	2	yes	—
87	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
88	C₁₆×C₆₇	—	2	2	no	—
89	C₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
90	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#131	1	2	yes	—
98	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×Dic₆₇	—	2	2	no	—
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	M₃₂	1	4	yes	—
103	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
104	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
105	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
106	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
107	Dic₁₃₄	—	2	8	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	C₄×Q₈	1	2	yes	—
110	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₆₇⋊C₈	—	2	2	no	—
113	C₆₇⋊C₈	—	2	2	no	—
114	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
115	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
116	C₂²×C₆₇	#504	1	2	yes	—
117	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	C₈×Q₈	1	2	yes	—
121	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
123	Dic₆₇	—	2	2	no	—
124	Dic₆₇	—	2	2	no	—
125	Dic₆₇	—	2	2	no	—
126	Dic₆₇	C₂×M₃₂	1	4	yes	—
127	C₂×#2183	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	#2183	1	4	yes	complemented
129	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
130	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
131	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
132	C₂×C₈×Q₈	—	67	1	no	—
133	#2183	—	67	8	no	—
134	C₂×C₄⋊C₁₆	—	67	1	no	—
135	C₂×C₄⋊C₁₆	—	67	1	no	—
136	C₂×C₄⋊C₁₆	—	67	1	no	—
137	C₂×#346	—	67	1	no	—
138	C₂×#346	D₆₇	1	1	yes	characteristic
139	C₂×C₄×C₁₆	—	67	1	no	—
140	C₆₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	—	67	1	no	—
142	C₂²×C₁₆	—	67	1	no	—
143	C₂²×C₁₆	—	67	1	no	—
144	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
145	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
146	C₈×Q₈	—	67	4	no	—
147	C₈×Q₈	—	67	4	no	—
148	C₂×C₄⋊C₈	—	67	1	no	—
149	C₂×C₄⋊C₈	—	67	1	no	—
150	C₂×C₄⋊C₈	—	67	1	no	—
151	C₂×C₄×C₈	—	67	1	no	—
152	C₂×C₄×C₈	—	67	1	no	—
153	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₄⋊C₁₆	—	67	4	no	—
155	C₄⋊C₁₆	—	67	4	no	—
156	C₄⋊C₁₆	—	67	4	no	—
157	#346	—	67	4	no	—
158	#346	C₂×D₆₇	1	4	yes	—
159	C₄×C₁₆	—	67	4	no	—
160	C₂²×Q₈	—	67	1	no	—
161	C₂²×C₈	—	67	1	no	—
162	C₂²×C₈	—	67	1	no	—
163	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
165	C₄×Q₈	—	67	4	no	—
166	C₄×Q₈	—	67	4	no	—
167	C₂×C₄⋊C₄	—	67	1	no	—
168	C₂×C₄⋊C₄	—	67	1	no	—
169	C₂×C₄⋊C₄	—	67	1	no	—
170	C₂×C₄²	—	67	1	no	—
171	C₂×C₄²	—	67	1	no	—
172	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	67	2	no	—
175	C₂×C₁₆	—	67	2	no	—
176	C₂×C₁₆	—	134	2	no	—
177	C₂×C₁₆	—	134	2	no	—
178	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
180	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
181	C₄⋊C₈	—	67	4	no	—
182	C₄⋊C₈	—	67	4	no	—
183	C₄⋊C₈	—	67	4	no	—
184	C₄×C₈	—	67	2	no	—
185	C₄×C₈	—	67	2	no	—
186	C₄×C₈	—	67	2	no	—
187	C₄×C₈	—	67	2	no	—
188	C₄×C₈	C₄×D₆₇	1	2	yes	—
189	C₄×C₈	C₂²×D₆₇	1	2	yes	—
190	C₂×Q₈	—	67	4	no	—
191	C₂×Q₈	—	134	4	no	—
192	C₂²×C₄	—	67	1	no	—
193	C₂²×C₄	—	67	1	no	—
194	C₂²×C₄	—	67	1	no	—
195	C₂²×C₄	—	67	1	no	—
196	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	67	2	no	—
201	C₂×C₈	—	67	2	no	—
202	C₂×C₈	—	134	2	no	—
203	C₂×C₈	—	134	2	no	—
204	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
206	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
207	C₂×C₈	#11761628	1	2	yes	—
208	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
209	C₄⋊C₄	—	67	4	no	—
210	C₄⋊C₄	—	67	4	no	—
211	C₄⋊C₄	—	67	4	no	—
212	C₄²	—	67	2	no	—
213	C₄²	—	67	2	no	—
214	C₄²	—	67	2	no	—
215	C₄²	—	67	2	no	—
216	C₄²	C₈×D₆₇	1	2	yes	—
217	C₄²	C₂×C₄×D₆₇	1	2	yes	—
218	C₁₆	—	2	2	no	—
219	C₁₆	—	134	2	no	—
220	C₁₆	—	134	2	no	—
221	C₁₆	C₂×Dic₁₃₄	1	4	yes	complemented
222	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
223	Q₈	—	134	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	67	4	no	—
227	C₂×C₄	—	67	2	no	—
228	C₂×C₄	—	67	2	no	—
229	C₂×C₄	—	67	2	no	—
230	C₂×C₄	—	67	2	no	—
231	C₂×C₄	—	134	2	no	—
232	C₂×C₄	—	134	2	no	—
233	C₂×C₄	—	134	2	no	—
234	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
235	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
236	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
237	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
238	C₂×C₄	#14524049	1	2	yes	—
239	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	134	2	no	—
242	C₈	—	134	2	no	—
243	C₈	C₄×Dic₁₃₄	1	2	yes	—
244	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
245	C₈	C₂×#11761628	1	1	yes	characteristic
246	C₂²	C₈×Dic₁₃₄	1	2	yes	—
247	C₂²	M₃₂×D₆₇	1	2	yes	—
248	C₂²	#16863814	1	2	yes	—
249	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	67	4	no	—
253	C₄	—	134	2	no	—
254	C₄	—	134	2	no	—
255	C₄	—	134	2	no	—
256	C₄	C₈×Dic₁₃₄	1	2	yes	—
257	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
258	C₄	C₂×#14524049	1	1	yes	characteristic
259	C₂	#22562115	1	4	yes	direct product factor
260	C₂	C₂×C₈×Dic₁₃₄	1	1	yes	characteristic
261	C₂	C₂×M₃₂×D₆₇	1	1	yes	characteristic
262	C₂	C₂×#16863814	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	C₂×#2183
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #22562115	C₂	direct (N × H)
53. C₂×C₁₆×C₆₇	Q₈	semidirect
89. C₁₆×C₆₇	C₂×Q₈	semidirect
128. C₂×C₆₇	#2183	semidirect
140. C₆₇	C₂×#2183	semidirect
178. C₂×C₁₆	Dic₁₃₄	semidirect
221. C₁₆	C₂×Dic₁₃₄	semidirect
259. C₂	#22562115	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×C₈×Dic₁₃₄	2
#22562115	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×Dic₆₇⋊C₁₆	2
C₂×#16863171	2
C₂×C₁₆×Dic₆₇	2
C₂×#346×C₆₇	2
C₂×#2183	67

Subgroups — C2×#22562115 #31009316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562115 #31009316