Subgroups — C₂×#22562256 #31010078

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523989	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22562256	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863200	C₂	1	2	yes	maximal
6	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₄	—	2	1	no	—
9	C₂×#14523989	—	2	8	no	—
10	C₂×#14523989	C₂²	1	8	yes	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄×Dic₂₆₈	—	2	2	no	—
13	C₄×Dic₂₆₈	C₂²	1	4	yes	—
14	C₂×#14523977	C₂²	1	1	yes	characteristic
15	C₂×#14523968	—	2	1	no	—
16	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
17	#16863200	C₂²	1	8	yes	—
18	C₂²×Q₁₆×C₆₇	—	2	1	no	—
19	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
21	#494×C₆₇	—	2	2	no	—
22	#494×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₃₄	—	2	1	no	—
25	C₂×Dic₂₆₈	—	2	1	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	D₄	1	4	yes	—
28	C₂×Dic₂₆₈	C₂³	1	2	yes	—
29	C₄×Dic₁₃₄	—	2	4	no	—
30	C₄×Dic₁₃₄	—	4	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×Dic₆₇⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₆₇	—	4	1	no	—
34	#14523989	—	2	16	no	—
35	C₂×C₆₇⋊C₁₆	—	2	4	no	—
36	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
37	#14523977	—	2	1	no	—
38	#14523977	C₂³	1	2	yes	—
39	#14523968	—	2	4	no	—
40	C₂²×Q₈×C₆₇	—	2	2	no	—
41	C₂×Q₁₆×C₆₇	—	2	4	no	—
42	C₂×Q₁₆×C₆₇	—	2	2	no	—
43	C₂×Q₁₆×C₆₇	—	2	4	no	—
44	C₂×Q₁₆×C₆₇	—	2	4	no	—
45	C₂×Q₁₆×C₆₇	—	4	4	no	—
46	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	—	2	4	no	—
50	C₄⋊Q₈×C₆₇	—	4	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂×Dic₁₃₄	—	2	4	no	—
57	C₂×Dic₁₃₄	—	2	2	no	—
58	C₂×Dic₁₃₄	—	4	1	no	—
59	C₂×Dic₁₃₄	—	4	2	no	—
60	C₂²×Dic₆₇	—	4	1	no	—
61	C₂²×Dic₆₇	—	4	1	no	—
62	Dic₂₆₈	—	2	4	no	—
63	Dic₂₆₈	—	2	2	no	—
64	Dic₂₆₈	C₂×D₄	1	4	yes	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	—	4	1	no	—
67	Dic₆₇⋊C₄	—	4	4	no	—
68	C₄×Dic₆₇	—	4	4	no	—
69	C₆₇⋊C₁₆	—	2	4	no	—
70	C₂×Q₈×C₆₇	—	2	4	no	—
71	C₂×Q₈×C₆₇	—	4	2	no	—
72	C₂×Q₈×C₆₇	—	4	4	no	—
73	C₂×Q₈×C₆₇	—	4	4	no	—
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₆₇	—	2	4	no	—
79	Q₁₆×C₆₇	—	2	4	no	—
80	Q₁₆×C₆₇	—	4	8	no	—
81	Q₁₆×C₆₇	—	4	4	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	#41	1	2	yes	—
87	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	—	4	4	no	—
89	C₄⋊C₄×C₆₇	—	4	4	no	—
90	C₄²×C₆₇	D₈	1	2	yes	—
91	C₄²×C₆₇	C₂×D₄	1	2	yes	—
92	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₆₇	—	4	4	no	—
94	Q₈×C₆₇	—	4	4	no	—
95	Q₈×C₆₇	—	8	4	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	8	4	no	—
99	C₂×C₄×C₆₇	Q₃₂	1	4	yes	complemented
100	C₂×C₄×C₆₇	#121	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₆₇	—	4	2	no	—
107	C₂×Dic₆₇	—	4	4	no	—
108	C₂×Dic₆₇	—	4	2	no	—
109	C₂×Dic₆₇	—	8	2	no	—
110	Dic₁₃₄	—	2	4	no	—
111	Dic₁₃₄	—	4	2	no	—
112	Dic₁₃₄	—	4	4	no	—
113	C₈×C₆₇	—	2	2	no	—
114	C₈×C₆₇	#121	1	2	yes	—
115	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂²×C₆₇	#459	1	2	yes	—
118	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
119	C₂²×C₆₇	#510	1	2	yes	—
120	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	8	4	no	—
123	C₄×C₆₇	#459	1	2	yes	—
124	C₄×C₆₇	C₂×Q₃₂	1	4	yes	complemented
125	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
126	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	Dic₆₇	—	4	4	no	—
128	Dic₆₇	—	8	2	no	—
129	C₂×#2209	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	#2209	1	4	yes	complemented
131	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
132	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
134	C₂²×Q₃₂	—	67	2	no	—
135	C₂×#494	D₆₇	1	1	yes	characteristic
136	C₂×C₄×Q₁₆	—	67	1	no	—
137	#2209	—	67	8	no	—
138	C₂×C₄⋊C₁₆	—	67	1	no	—
139	C₂×#358	—	67	2	no	—
140	C₆₇	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₁₆	—	2	1	no	—
142	C₂²×Q₁₆	—	67	1	no	—
143	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
144	C₂×C₄⋊Q₈	—	2	1	no	—
145	C₂×C₄×Q₈	—	134	1	no	—
146	C₂×Q₃₂	—	67	8	no	—
147	C₂×Q₃₂	—	134	8	no	—
148	C₂²×C₁₆	—	67	2	no	—
149	#494	—	2	2	no	—
150	#494	C₂×D₆₇	1	4	yes	—
151	C₄×Q₁₆	—	67	4	no	—
152	C₄×Q₁₆	—	134	2	no	—
153	C₂×#107	—	67	1	no	—
154	C₂×#103	—	134	1	no	—
155	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
156	C₄⋊C₁₆	—	67	4	no	—
157	#358	—	67	8	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	—	134	1	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×Q₁₆	—	67	4	no	—
166	C₂×Q₁₆	—	67	2	no	—
167	C₂×Q₁₆	—	134	1	no	—
168	C₂×Q₁₆	—	134	2	no	—
169	C₂×Q₁₆	C₂²×D₆₇	1	4	yes	—
170	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	—	4	2	no	—
174	C₄×Q₈	—	268	2	no	—
175	C₄×Q₈	—	134	4	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	268	1	no	—
179	C₂×C₄⋊C₄	—	134	1	no	—
180	C₂×C₄²	—	268	1	no	—
181	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
182	Q₃₂	—	134	16	no	—
183	C₂×C₁₆	—	67	4	no	—
184	C₂×C₁₆	—	134	4	no	—
185	#107	—	67	2	no	—
186	#107	—	134	1	no	—
187	#103	—	134	4	no	—
188	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
189	C₄×C₈	C₂²×D₆₇	1	2	yes	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	268	1	no	—
195	C₂×Q₈	—	268	2	no	—
196	C₂×Q₈	—	134	4	no	—
197	C₂×Q₈	—	134	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	268	1	no	—
200	C₂²×C₄	—	268	1	no	—
201	C₂²×C₄	#11761613	1	1	yes	characteristic
202	C₂²×C₄	#11761613	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	8	no	—
207	Q₁₆	—	4	4	no	—
208	Q₁₆	—	67	4	no	—
209	Q₁₆	—	134	4	no	—
210	Q₁₆	—	134	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	D₄×D₆₇	1	2	yes	—
213	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
214	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂×C₈	#11761627	1	2	yes	—
216	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	268	4	no	—
220	C₄⋊C₄	—	268	1	no	—
221	C₄⋊C₄	—	134	2	no	—
222	C₄²	—	268	4	no	—
223	C₄²	#11761613	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
225	C₁₆	—	134	4	no	—
226	C₂³	C₂×#11761613	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	8	4	no	—
230	Q₈	—	268	4	no	—
231	Q₈	—	268	2	no	—
232	Q₈	—	134	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	268	2	no	—
238	C₂×C₄	—	268	2	no	—
239	C₂×C₄	—	536	2	no	—
240	C₂×C₄	#14523989	1	4	yes	complemented
241	C₂×C₄	#14524014	1	2	yes	—
242	C₂×C₄	C₂×#11761613	1	2	yes	—
243	C₂×C₄	C₂×#11761613	1	2	yes	—
244	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
245	C₂×C₄	#14524069	1	2	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#14524014	1	2	yes	—
249	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
250	C₈	C₂×#11761627	1	1	yes	characteristic
251	C₂²	#16863588	1	2	yes	—
252	C₂²	C₂×#14523989	1	2	yes	—
253	C₂²	#16863837	1	2	yes	—
254	C₂²	C₂²×#11761613	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	8	4	no	—
257	C₄	—	268	4	no	—
258	C₄	—	536	2	no	—
259	C₄	#16863588	1	2	yes	—
260	C₄	C₂×#14523989	1	4	yes	complemented
261	C₄	C₂×#14524014	1	1	yes	characteristic
262	C₄	C₂×#14524069	1	1	yes	characteristic
263	C₂	#22562256	1	4	yes	direct product factor
264	C₂	C₂×#16863588	1	1	yes	characteristic
265	C₂	C₂²×#14523989	1	1	yes	characteristic
266	C₂	C₂×#16863837	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	C₂×#2209
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #22562256	C₂	direct (N × H)
99. C₂×C₄×C₆₇	Q₃₂	semidirect
124. C₄×C₆₇	C₂×Q₃₂	semidirect
130. C₂×C₆₇	#2209	semidirect
140. C₆₇	C₂×#2209	semidirect
240. C₂×C₄	#14523989	semidirect
260. C₄	C₂×#14523989	semidirect
263. C₂	#22562256	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523989	2
C₂×C₄×Dic₂₆₈	2
#22562256	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863200	2
C₂×#494×C₆₇	2
C₂×#2209	67

Subgroups — C2×#22562256 #31010078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562256 #31010078