Subgroups — C₂×#22562352 #31010455

Order 17152.

574 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22562352	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863224	C₂	1	2	yes	maximal
5	C₂×#16863201	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
8	C₂×#14524043	—	2	1	no	—
9	C₂×C₄×Dic₁₃₄	—	2	1	no	—
10	#16863801	—	2	2	no	—
11	#16863801	C₂²	1	4	yes	—
12	C₄×Dic₂₆₈	—	2	2	no	—
13	C₄×Dic₂₆₈	C₂²	1	4	yes	—
14	C₂×#14523977	—	2	1	no	—
15	C₂×#14523977	C₂²	1	1	yes	characteristic
16	C₂×#14523977	C₂²	1	2	yes	—
17	C₂×#14523968	—	2	1	no	—
18	#16863224	C₂²	1	8	yes	—
19	#16863201	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
23	C₂²×Dic₁₃₄	—	2	1	no	—
24	C₂²×Dic₁₃₄	—	4	1	no	—
25	C₂×Dic₂₆₈	—	2	1	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	Q₈	1	4	yes	complemented
28	C₂×Dic₂₆₈	C₂³	1	2	yes	—
29	#14524043	—	2	4	no	—
30	#14524043	—	4	2	no	—
31	C₄×Dic₁₃₄	—	2	4	no	—
32	C₄×Dic₁₃₄	—	4	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	1	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	—	4	1	no	—
36	C₂×Dic₆₇⋊C₄	—	4	1	no	—
37	C₂×C₄×Dic₆₇	—	4	1	no	—
38	#14523977	—	2	1	no	—
39	#14523977	—	2	2	no	—
40	#14523977	—	2	4	no	—
41	#14523977	C₂³	1	2	yes	—
42	#14523977	C₂³	1	4	yes	—
43	#14523968	—	2	4	no	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₇	—	2	4	no	—
48	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₄×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂×Dic₁₃₄	—	2	4	no	—
52	C₂×Dic₁₃₄	—	2	2	no	—
53	C₂×Dic₁₃₄	—	4	1	no	—
54	C₂×Dic₁₃₄	—	4	2	no	—
55	C₂×Dic₁₃₄	—	4	4	no	—
56	C₂×Dic₁₃₄	—	8	4	no	—
57	C₂²×Dic₆₇	—	4	1	no	—
58	C₂²×Dic₆₇	—	4	1	no	—
59	C₂²×Dic₆₇	—	4	2	no	—
60	Dic₂₆₈	—	2	4	no	—
61	Dic₂₆₈	—	2	2	no	—
62	Dic₂₆₈	C₂×Q₈	1	4	yes	complemented
63	C₄⋊Dic₆₇	—	2	2	no	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	—	4	1	no	—
66	C₄⋊Dic₆₇	—	4	2	no	—
67	C₄⋊Dic₆₇	—	4	4	no	—
68	Dic₆₇⋊C₄	—	4	4	no	—
69	C₄×Dic₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₆₇	#41	1	2	yes	—
78	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₄²×C₆₇	D₈	1	2	yes	—
80	C₄²×C₆₇	C₂×D₄	1	2	yes	—
81	C₁₆×C₆₇	—	2	4	no	—
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
85	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₆₇	#137	1	2	yes	—
90	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×Dic₆₇	—	4	2	no	—
92	C₂×Dic₆₇	—	4	4	no	—
93	C₂×Dic₆₇	—	4	2	no	—
94	C₂×Dic₆₇	—	4	4	no	—
95	C₂×Dic₆₇	—	8	4	no	—
96	C₂×Dic₆₇	—	8	2	no	—
97	Dic₁₃₄	—	2	4	no	—
98	Dic₁₃₄	—	4	2	no	—
99	Dic₁₃₄	—	4	4	no	—
100	Dic₁₃₄	—	8	8	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₆₇	#474	1	2	yes	—
106	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
107	C₂²×C₆₇	#510	1	2	yes	—
108	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	#474	1	2	yes	—
111	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
112	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
114	Dic₆₇	—	4	4	no	—
115	Dic₆₇	—	8	4	no	—
116	Dic₆₇	—	8	2	no	—
117	C₂×#2225	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	#2225	1	4	yes	complemented
119	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
120	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
122	C₂×#500	—	67	1	no	—
123	C₂×C₄×Q₁₆	—	67	1	no	—
124	#2225	—	67	8	no	—
125	C₂×#367	—	67	2	no	—
126	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
127	C₂×#358	—	67	2	no	—
128	C₆₇	C₂×#2225	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	67	1	no	—
130	C₂×C₄⋊Q₈	—	134	1	no	—
131	C₂×C₄×Q₈	—	134	1	no	—
132	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
133	#500	—	67	4	no	—
134	#500	—	134	2	no	—
135	C₄×Q₁₆	—	67	4	no	—
136	C₄×Q₁₆	—	134	2	no	—
137	C₂×#107	—	67	2	no	—
138	C₂×#107	—	67	1	no	—
139	C₂×#107	—	134	1	no	—
140	C₂×#103	—	134	1	no	—
141	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
142	#367	—	67	8	no	—
143	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	—
144	#358	—	67	8	no	—
145	C₂²×Q₈	—	268	1	no	—
146	C₂²×Q₈	—	134	1	no	—
147	C₂×Q₁₆	—	67	4	no	—
148	C₂×Q₁₆	—	67	2	no	—
149	C₂×Q₁₆	—	134	1	no	—
150	C₂×Q₁₆	—	134	2	no	—
151	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
153	C₄⋊Q₈	—	268	2	no	—
154	C₄⋊Q₈	—	134	4	no	—
155	C₄×Q₈	—	268	2	no	—
156	C₄×Q₈	—	134	4	no	—
157	C₂×C₄⋊C₄	—	268	1	no	—
158	C₂×C₄⋊C₄	—	268	1	no	—
159	C₂×C₄⋊C₄	—	134	1	no	—
160	C₂×C₄⋊C₄	—	134	2	no	—
161	C₂×C₄²	—	268	1	no	—
162	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
163	C₂×C₁₆	—	2	4	no	—
164	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
165	#107	—	67	4	no	—
166	#107	—	67	2	no	—
167	#107	—	134	1	no	—
168	#107	—	134	2	no	—
169	#107	—	134	4	no	—
170	#103	—	134	4	no	—
171	C₄×C₈	D₂₆₈	1	2	yes	—
172	C₄×C₈	C₂²×D₆₇	1	2	yes	—
173	C₂×Q₈	—	268	1	no	—
174	C₂×Q₈	—	268	4	no	—
175	C₂×Q₈	—	268	2	no	—
176	C₂×Q₈	—	536	4	no	—
177	C₂×Q₈	—	134	4	no	—
178	C₂×Q₈	—	134	2	no	—
179	C₂²×C₄	—	268	1	no	—
180	C₂²×C₄	—	268	1	no	—
181	C₂²×C₄	—	268	2	no	—
182	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
183	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
185	Q₁₆	—	67	4	no	—
186	Q₁₆	—	134	4	no	—
187	Q₁₆	—	134	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
190	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
191	C₂×C₈	Q₈×D₆₇	1	2	yes	—
192	C₂×C₈	#11761627	1	2	yes	—
193	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
194	C₄⋊C₄	—	268	4	no	—
195	C₄⋊C₄	—	268	4	no	—
196	C₄⋊C₄	—	268	1	no	—
197	C₄⋊C₄	—	268	2	no	—
198	C₄⋊C₄	—	134	2	no	—
199	C₄⋊C₄	—	134	4	no	—
200	C₄²	—	268	4	no	—
201	C₄²	D₅₃₆	1	2	yes	—
202	C₄²	C₂×D₂₆₈	1	2	yes	—
203	C₁₆	—	2	4	no	—
204	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
205	Q₈	—	268	4	no	—
206	Q₈	—	268	2	no	—
207	Q₈	—	536	8	no	—
208	Q₈	—	134	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	268	2	no	—
211	C₂×C₄	—	268	4	no	—
212	C₂×C₄	—	268	2	no	—
213	C₂×C₄	—	268	4	no	—
214	C₂×C₄	—	536	4	no	—
215	C₂×C₄	—	536	2	no	—
216	C₂×C₄	#14523988	1	4	yes	—
217	C₂×C₄	#14524023	1	2	yes	—
218	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
219	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
220	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
221	C₂×C₄	#14524076	1	2	yes	—
222	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	#14524023	1	2	yes	—
225	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
226	C₈	C₂×#11761627	1	1	yes	characteristic
227	C₂²	#16863677	1	2	yes	—
228	C₂²	C₂×#14523988	1	2	yes	—
229	C₂²	#16863839	1	2	yes	—
230	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	268	4	no	—
233	C₄	—	536	4	no	—
234	C₄	—	536	2	no	—
235	C₄	#16863677	1	2	yes	—
236	C₄	C₂×#14523988	1	4	yes	—
237	C₄	C₂×#14524023	1	1	yes	characteristic
238	C₄	C₂×#14524076	1	1	yes	characteristic
239	C₂	#22562352	1	4	yes	direct product factor
240	C₂	C₂×#16863677	1	1	yes	characteristic
241	C₂	C₂²×#14523988	1	1	yes	characteristic
242	C₂	C₂×#16863839	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2225
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22562352	C₂	direct (N × H)
27. C₂×Dic₂₆₈	Q₈	semidirect
62. Dic₂₆₈	C₂×Q₈	semidirect
118. C₂×C₆₇	#2225	semidirect
128. C₆₇	C₂×#2225	semidirect
239. C₂	#22562352	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×#16863801	2
C₂×C₄×Dic₂₆₈	2
#22562352	2
C₂×#16863224	2
C₂×#16863201	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#2225	67

Subgroups — C2×#22562352 #31010455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562352 #31010455