Subgroups — C₂×#22562358 #31010457

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×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#22562358	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863221	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863199	C₂	1	2	yes	maximal
6	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₄	—	2	1	no	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄×Dic₂₆₈	—	2	2	no	—
11	C₄×Dic₂₆₈	C₂²	1	4	yes	—
12	C₂×#14523977	C₂²	1	1	yes	characteristic
13	C₂×#14523968	—	2	1	no	—
14	#16863221	C₂²	1	8	yes	—
15	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
16	#16863199	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
18	#500×C₆₇	—	2	2	no	—
19	#500×C₆₇	C₂²	1	4	yes	—
20	C₂×#107×C₆₇	—	2	1	no	—
21	C₂×#107×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₂₆₈	—	2	1	no	—
25	C₂×Dic₂₆₈	—	2	2	no	—
26	C₂×Dic₂₆₈	Q₈	1	4	yes	complemented
27	C₂×Dic₂₆₈	C₂³	1	2	yes	—
28	C₄×Dic₁₃₄	—	2	4	no	—
29	C₄×Dic₁₃₄	—	4	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×Dic₆₇⋊C₄	—	4	1	no	—
32	C₂×C₄×Dic₆₇	—	4	1	no	—
33	C₂×C₆₇⋊C₁₆	—	2	4	no	—
34	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
35	#14523977	—	2	1	no	—
36	#14523977	C₂³	1	2	yes	—
37	#14523968	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	4	1	no	—
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	—	2	4	no	—
42	C₄⋊Q₈×C₆₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	#107×C₆₇	—	2	2	no	—
47	#107×C₆₇	—	2	4	no	—
48	#107×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	C₂²×Dic₆₇	—	4	1	no	—
56	C₂²×Dic₆₇	—	4	1	no	—
57	Dic₂₆₈	—	2	4	no	—
58	Dic₂₆₈	—	2	2	no	—
59	Dic₂₆₈	C₂×Q₈	1	4	yes	complemented
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	—	4	1	no	—
62	Dic₆₇⋊C₄	—	4	4	no	—
63	C₄×Dic₆₇	—	4	4	no	—
64	C₆₇⋊C₁₆	—	2	4	no	—
65	C₂×Q₈×C₆₇	—	4	4	no	—
66	C₂×Q₈×C₆₇	—	8	4	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₆₇	#41	1	2	yes	—
76	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	4	no	—
80	C₄²×C₆₇	D₈	1	2	yes	—
81	C₄²×C₆₇	C₂×D₄	1	2	yes	—
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₆₇	—	8	8	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	8	4	no	—
87	C₂×C₄×C₆₇	Q₃₂	1	4	yes	—
88	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₆₇	#137	1	2	yes	—
93	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₆₇	—	4	2	no	—
95	C₂×Dic₆₇	—	4	4	no	—
96	C₂×Dic₆₇	—	4	2	no	—
97	C₂×Dic₆₇	—	8	2	no	—
98	Dic₁₃₄	—	2	4	no	—
99	Dic₁₃₄	—	4	2	no	—
100	Dic₁₃₄	—	4	4	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×Q₃₂	1	2	yes	—
107	C₂²×C₆₇	#509	1	2	yes	—
108	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	8	4	no	—
111	C₄×C₆₇	#474	1	2	yes	—
112	C₄×C₆₇	C₂×Q₃₂	1	4	yes	—
113	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
114	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
115	Dic₆₇	—	4	4	no	—
116	Dic₆₇	—	8	2	no	—
117	C₂×#2227	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	#2227	1	4	yes	complemented
119	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
120	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
122	C₂×#500	D₆₇	1	1	yes	characteristic
123	C₂×C₄×Q₁₆	—	67	1	no	—
124	#2227	—	67	8	no	—
125	C₂×#366	—	67	2	no	—
126	C₂×C₄⋊C₁₆	—	67	1	no	—
127	C₂×#358	—	67	2	no	—
128	C₆₇	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	67	1	no	—
130	C₂×C₄⋊Q₈	—	2	1	no	—
131	C₂×C₄×Q₈	—	134	1	no	—
132	C₂²×C₁₆	—	67	2	no	—
133	#500	—	2	2	no	—
134	#500	C₂×D₆₇	1	4	yes	—
135	C₄×Q₁₆	—	67	4	no	—
136	C₄×Q₁₆	—	134	2	no	—
137	C₂×#107	—	2	1	no	—
138	C₂×#107	—	67	1	no	—
139	C₂×#107	C₂×D₆₇	1	2	yes	—
140	C₂×#103	—	134	1	no	—
141	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
142	#366	—	67	8	no	—
143	C₄⋊C₁₆	—	67	4	no	—
144	#358	—	67	8	no	—
145	C₂²×Q₈	—	4	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	C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	4	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄×Q₈	—	268	2	no	—
156	C₄×Q₈	—	134	4	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	4	1	no	—
159	C₂×C₄⋊C₄	—	268	1	no	—
160	C₂×C₄⋊C₄	—	134	1	no	—
161	C₂×C₄²	—	268	1	no	—
162	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂×C₁₆	—	67	4	no	—
164	C₂×C₁₆	—	134	4	no	—
165	#107	—	2	4	no	—
166	#107	—	2	2	no	—
167	#107	—	67	2	no	—
168	#107	—	134	1	no	—
169	#107	C₂²×D₆₇	1	4	yes	—
170	#103	—	134	4	no	—
171	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
172	C₄×C₈	C₂²×D₆₇	1	2	yes	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	8	4	no	—
175	C₂×Q₈	—	268	1	no	—
176	C₂×Q₈	—	268	2	no	—
177	C₂×Q₈	—	134	4	no	—
178	C₂×Q₈	—	134	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	268	1	no	—
181	C₂²×C₄	—	268	1	no	—
182	C₂²×C₄	#11761613	1	1	yes	characteristic
183	C₂²×C₄	#11761613	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₆₇	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×C₂²⋊D₆₇	1	2	yes	—
190	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂×C₈	Q₈×D₆₇	1	2	yes	—
192	C₂×C₈	#11761629	1	2	yes	—
193	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	268	4	no	—
198	C₄⋊C₄	—	268	1	no	—
199	C₄⋊C₄	—	134	2	no	—
200	C₄²	—	268	4	no	—
201	C₄²	#11761613	1	2	yes	—
202	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
203	C₁₆	—	134	4	no	—
204	C₂³	C₂×#11761613	1	1	yes	characteristic
205	Q₈	—	8	8	no	—
206	Q₈	—	268	4	no	—
207	Q₈	—	268	2	no	—
208	Q₈	—	134	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	—	268	2	no	—
213	C₂×C₄	—	268	4	no	—
214	C₂×C₄	—	268	2	no	—
215	C₂×C₄	—	536	2	no	—
216	C₂×C₄	#14523989	1	4	yes	—
217	C₂×C₄	#14524019	1	2	yes	—
218	C₂×C₄	C₂×#11761613	1	2	yes	—
219	C₂×C₄	C₂×#11761613	1	2	yes	—
220	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
221	C₂×C₄	#14524074	1	2	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	#14524019	1	2	yes	—
225	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
226	C₈	C₂×#11761629	1	1	yes	characteristic
227	C₂²	#16863673	1	2	yes	—
228	C₂²	C₂×#14523989	1	2	yes	—
229	C₂²	#16863832	1	2	yes	—
230	C₂²	C₂²×#11761613	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	4	no	—
233	C₄	—	268	4	no	—
234	C₄	—	536	2	no	—
235	C₄	#16863673	1	2	yes	—
236	C₄	C₂×#14523989	1	4	yes	—
237	C₄	C₂×#14524019	1	1	yes	characteristic
238	C₄	C₂×#14524074	1	1	yes	characteristic
239	C₂	#22562358	1	4	yes	direct product factor
240	C₂	C₂×#16863673	1	1	yes	characteristic
241	C₂	C₂²×#14523989	1	1	yes	characteristic
242	C₂	C₂×#16863832	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₂×#2227
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #22562358	C₂	direct (N × H)
26. C₂×Dic₂₆₈	Q₈	semidirect
59. Dic₂₆₈	C₂×Q₈	semidirect
118. C₂×C₆₇	#2227	semidirect
128. C₆₇	C₂×#2227	semidirect
239. C₂	#22562358	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×Dic₂₆₈	2
#22562358	2
C₂×#16863221	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863199	2
C₂×#500×C₆₇	2
C₂×#2227	67

Subgroups — C2×#22562358 #31010457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562358 #31010457