Subgroups — C₂×#22562489 #31011315

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 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	#22562489	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863223	C₂	1	4	yes	maximal
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	—	2	1	no	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#16863801	—	2	2	no	—
9	#16863801	C₂²	1	4	yes	—
10	C₂×#14523977	—	2	1	no	—
11	C₂×#14523977	C₂²	1	2	yes	—
12	#16863223	C₂²	1	16	yes	—
13	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
15	#500×C₆₇	—	2	2	no	—
16	#500×C₆₇	C₂²	1	4	yes	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₃₄	—	4	1	no	—
21	#14524043	—	2	4	no	—
22	#14524043	—	4	2	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	—	4	1	no	—
25	C₂×C₆₇⋊C₁₆	Q₈	1	8	yes	complemented
26	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
27	#14523977	—	2	2	no	—
28	#14523977	—	2	4	no	—
29	#14523977	C₂³	1	4	yes	—
30	C₂²×Q₈×C₆₇	—	4	1	no	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	C₄⋊Q₈×C₆₇	—	4	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	#107×C₆₇	—	2	2	no	—
39	#107×C₆₇	—	2	4	no	—
40	#107×C₆₇	C₂³	1	4	yes	—
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂×Dic₁₃₄	—	4	4	no	—
44	C₂×Dic₁₃₄	—	8	4	no	—
45	C₂²×Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	2	4	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₄⋊Dic₆₇	—	4	4	no	—
49	C₆₇⋊C₁₆	C₂×Q₈	1	8	yes	complemented
50	C₂×Q₈×C₆₇	—	4	4	no	—
51	C₂×Q₈×C₆₇	—	8	4	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄²×C₆₇	D₈	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	2	yes	—
66	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₆₇	—	8	8	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
71	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
72	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₆₇	—	4	4	no	—
79	C₂×Dic₆₇	—	8	4	no	—
80	Dic₁₃₄	—	8	8	no	—
81	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
82	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
83	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
84	C₂²×C₆₇	#500	1	2	yes	—
85	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
86	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₆₇	—	8	4	no	—
89	C₄×C₆₇	#500	1	2	yes	—
90	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
91	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
92	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₆₇	—	8	4	no	—
95	C₂×#2254	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	#2254	1	4	yes	complemented
97	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
98	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
99	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
100	C₂×#500	—	67	1	no	—
101	C₂×#500	D₆₇	1	1	yes	characteristic
102	#2254	—	67	8	no	—
103	C₂×#367	—	67	4	no	—
104	C₂×C₄×C₁₆	—	67	1	no	—
105	C₆₇	C₂×#2254	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×C₄⋊Q₈	—	134	1	no	—
108	C₂²×C₁₆	—	67	2	no	—
109	#500	—	2	2	no	—
110	#500	—	67	4	no	—
111	#500	—	134	2	no	—
112	#500	C₂×D₆₇	1	4	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	—	67	2	no	—
115	C₂×#107	—	134	1	no	—
116	C₂×#107	C₂×D₆₇	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	#367	—	67	16	no	—
119	C₄×C₁₆	—	67	4	no	—
120	C₂²×Q₈	—	4	1	no	—
121	C₂²×Q₈	—	268	1	no	—
122	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	C₄⋊Q₈	—	268	2	no	—
127	C₄⋊Q₈	—	134	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	268	1	no	—
131	C₂×C₄⋊C₄	—	134	2	no	—
132	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂×C₁₆	—	67	8	no	—
134	C₂×C₁₆	—	67	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	2	no	—
137	#107	—	67	4	no	—
138	#107	—	134	2	no	—
139	#107	—	134	4	no	—
140	#107	C₂²×D₆₇	1	4	yes	—
141	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
142	C₄×C₈	C₂²×D₆₇	1	2	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	8	4	no	—
145	C₂×Q₈	—	268	4	no	—
146	C₂×Q₈	—	536	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	268	2	no	—
149	C₂²×C₄	#11761613	1	1	yes	characteristic
150	C₂²×C₄	#11761613	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₈	#11761618	1	4	yes	—
153	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
155	C₂×C₈	Q₈×D₆₇	1	4	yes	—
156	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	268	4	no	—
161	C₄⋊C₄	—	268	2	no	—
162	C₄⋊C₄	—	134	4	no	—
163	C₄²	#11761613	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
165	C₁₆	—	67	8	no	—
166	C₂³	C₂×#11761613	1	1	yes	characteristic
167	Q₈	—	8	8	no	—
168	Q₈	—	536	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	268	4	no	—
172	C₂×C₄	—	536	4	no	—
173	C₂×C₄	#14523986	1	4	yes	—
174	C₂×C₄	#14523987	1	4	yes	—
175	C₂×C₄	#14524041	1	2	yes	—
176	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
177	C₂×C₄	C₂×#11761613	1	2	yes	—
178	C₂×C₄	C₂×#11761613	1	2	yes	—
179	C₂×C₄	C₂×#11761618	1	2	yes	—
180	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
181	C₈	#14524041	1	2	yes	—
182	C₈	C₂×#11761618	1	4	yes	—
183	C₈	C₂×Q₈×D₆₇	1	2	yes	—
184	C₂²	#16863799	1	2	yes	—
185	C₂²	C₂×#14523986	1	2	yes	—
186	C₂²	C₂×#14523987	1	2	yes	—
187	C₂²	C₂²×#11761613	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	536	4	no	—
190	C₄	#16863799	1	2	yes	—
191	C₄	C₂×#14523986	1	4	yes	—
192	C₄	C₂×#14523987	1	4	yes	—
193	C₄	C₂×#14524041	1	1	yes	characteristic
194	C₄	C₂²×#11761618	1	1	yes	characteristic
195	C₂	#22562489	1	4	yes	direct product factor
196	C₂	C₂×#16863799	1	1	yes	characteristic
197	C₂	C₂²×#14523986	1	1	yes	characteristic
198	C₂	C₂²×#14523987	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×#2254
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. #22562489	C₂	direct (N × H)
25. C₂×C₆₇⋊C₁₆	Q₈	semidirect
49. C₆₇⋊C₁₆	C₂×Q₈	semidirect
96. C₂×C₆₇	#2254	semidirect
105. C₆₇	C₂×#2254	semidirect
195. C₂	#22562489	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
#22562489	2
C₂×#16863223	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×#500×C₆₇	2
C₂×#2254	67

Subgroups — C2×#22562489 #31011315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562489 #31011315