Subgroups — C₂×#22562004 #31008887

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	2	yes	maximal
2	#22562004	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863226	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	2	no	—
6	C₂×#14523979	—	2	4	no	—
7	C₂×#14523979	—	2	2	no	—
8	C₂×#14523979	C₂²	1	4	yes	—
9	C₂×#14523979	C₂²	1	8	yes	—
10	#16863226	C₂²	1	16	yes	complemented
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₆₇	C₄	1	4	yes	—
13	C₂×M₃₂×C₆₇	C₄	1	4	yes	—
14	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×#11761612	—	2	4	no	—
17	C₂×#11761612	—	2	8	no	—
18	C₂×#11761612	—	4	2	no	—
19	C₂×#11761612	—	4	4	no	—
20	C₂²×C₆₇⋊C₈	—	4	2	no	—
21	#14523979	—	2	4	no	—
22	#14523979	—	2	8	no	—
23	#14523979	C₂³	1	8	yes	—
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
30	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	M₃₂×C₆₇	C₂×C₄	1	16	yes	—
34	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
37	#11761612	—	2	8	no	—
38	#11761612	—	4	8	no	—
39	#11761612	—	4	4	no	—
40	C₂×C₆₇⋊C₈	—	4	8	no	—
41	C₂×C₆₇⋊C₈	—	4	4	no	—
42	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	D₈	1	2	yes	—
46	C₂²×C₄×C₆₇	D₈	1	2	yes	—
47	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
65	C₂³×C₆₇	—	4	2	no	—
66	C₂³×C₆₇	#107	1	2	yes	—
67	C₂³×C₆₇	#107	1	2	yes	—
68	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₆₇	#107	1	4	yes	—
71	C₂×C₄×C₆₇	#107	1	2	yes	—
72	C₂×C₄×C₆₇	#107	1	4	yes	—
73	C₂×C₄×C₆₇	#107	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
88	C₆₇⋊C₈	—	4	8	no	—
89	C₂²×C₆₇	—	2	4	no	—
90	C₂²×C₆₇	—	2	2	no	—
91	C₂²×C₆₇	—	4	2	no	—
92	C₂²×C₆₇	—	4	2	no	—
93	C₂²×C₆₇	C₂×#107	1	4	yes	—
94	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
95	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
96	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#107	1	4	yes	—
98	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
99	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
100	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
102	C₂×#2161	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	1	no	—
104	C₂×C₆₇	—	2	4	no	—
105	C₂×C₆₇	—	2	1	no	—
106	C₂×C₆₇	#2161	1	2	yes	complemented
107	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
108	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
109	C₂²×#108	—	67	2	no	—
110	#2161	—	67	8	no	—
111	C₂×#368	—	67	4	no	—
112	C₆₇	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
113	C₂²×M₁₆	—	134	2	no	—
114	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₂×M₃₂	Dic₆₇	1	4	yes	—
116	C₂×M₃₂	Dic₆₇	1	4	yes	—
117	C₂×M₃₂	C₂×D₆₇	1	4	yes	—
118	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
119	C₂×#108	—	67	4	no	—
120	C₂×#108	—	67	8	no	—
121	C₂×#108	—	134	4	no	—
122	C₂×#108	—	134	2	no	—
123	#368	—	67	16	no	—
124	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
125	C₂×M₁₆	—	268	4	no	—
126	C₂×M₁₆	—	268	2	no	—
127	C₂×M₁₆	—	134	8	no	—
128	C₂×M₁₆	—	134	4	no	—
129	C₂²×C₈	—	268	2	no	—
130	C₂²×C₈	Dic₁₃₄	1	2	yes	—
131	C₂²×C₈	Dic₁₃₄	1	2	yes	—
132	C₂²×C₈	D₂₆₈	1	2	yes	—
133	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
135	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
136	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
138	M₃₂	C₂×Dic₆₇	1	16	yes	—
139	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
140	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
141	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
142	#108	—	67	8	no	—
143	#108	—	134	4	no	—
144	#108	—	134	8	no	—
145	C₂⁴	Dic₂₆₈	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
147	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
148	C₂²×C₄	D₅₃₆	1	2	yes	—
149	C₂²×C₄	D₅₃₆	1	2	yes	—
150	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
151	C₂²×C₄	Dic₂₆₈	1	2	yes	—
152	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
153	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
154	M₁₆	—	268	8	no	—
155	M₁₆	—	268	4	no	—
156	M₁₆	—	134	8	no	—
157	C₂×C₈	—	268	8	no	—
158	C₂×C₈	—	268	4	no	—
159	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
160	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
161	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
163	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
166	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
170	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
171	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
172	C₁₆	C₂²×Dic₆₇	1	8	yes	—
173	C₂³	—	4	2	no	—
174	C₂³	#14523977	1	2	yes	—
175	C₂³	#14523977	1	2	yes	—
176	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
177	C₂³	C₂×Dic₂₆₈	1	2	yes	—
178	C₂×C₄	#14523977	1	2	yes	—
179	C₂×C₄	#14523977	1	4	yes	—
180	C₂×C₄	#14523977	1	2	yes	—
181	C₂×C₄	#14523977	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
185	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
186	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
187	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
189	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
191	C₈	—	268	8	no	—
192	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
193	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
194	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
195	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
196	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
197	C₂²	—	2	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	C₂×#14523977	1	4	yes	—
202	C₂²	C₂×#14523977	1	1	yes	characteristic
203	C₂²	C₂×#14523977	1	1	yes	characteristic
204	C₂²	C₂²×Dic₂₆₈	1	1	yes	characteristic
205	C₄	C₂×#14523977	1	4	yes	—
206	C₄	C₂×#14523977	1	1	yes	characteristic
207	C₄	C₂×#14523977	1	1	yes	characteristic
208	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
209	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	4	no	—
212	C₂	—	2	1	no	—
213	C₂	#22562004	1	2	yes	direct product factor
214	C₂	C₂²×#14523977	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#2161
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22562004	C₂	direct (N × H)
3. C₂×#16863226	C₂	semidirect
10. #16863226	C₂²	semidirect
106. C₂×C₆₇	#2161	semidirect
112. C₆₇	C₂×#2161	semidirect
213. C₂	#22562004	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×#14523979	2
#22562004	2
C₂×#16863226	2
C₂²×M₃₂×C₆₇	2
C₂×#2161	67

Subgroups — C2×#22562004 #31008887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562004 #31008887