Subgroups — C₂×#22558301 #30770708

Order 17152.

694 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	C₂²×#14523977	C₂	1	2	yes	maximal
2	#22558301	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₆₇	—	2	2	no	—
5	C₂×#14523977	—	2	2	no	—
6	C₂×#14523977	—	2	4	no	—
7	C₂×#14523977	C₄	1	8	yes	complemented
8	C₂×#14523977	C₂²	1	4	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₇	—	2	4	no	—
11	C₂²×C₁₆×C₆₇	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂³×Dic₆₇	—	4	2	no	—
14	C₂×C₄⋊Dic₆₇	—	2	8	no	—
15	C₂×C₄⋊Dic₆₇	—	2	4	no	—
16	C₂×C₄⋊Dic₆₇	—	4	2	no	—
17	C₂×C₄⋊Dic₆₇	—	4	4	no	—
18	#14523977	—	2	8	no	—
19	#14523977	—	2	4	no	—
20	#14523977	C₂×C₄	1	8	yes	complemented
21	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	D₄	1	4	yes	—
26	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	—	2	8	no	—
29	C₂×C₁₆×C₆₇	—	2	4	no	—
30	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
31	C₂²×Dic₆₇	—	4	8	no	—
32	C₂²×Dic₆₇	—	4	4	no	—
33	C₂²×Dic₆₇	—	8	4	no	—
34	C₂²×Dic₆₇	—	8	4	no	—
35	C₄⋊Dic₆₇	—	2	8	no	—
36	C₄⋊Dic₆₇	—	4	8	no	—
37	C₄⋊Dic₆₇	—	4	4	no	—
38	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
43	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₄²	1	4	yes	—
46	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
49	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₁₆×C₆₇	—	2	4	no	—
55	C₂³×C₆₇	#102	1	4	yes	—
56	C₂³×C₆₇	#107	1	2	yes	—
57	C₂³×C₆₇	D₁₆	1	2	yes	—
58	C₂³×C₆₇	SD₃₂	1	4	yes	—
59	C₂³×C₆₇	Q₃₂	1	2	yes	—
60	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#95	1	4	yes	—
62	C₂×C₄×C₆₇	#102	1	4	yes	—
63	C₂×C₄×C₆₇	#103	1	8	yes	—
64	C₂×C₄×C₆₇	#106	1	4	yes	—
65	C₂×C₄×C₆₇	#107	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₆₇	—	4	8	no	—
71	C₂×Dic₆₇	—	8	16	no	—
72	C₂×Dic₆₇	—	8	4	no	—
73	C₂×Dic₆₇	—	8	4	no	—
74	C₈×C₆₇	#95	1	4	yes	—
75	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂²×C₆₇	#340	1	4	yes	—
79	C₂²×C₆₇	#357	1	8	yes	—
80	C₂²×C₆₇	#358	1	8	yes	—
81	C₂²×C₆₇	#366	1	4	yes	—
82	C₂²×C₆₇	#367	1	4	yes	—
83	C₂²×C₆₇	C₂×#102	1	2	yes	—
84	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
88	C₄×C₆₇	#340	1	4	yes	—
89	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
90	C₄×C₆₇	C₂×#103	1	2	yes	—
91	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
92	Dic₆₇	—	8	8	no	—
93	C₂×#1380	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	#1380	1	8	yes	complemented
95	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#357	1	2	yes	—
97	C₂×C₆₇	C₂×#358	1	2	yes	—
98	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
99	C₂×C₆₇	C₂×#367	1	1	yes	characteristic
100	C₂³×C₁₆	D₆₇	1	1	yes	characteristic
101	C₂²×#107	—	67	2	no	—
102	#1380	—	67	4	no	—
103	C₆₇	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₂²×C₄⋊C₄	—	134	2	no	—
106	C₂²×C₁₆	—	2	4	no	—
107	C₂²×C₁₆	Dic₆₇	1	4	yes	complemented
108	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
109	C₂×#107	—	67	8	no	—
110	C₂×#107	—	67	4	no	—
111	C₂×#107	—	134	2	no	—
112	C₂×#107	—	134	4	no	—
113	C₂³×C₄	—	268	2	no	—
114	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
115	C₂²×C₈	Dic₁₃₄	1	2	yes	—
116	C₂²×C₈	C₄×D₆₇	1	4	yes	—
117	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
119	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
120	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	268	2	no	—
122	C₂×C₄⋊C₄	—	268	4	no	—
123	C₂×C₄⋊C₄	—	134	8	no	—
124	C₂×C₄⋊C₄	—	134	4	no	—
125	C₂×C₁₆	—	2	8	no	—
126	C₂×C₁₆	—	2	4	no	—
127	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	complemented
128	#107	—	67	8	no	—
129	#107	—	134	8	no	—
130	#107	—	134	4	no	—
131	C₂⁴	D₅₃₆	1	1	yes	characteristic
132	C₂²×C₄	—	268	8	no	—
133	C₂²×C₄	—	268	4	no	—
134	C₂²×C₄	—	536	4	no	—
135	C₂²×C₄	—	536	4	no	—
136	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
137	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
138	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
139	C₂²×C₄	#11761617	1	4	yes	—
140	C₂²×C₄	Dic₂₆₈	1	2	yes	—
141	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
142	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
143	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
144	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
145	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
146	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
147	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
148	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
149	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
150	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
151	C₄⋊C₄	—	268	8	no	—
152	C₄⋊C₄	—	268	4	no	—
153	C₄⋊C₄	—	134	8	no	—
154	C₁₆	—	2	4	no	—
155	C₂³	#14523965	1	4	yes	—
156	C₂³	#14523977	1	2	yes	—
157	C₂³	D₁₀₇₂	1	2	yes	—
158	C₂³	#14523988	1	4	yes	—
159	C₂³	Dic₅₃₆	1	2	yes	—
160	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
161	C₂×C₄	—	268	8	no	—
162	C₂×C₄	—	536	4	no	—
163	C₂×C₄	—	536	4	no	—
164	C₂×C₄	—	536	16	no	—
165	C₂×C₄	C₆₇⋊#95	1	4	yes	—
166	C₂×C₄	#14523965	1	4	yes	—
167	C₂×C₄	#14523968	1	8	yes	—
168	C₂×C₄	#14523975	1	4	yes	—
169	C₂×C₄	#14523977	1	2	yes	—
170	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×#11761617	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
174	C₈	C₆₇⋊#95	1	4	yes	—
175	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
176	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
177	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
178	C₂²	#16863159	1	4	yes	—
179	C₂²	#16863198	1	8	yes	—
180	C₂²	#16863201	1	8	yes	—
181	C₂²	#16863222	1	4	yes	—
182	C₂²	#16863224	1	4	yes	—
183	C₂²	C₂×#14523965	1	2	yes	—
184	C₂²	C₂×#14523977	1	1	yes	characteristic
185	C₂²	C₂×D₁₀₇₂	1	1	yes	characteristic
186	C₂²	C₂×#14523988	1	2	yes	—
187	C₂²	C₂×Dic₅₃₆	1	1	yes	characteristic
188	C₄	—	536	8	no	—
189	C₄	#16863159	1	4	yes	—
190	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
191	C₄	C₂×#14523968	1	2	yes	—
192	C₄	C₂×#14523975	1	1	yes	characteristic
193	C₂	#22558301	1	8	yes	direct product factor
194	C₂	C₂×#16863159	1	1	yes	characteristic
195	C₂	C₂×#16863198	1	2	yes	—
196	C₂	C₂×#16863201	1	2	yes	—
197	C₂	C₂×#16863222	1	1	yes	characteristic
198	C₂	C₂×#16863224	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₃₆
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂³×C₁₆×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂³×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1380
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22558301	C₂	direct (N × H)
7. C₂×#14523977	C₄	semidirect
11. C₂²×C₁₆×C₆₇	C₄	semidirect
20. #14523977	C₂×C₄	semidirect
30. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
94. C₂×C₆₇	#1380	semidirect
103. C₆₇	C₂×#1380	semidirect
107. C₂²×C₁₆	Dic₆₇	semidirect
127. C₂×C₁₆	C₂×Dic₆₇	semidirect
193. C₂	#22558301	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×#14523977	2
#22558301	2
C₂³×C₁₆×C₆₇	2
C₂×#1380	67

Subgroups — C2×#22558301 #30770708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558301 #30770708