Subgroups — C₂×#22560350 #30856061

Order 17152.

830 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	2	yes	maximal
2	#22560350	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863161	C₂	1	4	yes	maximal
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₂²	1	4	yes	—
6	#16863420	—	2	4	no	—
7	#16863420	C₂²	1	8	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
9	#16863161	C₂²	1	16	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#97×C₆₇	C₄	1	4	yes	—
13	C₂×#97×C₆₇	C₄	1	4	yes	—
14	C₂×#97×C₆₇	C₂²	1	4	yes	—
15	C₂×#11761612	—	2	4	no	—
16	C₂×#11761612	—	2	16	no	—
17	C₂×#11761612	C₂³	1	8	yes	—
18	C₂²×C₆₇⋊C₈	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
20	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	—	2	8	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	2	yes	—
30	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
32	#97×C₆₇	C₂×C₄	1	16	yes	—
33	#11761612	—	2	16	no	—
34	#11761612	—	4	8	no	—
35	C₂×C₆₇⋊C₈	—	2	8	no	—
36	C₂×C₆₇⋊C₈	—	4	8	no	—
37	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	4	4	no	—
39	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	#41	1	8	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	8	no	—
50	C₂×C₈×C₆₇	—	2	8	no	—
51	C₂×C₈×C₆₇	—	2	8	no	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
53	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
54	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
56	C₄²×C₆₇	C₂×D₄	1	4	yes	—
57	C₂³×C₆₇	—	2	4	no	—
58	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₆₇	#125	1	4	yes	—
61	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	8	no	—
63	C₂×C₄×C₆₇	—	4	4	no	—
64	C₂×C₄×C₆₇	—	4	4	no	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#117	1	4	yes	—
69	C₂×C₄×C₆₇	#117	1	4	yes	—
70	C₂×C₄×C₆₇	#124	1	4	yes	—
71	C₂×C₄×C₆₇	#124	1	4	yes	—
72	C₂×C₄×C₆₇	#125	1	4	yes	—
73	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
76	C₈×C₆₇	—	2	8	no	—
77	C₆₇⋊C₈	—	4	8	no	—
78	C₂²×C₆₇	—	2	8	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	—	2	2	no	—
81	C₂²×C₆₇	#382	1	2	yes	—
82	C₂²×C₆₇	#382	1	2	yes	—
83	C₂²×C₆₇	#430	1	4	yes	—
84	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×#125	1	2	yes	—
86	C₄×C₆₇	—	2	8	no	—
87	C₄×C₆₇	#382	1	2	yes	—
88	C₄×C₆₇	#382	1	2	yes	—
89	C₄×C₆₇	C₂×#117	1	2	yes	—
90	C₄×C₆₇	C₂×#124	1	2	yes	—
91	C₂×#1838	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	—	2	2	no	—
93	C₂×C₆₇	—	2	2	no	—
94	C₂×C₆₇	#1838	1	4	yes	complemented
95	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#430	1	2	yes	—
97	C₂×#423	—	67	2	no	—
98	C₂²×#97	D₆₇	1	1	yes	characteristic
99	#1838	—	67	8	no	—
100	C₂×#341	—	67	4	no	—
101	C₆₇	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
102	C₂²×M₁₆	—	67	4	no	—
103	C₂³×C₈	C₂×D₆₇	1	2	yes	—
104	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
105	#423	—	67	8	no	—
106	#423	—	134	4	no	—
107	C₂×C₄⋊C₈	—	134	4	no	—
108	C₂×#97	Dic₆₇	1	4	yes	—
109	C₂×#97	Dic₆₇	1	4	yes	—
110	C₂×#97	C₂×D₆₇	1	4	yes	—
111	#341	—	67	16	no	—
112	C₂³×C₄	Dic₁₃₄	1	2	yes	—
113	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×M₁₆	—	67	8	no	—
115	C₂×M₁₆	—	134	16	no	—
116	C₂×M₁₆	—	134	4	no	—
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	—	134	4	no	—
119	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
120	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
121	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
122	C₂×C₄²	Dic₁₃₄	1	2	yes	—
123	C₂×C₄²	D₂₆₈	1	2	yes	—
124	C₂×C₄²	D₂₆₈	1	2	yes	—
125	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
126	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
127	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
128	C₄⋊C₈	—	134	16	no	—
129	#97	C₂×Dic₆₇	1	16	yes	—
130	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
133	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
134	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
139	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
140	C₂²×C₄	#11761628	1	8	yes	—
141	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
142	M₁₆	—	268	8	no	—
143	M₁₆	—	134	16	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	268	8	no	—
148	C₂×C₈	—	134	8	no	—
149	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
150	C₄²	C₄⋊Dic₆₇	1	4	yes	—
151	C₄²	C₄⋊Dic₆₇	1	4	yes	—
152	C₄²	C₂²×Dic₆₇	1	4	yes	—
153	C₄²	C₂×D₂₆₈	1	4	yes	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
156	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
157	C₂³	#14524035	1	4	yes	—
158	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
165	C₂×C₄	#14524002	1	4	yes	—
166	C₂×C₄	#14524002	1	4	yes	—
167	C₂×C₄	#14524031	1	4	yes	—
168	C₂×C₄	#14524031	1	4	yes	—
169	C₂×C₄	#14524035	1	4	yes	—
170	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
172	C₂×C₄	C₂×#11761628	1	4	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	268	8	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	#16863262	1	2	yes	—
179	C₂²	#16863262	1	2	yes	—
180	C₂²	#16863445	1	4	yes	—
181	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
182	C₂²	C₂×#14524035	1	2	yes	—
183	C₄	—	2	8	no	—
184	C₄	#16863262	1	2	yes	—
185	C₄	#16863262	1	2	yes	—
186	C₄	C₂×#14524002	1	2	yes	—
187	C₄	C₂×#14524031	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	#22560350	1	4	yes	direct product factor
191	C₂	C₂×#16863262	1	1	yes	characteristic
192	C₂	C₂×#16863445	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#1838
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22560350	C₂	direct (N × H)
94. C₂×C₆₇	#1838	semidirect
101. C₆₇	C₂×#1838	semidirect
190. C₂	#22560350	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×#16863420	2
#22560350	2
C₂×#16863161	2
C₂²×#97×C₆₇	2
C₂×#1838	67

Subgroups — C2×#22560350 #30856061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560350 #30856061