Subgroups — C₂×#22560329 #30855978

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
2	#22560329	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863151	C₂	1	4	yes	maximal
4	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
8	#16863151	C₂²	1	16	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#376×C₆₇	C₄	1	4	yes	—
12	#376×C₆₇	C₂²	1	4	yes	—
13	C₂×#95×C₆₇	C₄	1	4	yes	—
14	C₂²×C₆₇⋊C₈	—	2	8	no	—
15	C₂²×C₆₇⋊C₈	—	2	16	no	—
16	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
17	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	—	2	8	no	—
22	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₆₇	D₄	1	4	yes	—
25	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
28	#95×C₆₇	C₂×C₄	1	16	yes	—
29	C₂×C₆₇⋊C₈	—	2	8	no	—
30	C₂×C₆₇⋊C₈	—	2	16	no	—
31	C₂×C₆₇⋊C₈	—	4	16	no	—
32	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	—	2	8	no	—
34	C₂²×C₄×C₆₇	—	2	8	no	—
35	C₂²×C₄×C₆₇	—	2	16	no	—
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
42	C₂²×C₄×C₆₇	#41	1	8	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₄²×C₆₇	—	2	8	no	—
45	C₄²×C₆₇	—	2	8	no	—
46	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
47	C₄²×C₆₇	C₂×D₄	1	4	yes	—
48	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
49	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
50	C₂³×C₆₇	#117	1	4	yes	—
51	C₂³×C₆₇	#131	1	8	yes	—
52	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	2	8	no	—
54	C₂×C₄×C₆₇	—	2	16	no	—
55	C₂×C₄×C₆₇	—	2	8	no	—
56	C₂×C₄×C₆₇	—	2	16	no	—
57	C₂×C₄×C₆₇	—	4	16	no	—
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₆₇	#117	1	4	yes	—
61	C₂×C₄×C₆₇	#124	1	8	yes	—
62	C₂×C₄×C₆₇	#125	1	8	yes	—
63	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
66	C₆₇⋊C₈	—	4	8	no	—
67	C₂²×C₆₇	#382	1	4	yes	—
68	C₂²×C₆₇	#424	1	8	yes	—
69	C₂²×C₆₇	#433	1	16	yes	—
70	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₆₇	C₂×#117	1	2	yes	—
72	C₂²×C₆₇	C₂×#131	1	4	yes	—
73	C₄×C₆₇	—	2	8	no	—
74	C₄×C₆₇	—	4	8	no	—
75	C₄×C₆₇	#382	1	4	yes	—
76	C₄×C₆₇	C₂×#124	1	2	yes	—
77	C₄×C₆₇	C₂×#125	1	2	yes	—
78	C₂×#1832	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	#1832	1	8	yes	complemented
80	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
81	C₂×C₆₇	C₂×#424	1	2	yes	—
82	C₂×C₆₇	C₂×#433	1	4	yes	—
83	C₂²×C₄⋊C₈	—	67	2	no	—
84	C₂×#376	D₆₇	1	1	yes	characteristic
85	#1832	—	67	8	no	—
86	C₂×#336	—	67	4	no	—
87	C₆₇	C₂×#1832	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₈	—	67	4	no	—
89	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
90	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
91	C₂×C₄⋊C₈	—	67	8	no	—
92	C₂×C₄⋊C₈	—	134	8	no	—
93	#376	Dic₆₇	1	4	yes	—
94	#376	C₂×D₆₇	1	4	yes	—
95	C₂×#95	Dic₆₇	1	4	yes	—
96	#336	—	67	16	no	—
97	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
98	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
99	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	C₂²×C₈	—	67	8	no	—
101	C₂²×C₈	—	134	8	no	—
102	C₂²×C₈	—	134	16	no	—
103	C₂×C₄²	—	2	8	no	—
104	C₂×C₄²	Dic₁₃₄	1	4	yes	—
105	C₂×C₄²	D₂₆₈	1	4	yes	—
106	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
107	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
108	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
109	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
110	C₄⋊C₈	—	134	16	no	—
111	#95	C₂×Dic₆₇	1	16	yes	—
112	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	16	no	—
116	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
117	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
118	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
119	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
121	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
122	C₂²×C₄	#11761628	1	8	yes	—
123	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
124	C₂×C₈	—	268	16	no	—
125	C₂×C₈	—	134	8	no	—
126	C₂×C₈	—	134	16	no	—
127	C₄²	—	2	8	no	—
128	C₄²	—	2	8	no	—
129	C₄²	C₄⋊Dic₆₇	1	8	yes	—
130	C₄²	C₂×D₂₆₈	1	4	yes	—
131	C₄²	C₂×Dic₁₃₄	1	4	yes	—
132	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
133	C₂³	#14524002	1	4	yes	—
134	C₂³	#14524050	1	8	yes	—
135	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	16	no	—
140	C₂×C₄	—	4	16	no	—
141	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
143	C₂×C₄	#14524002	1	4	yes	—
144	C₂×C₄	#14524031	1	8	yes	—
145	C₂×C₄	#14524035	1	8	yes	—
146	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
148	C₂×C₄	C₂×#11761628	1	4	yes	—
149	C₈	—	268	8	no	—
150	C₂²	#16863262	1	4	yes	—
151	C₂²	#16863423	1	8	yes	—
152	C₂²	#16863457	1	16	yes	—
153	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
154	C₂²	C₂×#14524002	1	2	yes	—
155	C₂²	C₂×#14524050	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	#16863262	1	4	yes	—
159	C₄	C₂×#14524031	1	2	yes	—
160	C₄	C₂×#14524035	1	2	yes	—
161	C₂	#22560329	1	8	yes	direct product factor
162	C₂	C₂×#16863262	1	1	yes	characteristic
163	C₂	C₂×#16863423	1	2	yes	—
164	C₂	C₂×#16863457	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂³×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#376×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#376	D₆₇
hypocenter	C₆₇	C₂×#1832
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #22560329	C₂	direct (N × H)
79. C₂×C₆₇	#1832	semidirect
87. C₆₇	C₂×#1832	semidirect
161. C₂	#22560329	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×C₄⋊(C₆₇⋊C₈)	2
#22560329	2
C₂×#16863151	2
C₂×#376×C₆₇	2
C₂×#1832	67

Subgroups — C2×#22560329 #30855978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560329 #30855978