Subgroups — C₂×#22559962 #30853302

Order 17152.

1134 conjugacy classes of subgroups.

The subgroup lattice has 190 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	1	yes	maximal, characteristic
2	C₂×#16863242	C₂	1	1	yes	maximal, characteristic
3	#22559962	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863150	C₂	1	4	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₄	1	8	yes	complemented
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
10	#16863242	C₄	1	4	yes	—
11	#16863242	C₂²	1	4	yes	—
12	C₂×C₆₇⋊#95	C₄	1	4	yes	—
13	#16863150	C₂²	1	16	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#97×C₆₇	C₄	1	8	yes	complemented
17	C₂×#97×C₆₇	C₂²	1	4	yes	—
18	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
19	C₂²×C₆₇⋊C₈	—	2	8	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
21	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
22	C₂×C₄×Dic₆₇	—	2	8	no	—
23	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
24	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	16	yes	complemented
26	C₆₇⋊#95	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	8	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
36	#97×C₆₇	C₂×C₄	1	16	yes	complemented
37	C₂²×Dic₆₇	—	2	8	no	—
38	C₂²×Dic₆₇	—	2	16	no	—
39	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
40	C₂×C₆₇⋊C₈	—	2	16	no	—
41	C₂×C₆₇⋊C₈	—	2	8	no	—
42	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
43	C₄×Dic₆₇	—	2	8	no	—
44	C₄×Dic₆₇	—	2	8	no	—
45	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	—	2	8	no	—
47	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	#41	1	8	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	16	no	—
55	C₂×C₈×C₆₇	—	2	8	no	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
57	C₄²×C₆₇	C₄²	1	8	yes	—
58	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
59	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
61	C₂³×C₆₇	#117	1	4	yes	—
62	C₂³×C₆₇	#131	1	8	yes	—
63	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	—	2	8	no	—
65	C₂×C₄×C₆₇	—	2	16	no	—
66	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
67	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₆₇	#117	1	8	yes	—
69	C₂×C₄×C₆₇	#117	1	4	yes	—
70	C₂×C₄×C₆₇	#117	1	8	yes	—
71	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
74	C₂×Dic₆₇	—	2	8	no	—
75	C₂×Dic₆₇	—	2	16	no	—
76	C₂×Dic₆₇	—	4	16	no	—
77	C₈×C₆₇	—	2	8	no	—
78	C₆₇⋊C₈	—	2	8	no	—
79	C₂²×C₆₇	#376	1	4	yes	—
80	C₂²×C₆₇	#405	1	8	yes	—
81	C₂²×C₆₇	#433	1	16	yes	—
82	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂²×C₆₇	C₂×#117	1	2	yes	—
84	C₂²×C₆₇	C₂×#131	1	4	yes	—
85	C₄×C₆₇	—	2	8	no	—
86	C₄×C₆₇	#376	1	4	yes	—
87	C₄×C₆₇	C₂×#117	1	2	yes	—
88	C₄×C₆₇	C₂×#117	1	2	yes	—
89	Dic₆₇	—	4	8	no	—
90	C₂×#1746	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	#1746	1	8	yes	complemented
92	C₂×C₆₇	C₂×#376	1	1	yes	characteristic
93	C₂×C₆₇	C₂×#405	1	2	yes	—
94	C₂×C₆₇	C₂×#433	1	4	yes	—
95	C₂²×#97	D₆₇	1	1	yes	characteristic
96	C₂²×C₄×C₈	—	67	1	no	—
97	C₂×#376	—	67	1	no	—
98	#1746	—	67	8	no	—
99	C₂×#336	—	67	4	no	—
100	C₆₇	C₂×#1746	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₈	—	67	2	no	—
102	C₂³×C₈	C₂×D₆₇	1	2	yes	—
103	C₂²×C₄²	—	67	2	no	—
104	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
105	C₂×#97	Dic₆₇	1	8	yes	complemented
106	C₂×#97	C₂×D₆₇	1	4	yes	—
107	C₂×C₄×C₈	—	67	8	no	—
108	C₂×C₄×C₈	—	67	4	no	—
109	#376	—	67	4	no	—
110	#376	—	67	4	no	—
111	C₂×#95	—	67	4	no	—
112	#336	—	67	16	no	—
113	C₂³×C₄	—	67	4	no	—
114	C₂³×C₄	C₄×D₆₇	1	2	yes	—
115	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
116	C₂²×C₈	—	2	8	no	—
117	C₂²×C₈	—	67	8	no	—
118	C₂²×C₈	—	67	4	no	—
119	C₂²×C₈	—	134	8	no	—
120	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
121	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
122	C₂×C₄²	—	67	4	no	—
123	C₂×C₄²	—	67	4	no	—
124	C₂×C₄²	—	134	8	no	—
125	C₂×C₄²	C₄×D₆₇	1	2	yes	—
126	C₂×C₄²	C₄×D₆₇	1	4	yes	—
127	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
128	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
129	#97	C₂×Dic₆₇	1	16	yes	complemented
130	C₄×C₈	—	67	16	no	—
131	#95	—	67	16	no	—
132	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	67	8	no	—
135	C₂²×C₄	—	134	8	no	—
136	C₂²×C₄	—	134	16	no	—
137	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
138	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
139	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
142	C₂²×C₄	#11761628	1	8	yes	—
143	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
144	C₂×C₈	—	2	16	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	67	8	no	—
147	C₂×C₈	—	134	16	no	—
148	C₂×C₈	—	134	8	no	—
149	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
150	C₄²	—	134	8	no	—
151	C₄²	—	134	8	no	—
152	C₄²	C₄×Dic₆₇	1	8	yes	—
153	C₄²	C₂×C₄×D₆₇	1	4	yes	—
154	C₄²	C₂²×Dic₆₇	1	4	yes	—
155	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
156	C₂³	#14524000	1	4	yes	—
157	C₂³	#14524051	1	8	yes	—
158	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
159	C₂×C₄	—	2	16	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	268	16	no	—
162	C₂×C₄	—	134	8	no	—
163	C₂×C₄	—	134	16	no	—
164	C₂×C₄	C₂×C₄×Dic₆₇	1	8	yes	—
165	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
166	C₂×C₄	#14524000	1	4	yes	—
167	C₂×C₄	#14524000	1	8	yes	—
168	C₂×C₄	#14524002	1	8	yes	—
169	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
170	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	C₂×#11761628	1	4	yes	—
172	C₈	—	2	8	no	—
173	C₈	—	134	8	no	—
174	C₂²	#16863242	1	4	yes	—
175	C₂²	#16863354	1	8	yes	—
176	C₂²	#16863458	1	16	yes	—
177	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
178	C₂²	C₂×#14524000	1	2	yes	—
179	C₂²	C₂×#14524051	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	268	8	no	—
182	C₄	#16863242	1	4	yes	—
183	C₄	C₂×#14524000	1	2	yes	—
184	C₄	C₂×#14524002	1	2	yes	—
185	C₂	#22559962	1	8	yes	direct product factor
186	C₂	C₂×#16863242	1	1	yes	characteristic
187	C₂	C₂×#16863354	1	2	yes	—
188	C₂	C₂×#16863458	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#1746
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22559962	C₂	direct (N × H)
8. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
16. C₂×#97×C₆₇	C₄	semidirect
25. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
36. #97×C₆₇	C₂×C₄	semidirect
91. C₂×C₆₇	#1746	semidirect
100. C₆₇	C₂×#1746	semidirect
105. C₂×#97	Dic₆₇	semidirect
129. #97	C₂×Dic₆₇	semidirect
185. C₂	#22559962	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
C₂×#16863242	2
#22559962	2
C₂×#16863150	2
C₂²×#97×C₆₇	2
C₂×#1746	67

Subgroups — C2×#22559962 #30853302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559962 #30853302