Subgroups — C₂×C₄×#14523953 #30853154

Order 17152.

1790 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523953	C₂	1	4	yes	maximal
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#14523953	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₆₇	C₄	1	4	yes	—
8	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
9	C₂×#14523953	C₄	1	32	yes	direct product factor
10	C₂×#14523953	C₂²	1	16	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
13	C₄²×Dic₆₇	C₄	1	4	yes	—
14	C₄²×Dic₆₇	C₂²	1	4	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
19	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	16	yes	—
22	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₆₇	—	2	16	no	—
24	C₂×C₄×Dic₆₇	C₂×C₄	1	16	yes	—
25	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
26	C₂×C₄×Dic₆₇	C₂×C₄	1	16	yes	—
27	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
28	#14523953	C₂×C₄	1	64	yes	direct product factor
29	C₄×C₆₇⋊C₈	C₂×C₄	1	16	yes	complemented
30	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
40	C₂²×Dic₆₇	—	2	16	no	—
41	C₂²×Dic₆₇	C₄²	1	16	yes	—
42	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
43	C₂×C₆₇⋊C₈	C₄²	1	32	yes	complemented
44	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
45	C₂×C₆₇⋊C₈	C₂²×C₄	1	16	yes	—
46	C₄×Dic₆₇	—	2	32	no	—
47	C₄×Dic₆₇	C₄²	1	16	yes	—
48	C₄×Dic₆₇	C₂²×C₄	1	16	yes	—
49	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	16	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₄²	1	32	yes	complemented
59	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
61	C₄²×C₆₇	C₄²	1	8	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
65	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
66	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
67	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#97	1	32	yes	complemented
69	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
70	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
71	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
73	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	16	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
77	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×Dic₆₇	—	2	16	no	—
79	C₂×Dic₆₇	—	2	32	no	—
80	C₂×Dic₆₇	C₂×C₄²	1	16	yes	—
81	C₈×C₆₇	C₂×C₄²	1	16	yes	complemented
82	C₆₇⋊C₈	C₂×C₄²	1	16	yes	complemented
83	C₂²×C₆₇	C₄³	1	4	yes	—
84	C₂²×C₆₇	C₂×#97	1	8	yes	—
85	C₂²×C₆₇	C₄×M₁₆	1	16	yes	—
86	C₂²×C₆₇	C₂²×C₄²	1	2	yes	—
87	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
88	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
89	C₄×C₆₇	C₄³	1	4	yes	—
90	C₄×C₆₇	C₂×#97	1	16	yes	complemented
91	C₄×C₆₇	C₂²×C₄²	1	2	yes	—
92	C₄×C₆₇	C₂²×C₄²	1	2	yes	—
93	Dic₆₇	—	2	16	no	—
94	C₂×C₄×#97	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	C₄×#97	1	8	yes	complemented
96	C₂×C₆₇	C₂×C₄³	1	1	yes	characteristic
97	C₂×C₆₇	C₂²×#97	1	2	yes	—
98	C₂×C₆₇	C₂×C₄×M₁₆	1	4	yes	—
99	C₂²×#97	—	67	4	no	—
100	C₂²×C₄×C₈	—	67	1	no	—
101	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
102	C₂×C₄³	—	67	1	no	—
103	C₄×#97	—	67	8	no	—
104	C₆₇	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	—	67	2	no	—
106	C₂³×C₈	C₂×D₆₇	1	2	yes	—
107	C₂²×C₄²	—	67	2	no	—
108	C₂²×C₄²	—	67	4	no	—
109	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
110	C₂×#97	—	67	32	no	—
111	C₂×#97	—	67	16	no	—
112	C₂×C₄×C₈	—	67	8	no	—
113	C₂×C₄×C₈	—	67	4	no	—
114	C₂×C₄×C₈	Dic₆₇	1	8	yes	complemented
115	C₂×C₄×C₈	C₂×D₆₇	1	4	yes	—
116	C₄³	—	67	4	no	—
117	C₄³	—	67	4	no	—
118	C₂³×C₄	—	67	4	no	—
119	C₂³×C₄	C₄×D₆₇	1	2	yes	—
120	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
121	C₂²×C₈	—	67	8	no	—
122	C₂²×C₈	—	67	16	no	—
123	C₂²×C₈	—	67	4	no	—
124	C₂²×C₈	C₄×D₆₇	1	16	yes	—
125	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
126	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
127	C₂×C₄²	—	67	4	no	—
128	C₂×C₄²	—	67	16	no	—
129	C₂×C₄²	—	67	4	no	—
130	C₂×C₄²	—	67	16	no	—
131	C₂×C₄²	—	134	16	no	—
132	C₂×C₄²	C₄×D₆₇	1	2	yes	—
133	C₂×C₄²	C₄×D₆₇	1	4	yes	—
134	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
135	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
136	#97	—	67	64	no	—
137	C₄×C₈	—	67	16	no	—
138	C₄×C₈	C₂×Dic₆₇	1	16	yes	complemented
139	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
140	C₂²×C₄	—	67	16	no	—
141	C₂²×C₄	—	67	8	no	—
142	C₂²×C₄	—	134	16	no	—
143	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
144	C₂²×C₄	#11761611	1	16	yes	—
145	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₆₇	1	8	yes	—
148	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
150	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
151	C₂×C₈	—	67	32	no	—
152	C₂×C₈	—	67	8	no	—
153	C₂×C₈	—	67	16	no	—
154	C₂×C₈	C₄×Dic₆₇	1	32	yes	complemented
155	C₂×C₈	C₂×C₄×D₆₇	1	16	yes	—
156	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
157	C₄²	—	67	16	no	—
158	C₄²	—	67	16	no	—
159	C₄²	—	134	32	no	—
160	C₄²	C₄×Dic₆₇	1	8	yes	—
161	C₄²	C₂×C₄×D₆₇	1	4	yes	—
162	C₄²	C₂²×Dic₆₇	1	4	yes	—
163	C₂³	C₄²×D₆₇	1	4	yes	—
164	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
165	C₂³	C₂×#11761611	1	8	yes	—
166	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
167	C₂×C₄	—	67	16	no	—
168	C₂×C₄	—	134	32	no	—
169	C₂×C₄	—	134	16	no	—
170	C₂×C₄	#14523953	1	32	yes	direct product factor
171	C₂×C₄	C₄²×D₆₇	1	4	yes	—
172	C₂×C₄	C₄²×D₆₇	1	8	yes	—
173	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
174	C₂×C₄	C₂×C₄×Dic₆₇	1	8	yes	—
175	C₂×C₄	C₂×C₄×Dic₆₇	1	8	yes	—
176	C₂×C₄	C₂×#11761611	1	16	yes	—
177	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
179	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
180	C₈	—	67	16	no	—
181	C₈	C₂×C₄×Dic₆₇	1	16	yes	complemented
182	C₂²	C₄²×Dic₆₇	1	4	yes	—
183	C₂²	C₂×#14523953	1	8	yes	—
184	C₂²	C₄×#11761611	1	16	yes	—
185	C₂²	C₂×C₄²×D₆₇	1	2	yes	—
186	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
187	C₂²	C₂²×#11761611	1	4	yes	—
188	C₄	—	134	16	no	—
189	C₄	C₄²×Dic₆₇	1	4	yes	—
190	C₄	C₂×#14523953	1	16	yes	direct product factor
191	C₄	C₂×C₄²×D₆₇	1	2	yes	—
192	C₄	C₂²×C₄×Dic₆₇	1	2	yes	—
193	C₂	C₄×#14523953	1	8	yes	direct product factor
194	C₂	C₂×C₄²×Dic₆₇	1	1	yes	characteristic
195	C₂	C₂²×#14523953	1	2	yes	—
196	C₂	C₂×C₄×#11761611	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₆₇
commutator	C₂×C₆₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×C₄×#97
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×#14523953	C₂	direct (N × H)
9. C₂×#14523953	C₄	direct (N × H)
11. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
17. C₂×C₄×C₈×C₆₇	C₄	semidirect
28. #14523953	C₂×C₄	direct (N × H)
29. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
39. C₄×C₈×C₆₇	C₂×C₄	semidirect
43. C₂×C₆₇⋊C₈	C₄²	semidirect
58. C₂×C₈×C₆₇	C₄²	semidirect
68. C₂×C₄×C₆₇	#97	semidirect
81. C₈×C₆₇	C₂×C₄²	semidirect
82. C₆₇⋊C₈	C₂×C₄²	semidirect
90. C₄×C₆₇	C₂×#97	semidirect
95. C₂×C₆₇	C₄×#97	semidirect
104. C₆₇	C₂×C₄×#97	semidirect
114. C₂×C₄×C₈	Dic₆₇	semidirect
138. C₄×C₈	C₂×Dic₆₇	semidirect
154. C₂×C₈	C₄×Dic₆₇	semidirect
170. C₂×C₄	#14523953	direct (N × H)
181. C₈	C₂×C₄×Dic₆₇	semidirect
190. C₄	C₂×#14523953	direct (N × H)
193. C₂	C₄×#14523953	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×#14523953	2
C₂²×C₄×C₆₇⋊C₈	2
C₂×C₄²×Dic₆₇	2
C₄×#14523953	2
C₂²×C₄×C₈×C₆₇	2
C₂×C₄×#97	67

Subgroups — C2×C4×#14523953 #30853154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14523953 #30853154