Subgroups — #22563164

Order 8576.

480 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×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16863302	C₂	1	2	yes	maximal
3	#16863283	C₂	1	4	yes	maximal
4	#16863266	C₂	1	1	yes	maximal, characteristic
5	#16863261	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₆₇	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₃₄	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
17	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
18	C₆₇⋊#95	C₂²	1	4	yes	—
19	C₆₇⋊#95	C₂²	1	4	yes	—
20	C₆₇⋊#95	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#95×C₆₇	C₂²	1	2	yes	—
26	C₂×Dic₁₃₄	—	2	2	no	—
27	C₂×Dic₁₃₄	—	2	2	no	—
28	C₂×Dic₁₃₄	C₂×C₄	1	4	yes	—
29	C₂²×Dic₆₇	C₂³	1	4	yes	—
30	C₂²×Dic₆₇	C₂³	1	2	yes	—
31	C₂²×Dic₆₇	C₂³	1	2	yes	—
32	C₄⋊Dic₆₇	—	2	4	no	—
33	C₄⋊Dic₆₇	—	2	4	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
36	Dic₆₇⋊C₄	C₂×C₄	1	8	yes	—
37	C₄×Dic₆₇	—	4	4	no	—
38	C₄×Dic₆₇	C₂×C₄	1	8	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₄⋊C₄×C₆₇	Q₈	1	4	yes	complemented
46	C₄²×C₆₇	—	2	4	no	—
47	C₄²×C₆₇	C₂×C₄	1	4	yes	—
48	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₆₇	#41	1	2	yes	—
62	C₂×C₄×C₆₇	#41	1	2	yes	—
63	C₂×C₄×C₆₇	#41	1	2	yes	—
64	C₂×Dic₆₇	—	2	2	no	—
65	C₂×Dic₆₇	—	2	2	no	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	2	4	no	—
69	C₂×Dic₆₇	—	2	4	no	—
70	C₂×Dic₆₇	—	2	4	no	—
71	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
72	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
73	Dic₁₃₄	—	2	8	no	—
74	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
77	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	2	4	no	—
83	C₄×C₆₇	—	4	2	no	—
84	C₄×C₆₇	#117	1	4	yes	—
85	C₄×C₆₇	C₄×Q₈	1	4	yes	—
86	Dic₆₇	—	2	4	no	—
87	Dic₆₇	—	2	4	no	—
88	Dic₆₇	—	4	4	no	—
89	C₂×C₆₇	C₂×#117	1	1	yes	characteristic
90	C₂×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₆₇	#520	1	1	yes	characteristic
92	C₂×C₆₇	#554	1	1	yes	characteristic
93	C₂×C₆₇	#556	1	1	yes	characteristic
94	C₂×C₆₇	#557	1	1	yes	characteristic
95	C₂×C₆₇	#559	1	1	yes	characteristic
96	#2355	—	67	1	no	maximal, Sylow
97	C₆₇	#2355	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×Q₈	—	67	1	no	—
99	#391	—	67	2	no	—
100	#387	—	67	4	no	—
101	#384	—	67	1	no	—
102	#382	—	67	2	no	—
103	C₄×C₄⋊C₄	—	67	2	no	—
104	C₄×C₄⋊C₄	—	67	2	no	—
105	C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
106	C₂²×Q₈	—	67	1	no	—
107	C₄×Q₈	—	67	8	no	—
108	C₂×C₄⋊C₄	—	67	2	no	—
109	C₂×C₄⋊C₄	—	67	2	no	—
110	C₂×C₄⋊C₄	—	67	1	no	—
111	C₂×C₄⋊C₄	—	67	2	no	—
112	C₂×C₄⋊C₄	—	67	2	no	—
113	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂×C₄²	—	67	4	no	—
116	C₂×C₄²	—	67	2	no	—
117	C₂×C₄²	C₂×D₆₇	1	2	yes	—
118	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
119	#95	—	67	4	no	—
120	#95	—	67	4	no	—
121	#95	—	67	4	no	—
122	#95	C₂×D₆₇	1	2	yes	—
123	C₂×Q₈	—	67	4	no	—
124	C₂×Q₈	—	134	2	no	—
125	C₂×Q₈	—	134	2	no	—
126	C₂²×C₄	—	67	2	no	—
127	C₂²×C₄	—	67	2	no	—
128	C₂²×C₄	—	67	4	no	—
129	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
130	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
132	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	268	2	no	—
136	C₄⋊C₄	—	67	8	no	—
137	C₄⋊C₄	—	67	4	no	—
138	C₄⋊C₄	—	134	4	no	—
139	C₄⋊C₄	—	134	4	no	—
140	C₄⋊C₄	Dic₁₃₄	1	4	yes	complemented
141	C₄²	—	2	4	no	—
142	C₄²	—	268	4	no	—
143	C₄²	—	67	8	no	—
144	C₄²	C₄×D₆₇	1	4	yes	—
145	C₂³	C₂³×D₆₇	1	1	yes	characteristic
146	Q₈	—	134	8	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	67	4	no	—
155	C₂×C₄	—	67	4	no	—
156	C₂×C₄	—	134	4	no	—
157	C₂×C₄	—	134	4	no	—
158	C₂×C₄	—	134	4	no	—
159	C₂×C₄	—	134	2	no	—
160	C₂×C₄	—	134	2	no	—
161	C₂×C₄	—	134	2	no	—
162	C₂×C₄	—	134	2	no	—
163	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
164	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
166	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
167	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
168	C₂×C₄	#11761627	1	2	yes	—
169	C₂×C₄	#11761628	1	2	yes	—
170	C₂×C₄	#11761629	1	2	yes	—
171	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
172	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
173	C₂²	C₂×#11761627	1	1	yes	characteristic
174	C₂²	C₂×#11761628	1	1	yes	characteristic
175	C₂²	C₂×#11761629	1	1	yes	characteristic
176	C₂²	#14524089	1	1	yes	characteristic
177	C₂²	#14524090	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	268	4	no	—
182	C₄	—	134	4	no	—
183	C₄	—	134	4	no	—
184	C₄	#14524001	1	4	yes	—
185	C₄	C₄×Dic₁₃₄	1	4	yes	—
186	C₂	C₂×#14524001	1	1	yes	characteristic
187	C₂	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
188	C₂	#16863876	1	1	yes	characteristic
189	C₂	#16864103	1	1	yes	characteristic
190	C₂	#16864116	1	1	yes	characteristic
191	C₂	#16864125	1	1	yes	characteristic
192	C₂	#16864131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#2355
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₁₃₄	C₄	semidirect
45. C₄⋊C₄×C₆₇	Q₈	semidirect
97. C₆₇	#2355	semidirect
140. C₄⋊C₄	Dic₁₃₄	semidirect