Subgroups — #22564820

Order 8576.

372 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇⋊#401	C₂	1	2	yes	maximal
2	#16863335	C₂	1	2	yes	maximal
3	#16863302	C₂	1	1	yes	maximal, characteristic
4	#16863301	C₂	1	1	yes	maximal, characteristic
5	#16863301	C₂	1	1	yes	maximal, characteristic
6	#16863282	C₂	1	2	yes	maximal
7	#16863263	C₂	1	2	yes	maximal
8	#16863261	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	#389×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
19	C₆₇⋊#95	C₂²	1	1	yes	characteristic
20	C₆₇⋊#95	C₂²	1	1	yes	characteristic
21	C₆₇⋊#95	C₂²	1	2	yes	—
22	C₆₇⋊#95	C₂²	1	2	yes	—
23	C₆₇⋊#95	C₂²	1	2	yes	—
24	C₆₇⋊#95	C₂²	1	2	yes	—
25	C₆₇⋊#95	C₂²	1	2	yes	—
26	C₆₇⋊#95	C₂²	1	2	yes	—
27	C₆₇⋊#95	C₂²	1	2	yes	—
28	C₆₇⋊#95	C₂²	1	2	yes	—
29	C₆₇⋊#95	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#95×C₆₇	C₂²	1	2	yes	—
35	C₂×Dic₁₃₄	—	2	2	no	—
36	C₂×Dic₁₃₄	—	2	2	no	—
37	C₂×Dic₁₃₄	—	2	2	no	—
38	C₂²×Dic₆₇	C₂³	1	2	yes	—
39	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₆₇	C₂³	1	2	yes	—
43	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	C₄⋊Dic₆₇	—	2	2	no	—
47	C₄×Dic₆₇	—	2	2	no	—
48	C₄×Dic₆₇	—	2	2	no	—
49	C₄×Dic₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	—	2	2	no	—
59	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	—	2	1	no	—
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	1	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	#41	1	2	yes	—
72	C₂×C₄×C₆₇	#41	1	2	yes	—
73	C₂×C₄×C₆₇	#41	1	2	yes	—
74	C₂×C₄×C₆₇	#41	1	2	yes	—
75	C₂×Dic₆₇	—	2	1	no	—
76	C₂×Dic₆₇	—	2	1	no	—
77	C₂×Dic₆₇	—	2	2	no	—
78	C₂×Dic₆₇	—	2	1	no	—
79	C₂×Dic₆₇	—	2	1	no	—
80	C₂×Dic₆₇	—	2	1	no	—
81	C₂×Dic₆₇	—	2	1	no	—
82	C₂×Dic₆₇	—	2	1	no	—
83	C₂×Dic₆₇	—	2	1	no	—
84	C₂×Dic₆₇	—	2	2	no	—
85	C₂×Dic₆₇	—	2	2	no	—
86	C₂×Dic₆₇	—	2	2	no	—
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	2	1	no	—
89	C₂×Dic₆₇	—	2	2	no	—
90	C₂×Dic₆₇	—	2	1	no	—
91	C₂×Dic₆₇	—	2	1	no	—
92	C₂×Dic₆₇	#41	1	2	yes	—
93	Dic₁₃₄	—	4	4	no	—
94	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
95	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
97	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
98	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	4	2	no	—
103	C₄×C₆₇	—	4	1	no	—
104	C₄×C₆₇	—	4	2	no	—
105	C₄×C₆₇	#126	1	4	yes	—
106	Dic₆₇	—	2	2	no	—
107	Dic₆₇	—	4	2	no	—
108	Dic₆₇	—	4	1	no	—
109	Dic₆₇	—	4	2	no	—
110	Dic₆₇	—	4	1	no	—
111	Dic₆₇	—	4	1	no	—
112	C₂×C₆₇	C₂×#126	1	1	yes	characteristic
113	C₂×C₆₇	#529	1	1	yes	characteristic
114	C₂×C₆₇	#541	1	1	yes	characteristic
115	C₂×C₆₇	#552	1	1	yes	characteristic
116	C₂×C₆₇	#556	1	1	yes	characteristic
117	C₂×C₆₇	#556	1	1	yes	characteristic
118	C₂×C₆₇	#556	1	1	yes	characteristic
119	#2514	—	67	1	no	maximal, Sylow
120	C₆₇	#2514	1	1	yes	characteristic, complemented, Sylow
121	#401	—	67	2	no	—
122	#400	—	67	2	no	—
123	#391	—	67	1	no	—
124	#391	—	67	1	no	—
125	#391	—	67	1	no	—
126	#389	D₆₇	1	1	yes	characteristic
127	#387	—	67	2	no	—
128	#383	—	67	2	no	—
129	#382	—	67	1	no	—
130	C₄×C₄⋊C₄	—	67	1	no	—
131	C₄×C₄⋊C₄	—	67	1	no	—
132	C₂²×Q₈	—	67	1	no	—
133	C₂×C₄⋊C₄	—	67	1	no	—
134	C₂×C₄⋊C₄	—	67	1	no	—
135	C₂×C₄⋊C₄	—	67	1	no	—
136	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
139	C₂×C₄²	—	67	1	no	—
140	C₂×C₄²	—	67	1	no	—
141	C₂×C₄²	—	67	2	no	—
142	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
143	#95	—	67	1	no	—
144	#95	—	67	2	no	—
145	#95	—	67	1	no	—
146	#95	—	67	2	no	—
147	#95	—	67	2	no	—
148	#95	—	67	2	no	—
149	#95	—	67	2	no	—
150	#95	—	67	2	no	—
151	#95	—	67	2	no	—
152	#95	—	67	2	no	—
153	#95	—	67	2	no	—
154	#95	C₂×D₆₇	1	2	yes	—
155	C₂×Q₈	—	134	2	no	—
156	C₂×Q₈	—	134	2	no	—
157	C₂×Q₈	—	134	2	no	—
158	C₂²×C₄	—	67	1	no	—
159	C₂²×C₄	—	67	2	no	—
160	C₂²×C₄	—	67	2	no	—
161	C₂²×C₄	—	67	1	no	—
162	C₂²×C₄	—	67	1	no	—
163	C₂²×C₄	—	67	1	no	—
164	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
166	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
167	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	134	2	no	—
173	C₄⋊C₄	—	134	2	no	—
174	C₄⋊C₄	—	134	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	134	4	no	—
177	C₄²	—	134	2	no	—
178	C₄²	—	134	2	no	—
179	C₂³	C₂³×D₆₇	1	1	yes	characteristic
180	Q₈	—	268	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	67	2	no	—
193	C₂×C₄	—	134	2	no	—
194	C₂×C₄	—	134	1	no	—
195	C₂×C₄	—	134	1	no	—
196	C₂×C₄	—	134	1	no	—
197	C₂×C₄	—	134	2	no	—
198	C₂×C₄	—	134	2	no	—
199	C₂×C₄	—	134	2	no	—
200	C₂×C₄	—	134	1	no	—
201	C₂×C₄	—	134	1	no	—
202	C₂×C₄	—	134	1	no	—
203	C₂×C₄	—	134	2	no	—
204	C₂×C₄	—	134	1	no	—
205	C₂×C₄	—	134	2	no	—
206	C₂×C₄	—	134	1	no	—
207	C₂×C₄	—	134	1	no	—
208	C₂×C₄	—	134	1	no	—
209	C₂×C₄	—	134	1	no	—
210	C₂×C₄	#11761627	1	2	yes	—
211	C₂×C₄	#11761627	1	2	yes	—
212	C₂×C₄	#11761628	1	2	yes	—
213	C₂×C₄	#11761629	1	2	yes	—
214	C₂²	#41×D₆₇	1	1	yes	characteristic
215	C₂²	C₂×#11761627	1	1	yes	characteristic
216	C₂²	C₂×#11761627	1	1	yes	characteristic
217	C₂²	C₂×#11761628	1	1	yes	characteristic
218	C₂²	C₂×#11761629	1	1	yes	characteristic
219	C₂²	#14524090	1	1	yes	characteristic
220	C₂²	#14524091	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	268	1	no	—
226	C₄	—	268	2	no	—
227	C₄	—	268	1	no	—
228	C₄	—	268	1	no	—
229	C₄	—	268	2	no	—
230	C₄	—	134	2	no	—
231	C₄	#14524038	1	4	yes	—
232	C₂	C₂×#14524038	1	1	yes	characteristic
233	C₂	#16863927	1	1	yes	characteristic
234	C₂	#16864006	1	1	yes	characteristic
235	C₂	#16864084	1	1	yes	characteristic
236	C₂	#16864116	1	1	yes	characteristic
237	C₂	#16864118	1	1	yes	characteristic
238	C₂	#16864120	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#389×C₆₇	C₂
hypercenter	#389	D₆₇
hypocenter	C₆₇	#2514
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
120. C₆₇	#2514	semidirect