Subgroups — #30775771

Order 17152.

310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863223	C₂	1	2	yes	maximal
2	#22560369	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	2	yes	maximal
4	#22558287	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1942	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14523977	—	2	1	no	—
8	C₂×#14523977	C₂²	1	2	yes	—
9	#16863266	—	2	1	no	—
10	#16863223	—	2	4	no	—
11	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#389×C₆₇	—	2	1	no	—
15	#340×C₆₇	—	2	1	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	4	1	no	—
18	C₂×C₄⋊Dic₆₇	—	4	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
22	#14523977	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523977	—	4	2	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
30	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
31	#107×C₆₇	—	2	1	no	—
32	#107×C₆₇	—	2	2	no	—
33	#107×C₆₇	D₄	1	2	yes	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	#95×C₆₇	—	4	1	no	—
36	C₂²×Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	8	2	no	—
40	C₄⋊Dic₆₇	—	8	2	no	—
41	C₆₇⋊C₁₆	—	2	4	no	—
42	C₂²×C₄×C₆₇	—	4	1	no	—
43	C₂²×C₄×C₆₇	—	4	1	no	—
44	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₆₇	#41	1	2	yes	—
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₄⋊C₄×C₆₇	—	4	1	no	—
56	C₄⋊C₄×C₆₇	—	4	1	no	—
57	C₄⋊C₄×C₆₇	—	4	1	no	—
58	C₄⋊C₄×C₆₇	—	8	2	no	—
59	C₄²×C₆₇	—	2	1	no	—
60	C₄²×C₆₇	—	2	1	no	—
61	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	—	2	1	no	—
64	C₂×C₄×C₆₇	—	2	1	no	—
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	—	8	1	no	—
68	C₂×C₄×C₆₇	—	8	1	no	—
69	C₂×C₄×C₆₇	—	8	1	no	—
70	C₂×C₄×C₆₇	—	8	1	no	—
71	C₂×C₄×C₆₇	—	8	1	no	—
72	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
73	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
74	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
80	C₂×Dic₆₇	—	8	2	no	—
81	C₂×Dic₆₇	—	8	2	no	—
82	C₂×Dic₆₇	—	8	2	no	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₆₇	#125	1	2	yes	—
86	C₂²×C₆₇	#459	1	1	yes	characteristic
87	C₂²×C₆₇	#480	1	1	yes	characteristic
88	C₂²×C₆₇	#500	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
90	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
91	C₂²×C₆₇	#509	1	1	yes	characteristic
92	C₂²×C₆₇	#510	1	1	yes	characteristic
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	2	2	no	—
95	C₄×C₆₇	—	8	2	no	—
96	C₄×C₆₇	—	16	1	no	—
97	C₄×C₆₇	#398	1	1	yes	characteristic
98	C₄×C₆₇	#462	1	1	yes	characteristic
99	C₄×C₆₇	#484	1	1	yes	characteristic
100	C₄×C₆₇	#501	1	1	yes	characteristic
101	Dic₆₇	—	16	2	no	—
102	#13506	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	#2083	1	1	yes	characteristic
104	C₂×C₆₇	#2207	1	1	yes	characteristic
105	C₂×C₆₇	#2208	1	1	yes	characteristic
106	C₂×C₆₇	#2233	1	1	yes	characteristic
107	C₂×C₆₇	#2235	1	1	yes	characteristic
108	C₂×C₆₇	#2254	1	1	yes	characteristic
109	C₂×C₆₇	#2255	1	1	yes	characteristic
110	C₂×#367	—	67	2	no	—
111	#1942	D₆₇	1	1	yes	characteristic
112	#1845	—	67	1	no	—
113	#1380	—	67	2	no	—
114	#1374	—	67	1	no	—
115	C₆₇	#13506	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₁₆	—	67	2	no	—
117	C₂×#107	—	67	2	no	—
118	C₂×#107	—	134	1	no	—
119	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
120	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
122	#389	—	2	1	no	—
123	#384	—	134	1	no	—
124	#367	—	134	4	no	—
125	#340	—	2	1	no	—
126	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	268	1	no	—
132	C₂×C₄⋊C₄	—	268	1	no	—
133	C₂×C₄⋊C₄	—	134	2	no	—
134	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂×C₁₆	—	67	4	no	—
136	C₂×C₁₆	—	134	2	no	—
137	C₂×C₁₆	—	134	2	no	—
138	#107	—	2	1	no	—
139	#107	—	2	2	no	—
140	#107	—	268	2	no	—
141	#107	—	134	2	no	—
142	#107	—	134	2	no	—
143	#107	D₂₆₈	1	2	yes	—
144	C₄×C₈	—	2	2	no	—
145	#95	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	268	2	no	—
149	C₂²×C₄	#11761613	1	1	yes	characteristic
150	C₂²×C₄	#11761613	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	#11761618	1	2	yes	—
155	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
156	C₂×C₈	Q₈×D₆₇	1	2	yes	—
157	C₂×C₈	#11761627	1	2	yes	—
158	C₂×C₈	#11761628	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	8	2	no	—
164	C₄⋊C₄	—	268	2	no	—
165	C₄⋊C₄	—	268	2	no	—
166	C₄⋊C₄	—	536	2	no	—
167	C₄⋊C₄	—	536	2	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₁₆	—	134	4	no	—
171	C₂³	C₂×#11761613	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	536	2	no	—
183	C₂×C₄	—	536	2	no	—
184	C₂×C₄	—	536	2	no	—
185	C₂×C₄	#14523986	1	2	yes	—
186	C₂×C₄	#14523987	1	2	yes	—
187	C₂×C₄	#14524017	1	1	yes	characteristic
188	C₂×C₄	#14524024	1	1	yes	characteristic
189	C₂×C₄	#14524041	1	1	yes	characteristic
190	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
191	C₂×C₄	#14524068	1	1	yes	characteristic
192	C₂×C₄	#14524073	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	#14524023	1	2	yes	—
195	C₈	#14524033	1	2	yes	—
196	C₂²	#16863591	1	1	yes	characteristic
197	C₂²	#16863707	1	1	yes	characteristic
198	C₂²	#16863799	1	1	yes	characteristic
199	C₂²	C₂×#14523986	1	1	yes	characteristic
200	C₂²	C₂×#14523987	1	1	yes	characteristic
201	C₂²	#16863831	1	1	yes	characteristic
202	C₂²	#16863836	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	2	no	—
206	C₄	—	16	1	no	—
207	C₄	—	1072	2	no	—
208	C₄	#16863326	1	1	yes	characteristic
209	C₄	#16863606	1	1	yes	characteristic
210	C₄	#16863729	1	1	yes	characteristic
211	C₄	#16863806	1	1	yes	characteristic
212	C₂	#22561680	1	1	yes	characteristic
213	C₂	#22562249	1	1	yes	characteristic
214	C₂	#22562254	1	1	yes	characteristic
215	C₂	#22562392	1	1	yes	characteristic
216	C₂	#22562404	1	1	yes	characteristic
217	C₂	#22562489	1	1	yes	characteristic
218	C₂	#22562496	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1942	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#1942	D₆₇
hypocenter	C₆₇	#13506
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
115. C₆₇	#13506	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863223	2
#22560369	2
#22558300	2
#22558287	2
C₆₇×#1942	2
#13506	67