Subgroups — #30765167

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22562008	C₂	1	1	yes	maximal, characteristic
3	#22558148	C₂	1	4	yes	maximal
4	C₆₇×#2162	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₈×Dic₁₃₄	—	2	1	no	—
7	C₈×Dic₁₃₄	—	2	1	no	—
8	C₈×Dic₁₃₄	—	2	1	no	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
13	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
14	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄×C₆₇⋊C₁₆	—	2	1	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₆₇	—	2	1	no	—
18	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
19	C₄×C₁₆×C₆₇	—	2	1	no	—
20	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	—	2	1	no	—
25	C₄×Dic₁₃₄	—	2	1	no	—
26	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
28	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₆₇⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	Dic₆₇⋊C₈	—	2	2	no	—
33	Dic₆₇⋊C₈	—	2	2	no	—
34	C₈×Dic₆₇	—	2	2	no	—
35	C₈×Dic₆₇	—	2	2	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂×Dic₁₃₄	—	2	1	no	—
45	C₂×Dic₁₃₄	—	2	1	no	—
46	C₂×Dic₁₃₄	—	2	2	no	—
47	C₂×Dic₁₃₄	C₁₆	1	4	yes	complemented
48	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
49	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
50	C₂×C₆₇⋊C₈	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₄⋊Dic₆₇	—	2	1	no	—
53	C₄⋊Dic₆₇	—	2	1	no	—
54	Dic₆₇⋊C₄	—	2	2	no	—
55	Dic₆₇⋊C₄	—	2	2	no	—
56	C₄×Dic₆₇	—	2	2	no	—
57	C₄×Dic₆₇	—	2	2	no	—
58	C₆₇⋊C₁₆	—	4	2	no	—
59	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	4	1	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
71	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
72	C₁₆×C₆₇	—	4	2	no	—
73	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	4	1	no	—
77	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₆₇	—	2	2	no	—
86	C₂×Dic₆₇	—	2	2	no	—
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	4	2	no	—
89	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	—
90	Dic₁₃₄	—	2	4	no	—
91	Dic₁₃₄	—	2	4	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	#101	1	2	yes	—
96	C₆₇⋊C₈	—	4	2	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
101	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	C₂²⋊C₁₆	1	2	yes	—
106	C₄×C₆₇	C₂²⋊C₁₆	1	2	yes	—
107	C₄×C₆₇	#410	1	1	yes	characteristic
108	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
109	Dic₆₇	—	2	4	no	—
110	Dic₆₇	—	4	2	no	—
111	#11493	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	—	2	1	no	—
114	C₂×C₆₇	#1461	1	1	yes	characteristic
115	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
116	C₂×C₆₇	#2115	1	1	yes	characteristic
117	C₂×C₈×Q₈	—	67	1	no	—
118	#2162	—	67	1	no	—
119	#2162	D₆₇	1	1	yes	characteristic
120	#1322	—	67	4	no	—
121	C₆₇	#11493	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	—	67	1	no	—
123	C₈×Q₈	—	134	1	no	—
124	C₈×Q₈	—	134	1	no	—
125	C₈×Q₈	—	134	1	no	—
126	C₈×Q₈	—	134	1	no	—
127	C₂×C₄⋊C₈	—	67	2	no	—
128	C₂×C₄⋊C₈	—	67	1	no	—
129	C₂×C₄×C₈	—	67	2	no	—
130	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	1	no	—
132	C₄⋊C₁₆	—	134	1	no	—
133	C₂²⋊C₁₆	—	67	2	no	—
134	C₂²⋊C₁₆	C₂×D₆₇	1	2	yes	—
135	C₄×C₁₆	—	2	1	no	—
136	C₄×C₁₆	—	134	1	no	—
137	C₂²×Q₈	—	67	1	no	—
138	C₂²×C₈	—	67	2	no	—
139	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
141	C₄×Q₈	—	134	1	no	—
142	C₄×Q₈	—	134	1	no	—
143	C₄×Q₈	—	134	1	no	—
144	C₄×Q₈	—	134	1	no	—
145	C₂×C₄⋊C₄	—	67	2	no	—
146	C₂×C₄⋊C₄	—	67	1	no	—
147	C₂×C₄²	—	67	2	no	—
148	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	134	2	no	—
151	C₄⋊C₈	—	134	2	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₄⋊C₈	—	134	1	no	—
154	C₄⋊C₈	—	134	1	no	—
155	C₄×C₈	—	134	2	no	—
156	C₄×C₈	—	134	2	no	—
157	C₄×C₈	D₂₆₈	1	1	yes	characteristic
158	C₄×C₈	D₂₆₈	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
161	C₂×Q₈	—	67	4	no	—
162	C₂×Q₈	—	134	1	no	—
163	C₂×Q₈	—	134	1	no	—
164	C₂×Q₈	—	134	2	no	—
165	C₂²×C₄	—	67	2	no	—
166	C₂²×C₄	—	67	2	no	—
167	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	268	2	no	—
173	C₂×C₈	—	134	2	no	—
174	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	2	no	—
180	C₄⋊C₄	—	134	1	no	—
181	C₄⋊C₄	—	134	1	no	—
182	C₄²	—	134	2	no	—
183	C₄²	—	134	2	no	—
184	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
186	C₄²	#11761611	1	1	yes	characteristic
187	C₄²	#11761611	1	1	yes	characteristic
188	C₁₆	—	4	2	no	—
189	C₁₆	—	268	2	no	—
190	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
191	Q₈	—	134	4	no	—
192	Q₈	—	134	4	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	268	2	no	—
197	C₂×C₄	—	67	4	no	—
198	C₂×C₄	—	134	2	no	—
199	C₂×C₄	—	134	2	no	—
200	C₂×C₄	—	134	2	no	—
201	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
202	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
204	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
205	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
206	C₂×C₄	#14523982	1	2	yes	—
207	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	268	2	no	—
213	C₈	#14523961	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
217	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
218	C₂²	C₂×#14523982	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	268	2	no	—
223	C₄	—	134	4	no	—
224	C₄	(C₂×D₆₇)⋊C₁₆	1	2	yes	—
225	C₄	(C₂×D₆₇)⋊C₁₆	1	2	yes	—
226	C₄	#16863369	1	1	yes	characteristic
227	C₄	C₂×#14523961	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#22558532	1	1	yes	characteristic
231	C₂	C₂×(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
232	C₂	#22561829	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₆₇)⋊C₄
commutator	C₂²×C₆₇	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2162	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2162	D₆₇
hypocenter	C₆₇	#11493
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
47. C₂×Dic₁₃₄	C₁₆	semidirect
121. C₆₇	#11493	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22562008	2
#22558148	2
C₆₇×#2162	2
#11493	67