Subgroups — #30856442

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	2	yes	maximal
2	#22560381	C₂	1	2	yes	maximal, complemented
3	#22560347	C₂	1	2	yes	maximal, complemented
4	#22560247	C₂	1	4	yes	maximal
5	C₄×#14523979	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863453	—	2	4	no	—
8	#16863449	—	2	4	no	—
9	C₂×#14523979	C₂²	1	8	yes	—
10	#16863420	C₂²	1	4	yes	—
11	#16863360	C₂²	1	4	yes	—
12	C₄×#11761612	C₂²	1	4	yes	—
13	#16863161	C₂²	1	8	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#11761612	C₂³	1	8	yes	—
22	#14523979	—	2	16	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
25	#14523954	—	2	4	no	—
26	C₄×C₆₇⋊C₈	—	2	4	no	—
27	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	8	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
43	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	#11761612	—	2	16	no	—
46	C₂×C₆₇⋊C₈	—	2	8	no	—
47	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	8	no	—
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	16	no	—
62	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₇	#41	1	8	yes	—
65	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₆₇	—	2	2	no	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	#108	1	8	yes	complemented
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
96	C₈×C₆₇	—	2	4	no	—
97	C₈×C₆₇	#117	1	8	yes	—
98	C₆₇⋊C₈	—	4	8	no	—
99	C₂²×C₆₇	—	2	8	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	C₂×#108	1	4	yes	—
104	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₆₇	#533	1	2	yes	—
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	C₂×#108	1	4	yes	—
110	C₄×C₆₇	C₂×#108	1	4	yes	—
111	C₄×C₆₇	C₂×#117	1	2	yes	—
112	C₄×C₆₇	#532	1	2	yes	—
113	#20826	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	2	no	—
115	C₂×C₆₇	—	2	4	no	—
116	C₂×C₆₇	#2285	1	1	yes	characteristic
117	C₂×C₆₇	C₂²×#108	1	2	yes	—
118	#2920	—	67	2	no	—
119	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
120	#1848	—	67	2	no	—
121	#1837	—	67	2	no	—
122	#1810	—	67	4	no	—
123	C₄×#108	—	67	4	no	—
124	C₆₇	#20826	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₈	C₂×D₆₇	1	2	yes	—
126	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
127	#432	—	134	4	no	—
128	#431	—	134	4	no	—
129	C₂×#108	—	67	8	no	—
130	#423	—	67	4	no	—
131	#407	—	67	4	no	—
132	C₄×M₁₆	—	67	4	no	—
133	C₂×C₄×C₈	Dic₆₇	1	4	yes	—
134	C₂×C₄×C₈	Dic₆₇	1	4	yes	—
135	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
138	#341	—	67	8	no	—
139	C₂³×C₄	Dic₁₃₄	1	2	yes	—
140	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
141	C₂×M₁₆	—	67	8	no	—
142	C₂²×C₈	—	2	8	no	—
143	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
144	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
145	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
146	C₂×C₄²	Dic₁₃₄	1	2	yes	—
147	C₂×C₄²	D₂₆₈	1	2	yes	—
148	C₂×C₄²	D₂₆₈	1	2	yes	—
149	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
150	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
151	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
153	#108	—	134	16	no	—
154	C₄⋊C₈	—	134	8	no	—
155	C₂²⋊C₈	—	134	8	no	—
156	#97	—	134	4	no	—
157	C₄×C₈	—	134	4	no	—
158	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
159	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
160	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
161	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
162	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
163	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
167	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
171	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
172	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
175	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
176	M₁₆	—	134	16	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	16	no	—
179	C₂×C₈	—	134	8	no	—
180	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
181	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
182	C₂×C₈	#11761628	1	8	yes	—
183	C₄²	C₄⋊Dic₆₇	1	2	yes	—
184	C₄²	C₄⋊Dic₆₇	1	2	yes	—
185	C₄²	C₄⋊Dic₆₇	1	2	yes	—
186	C₄²	C₄⋊Dic₆₇	1	2	yes	—
187	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
188	C₄²	C₂²×Dic₆₇	1	2	yes	—
189	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
190	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
191	C₄²	C₂×D₂₆₈	1	2	yes	—
192	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
193	C₂³	—	2	2	no	—
194	C₂³	—	2	4	no	—
195	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
196	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	#14523979	1	8	yes	complemented
206	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
211	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
213	C₂×C₄	C₂×#11761628	1	4	yes	—
214	C₈	—	2	4	no	—
215	C₈	—	268	8	no	—
216	C₈	#14524002	1	8	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	C₂×#14523979	1	4	yes	—
222	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
223	C₂²	#16863953	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₂×#14523979	1	4	yes	—
228	C₄	C₂×#14523979	1	4	yes	—
229	C₄	C₂×#14524002	1	2	yes	—
230	C₄	#16863946	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	#22562616	1	1	yes	characteristic
234	C₂	C₂²×#14523979	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	#20826
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22560381	C₂	semidirect
3. #22560347	C₂	semidirect
5. C₄×#14523979	C₂	semidirect
87. C₂×C₄×C₆₇	#108	semidirect
124. C₆₇	#20826	semidirect
205. C₂×C₄	#14523979	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22560381	2
#22560347	2
#22560247	2
C₄×#14523979	2
C₂²×C₄×C₈×C₆₇	2
#20826	67