Subgroups — #30856464

Order 17152.

718 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	#22568847	C₂	1	2	yes	maximal
2	#22560384	C₂	1	2	yes	maximal, complemented
3	#22560347	C₂	1	2	yes	maximal, complemented
4	#22560251	C₂	1	2	yes	maximal
5	#22560247	C₂	1	2	yes	maximal
6	#22560115	C₂	1	2	yes	maximal, complemented
7	C₄×#14523979	C₂	1	2	yes	maximal, complemented
8	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863457	—	2	8	no	—
10	C₂×#14523979	C₂²	1	4	yes	—
11	C₂×#14523979	C₂²	1	4	yes	—
12	#16863420	C₂²	1	4	yes	—
13	#16863360	C₂²	1	4	yes	—
14	C₄×#11761612	C₂²	1	4	yes	—
15	#16863161	C₂²	1	4	yes	—
16	#16863161	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	#405×C₆₇	C₄	1	8	yes	—
21	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	C₂³	1	8	yes	—
26	#14523979	—	2	8	no	—
27	#14523979	—	2	8	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
30	#14523954	—	2	4	no	—
31	C₄×C₆₇⋊C₈	—	2	4	no	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₇	—	2	4	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#117×C₆₇	C₂×C₄	1	4	yes	—
42	#117×C₆₇	D₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
44	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#97×C₆₇	C₂×C₄	1	4	yes	—
48	#97×C₆₇	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	#11761612	—	2	8	no	—
52	#11761612	—	2	8	no	—
53	C₂×C₆₇⋊C₈	—	2	8	no	—
54	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₆₇	—	2	8	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	#41	1	4	yes	—
72	C₂×C₈×C₆₇	#41	1	4	yes	—
73	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
74	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	C₂×D₄	1	2	yes	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	4	2	no	—
81	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
96	C₈×C₆₇	—	2	2	no	—
97	C₈×C₆₇	—	2	2	no	—
98	C₈×C₆₇	#117	1	4	yes	—
99	C₈×C₆₇	#117	1	4	yes	—
100	C₆₇⋊C₈	—	4	8	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	4	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	4	4	no	—
105	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₆₇	#533	1	2	yes	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	2	4	no	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
111	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
112	C₄×C₆₇	#532	1	2	yes	—
113	#20829	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	2	no	—
115	C₂×C₆₇	—	4	2	no	—
116	C₂×C₆₇	#2285	1	1	yes	characteristic
117	C₂×C₆₇	#2916	1	2	yes	—
118	#2921	—	67	2	no	—
119	C₂×#405	D₆₇	1	1	yes	characteristic
120	#1849	—	67	2	no	—
121	#1837	—	67	2	no	—
122	#1811	—	67	2	no	—
123	#1810	—	67	2	no	—
124	#1778	—	67	2	no	—
125	C₄×#108	—	67	2	no	—
126	C₆₇	#20829	1	1	yes	characteristic, complemented, Sylow
127	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
128	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
130	#433	—	134	8	no	—
131	C₂×#108	—	67	4	no	—
132	C₂×#108	—	67	4	no	—
133	#423	—	67	4	no	—
134	#407	—	67	4	no	—
135	#405	Dic₆₇	1	8	yes	—
136	C₄×M₁₆	—	67	4	no	—
137	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
138	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#341	—	67	4	no	—
142	#341	—	67	4	no	—
143	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	67	8	no	—
146	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
149	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
153	#117	D₂₆₈	1	4	yes	—
154	#117	C₂×Dic₆₇	1	4	yes	—
155	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
156	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
157	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
159	#108	—	134	8	no	—
160	#108	—	134	8	no	—
161	C₄⋊C₈	—	134	8	no	—
162	C₂²⋊C₈	—	134	8	no	—
163	#97	—	134	4	no	—
164	#97	C₂×Dic₆₇	1	4	yes	—
165	#97	C₂×Dic₆₇	1	4	yes	—
166	C₄×C₈	—	134	4	no	—
167	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
168	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
169	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
173	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
176	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
177	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	134	8	no	—
180	M₁₆	—	134	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	134	8	no	—
185	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
186	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
187	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
188	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
189	C₂×C₈	#11761628	1	4	yes	—
190	C₂×C₈	#11761628	1	4	yes	—
191	C₄⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
192	C₂²⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
193	C₄²	C₂²×Dic₆₇	1	2	yes	—
194	C₄²	C₂²×Dic₆₇	1	2	yes	—
195	C₄²	C₂×D₂₆₈	1	2	yes	—
196	C₄²	C₂×D₂₆₈	1	2	yes	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
200	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
210	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
212	C₂×C₄	C₂×#11761628	1	2	yes	—
213	C₂×C₄	C₂×#11761628	1	2	yes	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	268	8	no	—
217	C₈	#14524002	1	4	yes	—
218	C₈	#14524002	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
224	C₂²	#16863953	1	2	yes	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	4	no	—
228	C₄	C₂×#14524002	1	1	yes	characteristic
229	C₄	C₂×#14524002	1	1	yes	characteristic
230	C₄	#16863946	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	#22562616	1	1	yes	characteristic
234	C₂	#22568820	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#20829
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22560384	C₂	semidirect
3. #22560347	C₂	semidirect
6. #22560115	C₂	semidirect
7. C₄×#14523979	C₂	semidirect
126. C₆₇	#20829	semidirect

Maximal subgroups

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

subgroup	index
#22568847	2
#22560384	2
#22560347	2
#22560251	2
#22560247	2
#22560115	2
C₄×#14523979	2
C₂×#405×C₆₇	2
#20829	67