Subgroups — #30864475

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570655	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863519	C₂	1	4	yes	maximal, complemented
3	#22568922	C₂	1	2	yes	maximal, complemented
4	#22561097	C₂	1	4	yes	maximal, complemented
5	#22561041	C₂	1	2	yes	maximal, complemented
6	#22560384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524065	C₂²	1	4	yes	—
9	C₂×#14524049	C₂²	1	4	yes	—
10	#16863946	C₂²	1	2	yes	—
11	#16863790	C₄	1	4	yes	complemented
12	#16863785	C₄	1	4	yes	complemented
13	#16863519	—	2	16	no	—
14	#16863475	—	2	4	no	—
15	C₈×D₂₆₈	—	2	4	no	—
16	C₂×#14523979	C₂²	1	4	yes	—
17	#16863420	C₂²	1	2	yes	—
18	C₂×#14523971	C₂²	1	8	yes	—
19	#16863362	C₂²	1	4	yes	—
20	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761628	C₂³	1	4	yes	—
26	#14524065	—	2	8	no	—
27	#14524065	—	2	8	no	—
28	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
29	C₂×#11761617	C₂×C₄	1	8	yes	—
30	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
31	#14524049	—	2	16	no	—
32	C₂×#11761612	C₂³	1	4	yes	—
33	C₂×#11761611	—	2	4	no	—
34	C₂×C₈×D₆₇	—	2	4	no	—
35	#14524043	C₂×C₄	1	2	yes	—
36	#14524040	C₂×C₄	1	2	yes	—
37	#14524031	C₂×C₄	1	4	yes	—
38	#14524017	—	2	4	no	—
39	C₄×D₂₆₈	—	2	4	no	—
40	#14523979	—	2	8	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
42	#14523971	—	2	16	no	—
43	(C₂×D₆₇)⋊C₈	—	2	8	no	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#97×C₆₇	C₂×C₄	1	4	yes	—
50	#97×C₆₇	D₄	1	8	yes	complemented
51	C₄×C₈×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	D₄	1	4	yes	—
53	#95×C₆₇	—	2	2	no	—
54	#11761628	—	2	8	no	—
55	#11761628	—	2	8	no	—
56	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
57	C₂×D₂₆₈	—	2	4	no	—
58	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
59	C₂×C₂²⋊D₆₇	—	2	4	no	—
60	C₂×C₄×D₆₇	—	2	4	no	—
61	Dic₂₆₈	—	2	8	no	—
62	#11761617	—	2	16	no	—
63	D₅₃₆	—	2	8	no	—
64	#11761612	—	2	8	no	—
65	#11761611	—	4	8	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₈×D₆₇	—	4	8	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	(C₂×D₆₇)⋊C₄	—	2	8	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	#41	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₆₇	#41	1	4	yes	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₆₇	C₂×D₄	1	4	yes	—
83	C₄²×C₆₇	C₂×D₄	1	2	yes	—
84	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
85	C₂²×D₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
94	C₂²⋊D₆₇	—	4	8	no	—
95	C₂×Dic₆₇	—	2	4	no	—
96	D₂₆₈	—	2	8	no	—
97	D₂₆₈	—	4	8	no	—
98	C₄×D₆₇	—	4	4	no	—
99	C₄×D₆₇	—	4	4	no	—
100	Dic₁₃₄	—	2	8	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	C₄×D₄	1	8	yes	—
103	C₈×C₆₇	#127	1	4	yes	—
104	C₆₇⋊C₈	—	4	4	no	—
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂²×C₆₇	#532	1	2	yes	—
108	C₂×D₆₇	—	4	4	no	—
109	C₂×D₆₇	—	4	8	no	—
110	C₄×C₆₇	—	2	4	no	—
111	C₄×C₆₇	—	4	4	no	—
112	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
113	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
114	C₄×C₆₇	#532	1	1	yes	characteristic
115	Dic₆₇	—	4	4	no	—
116	#21709	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
119	C₂×C₆₇	#2955	1	2	yes	—
120	D₆₇	—	8	4	no	—
121	#3150	—	67	1	no	—
122	C₂×#444	—	67	4	no	—
123	#2931	—	67	2	no	—
124	#1974	—	67	4	no	—
125	#1963	—	67	2	no	—
126	#1849	—	67	1	no	—
127	#1726	D₆₇	1	1	yes	characteristic, complemented
128	C₆₇	#21709	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	67	4	no	—
130	C₂×#131	—	67	4	no	—
131	#532	—	67	2	no	—
132	#497	—	67	4	no	—
133	#496	—	67	4	no	—
134	#444	—	134	16	no	—
135	#436	—	134	4	no	—
136	C₈×D₄	—	134	4	no	—
137	C₂×#108	—	67	4	no	—
138	#423	—	67	2	no	—
139	C₂×#104	—	67	8	no	—
140	#408	—	67	4	no	—
141	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
142	C₂×#97	C₂×D₆₇	1	2	yes	—
143	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
144	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
145	#376	C₂×D₆₇	1	1	yes	characteristic
146	C₂×#41	—	67	4	no	—
147	#135	—	134	8	no	—
148	#135	—	134	8	no	—
149	C₂×Q₁₆	—	67	4	no	—
150	C₂×SD₁₆	—	67	8	no	—
151	C₂×D₈	—	67	4	no	—
152	#131	—	134	16	no	—
153	C₂×M₁₆	—	67	4	no	—
154	C₂×M₁₆	—	134	4	no	—
155	C₂²×C₈	—	134	4	no	—
156	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
158	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₄⋊Q₈	—	67	2	no	—
160	#127	—	67	2	no	—
161	#124	—	67	4	no	—
162	#121	—	134	4	no	—
163	C₄×D₄	—	134	4	no	—
164	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	#108	—	134	8	no	—
167	C₄⋊C₈	—	134	4	no	—
168	#104	—	134	16	no	—
169	C₂²⋊C₈	—	134	8	no	—
170	#97	C₄×D₆₇	1	4	yes	—
171	#97	D₂₆₈	1	8	yes	complemented
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	D₂₆₈	1	4	yes	—
174	#95	—	2	2	no	—
175	#41	—	134	8	no	—
176	#41	—	134	8	no	—
177	C₂×Q₈	—	67	4	no	—
178	C₂×D₄	—	67	4	no	—
179	C₂×D₄	—	134	4	no	—
180	C₂×D₄	—	134	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	134	4	no	—
183	C₂²×C₄	#11761628	1	2	yes	—
184	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
185	Q₁₆	—	134	8	no	—
186	SD₁₆	—	134	16	no	—
187	D₈	—	134	8	no	—
188	M₁₆	—	268	8	no	—
189	M₁₆	—	134	8	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	268	8	no	—
192	C₂×C₈	—	134	4	no	—
193	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
194	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
195	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
196	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
197	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
198	C₂×C₈	#11761628	1	4	yes	—
199	C₄⋊C₄	—	134	4	no	—
200	C₂²⋊C₄	—	134	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₂×C₄×D₆₇	1	2	yes	—
203	C₄²	C₂×D₂₆₈	1	4	yes	—
204	C₄²	C₂×D₂₆₈	1	2	yes	—
205	C₂³	—	134	4	no	—
206	C₂³	C₂×#11761628	1	1	yes	characteristic
207	Q₈	—	134	8	no	—
208	D₄	—	268	8	no	—
209	D₄	—	268	8	no	—
210	D₄	—	134	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	268	4	no	—
215	C₂×C₄	—	268	4	no	—
216	C₂×C₄	—	134	4	no	—
217	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
218	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
220	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
221	C₂×C₄	C₂×#11761628	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	268	4	no	—
224	C₈	C₄×D₂₆₈	1	8	yes	—
225	C₈	#14524040	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	268	8	no	—
228	C₂²	—	268	4	no	—
229	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
230	C₂²	#16863946	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	4	no	—
233	C₄	—	268	4	no	—
234	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
235	C₄	C₂×#14524040	1	1	yes	characteristic
236	C₄	#16863946	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	536	4	no	—
239	C₂	C₄×#14524040	1	1	yes	characteristic
240	C₂	#22569113	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570655	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21709
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#16863519	C₂	semidirect
3. #22568922	C₂	semidirect
4. #22561097	C₂	semidirect
5. #22561041	C₂	semidirect
6. #22560384	C₂	semidirect
7. C₆₇×#1726	C₂	semidirect
11. #16863790	C₄	semidirect
12. #16863785	C₄	semidirect
50. #97×C₆₇	D₄	semidirect
127. #1726	D₆₇	semidirect
128. C₆₇	#21709	semidirect
171. #97	D₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22570655	2
C₂×#16863519	2
#22568922	2
#22561097	2
#22561041	2
#22560384	2
C₆₇×#1726	2
#21709	67