Subgroups — #31008868

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523982	C₂	1	1	yes	maximal, characteristic
2	#427×D₆₇	C₂	1	2	yes	maximal
3	#22561996	C₂	1	4	yes	maximal, complemented
4	#22561994	C₂	1	4	yes	maximal, complemented
5	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561991	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	#117×D₆₇	—	2	2	no	—
10	C₂×#14523982	C₄	1	8	yes	complemented
11	C₂×#14523982	C₂²	1	2	yes	—
12	C₂×#14523982	C₂²	1	2	yes	—
13	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#16863434	C₂²	1	2	yes	—
15	#16863433	C₂²	1	4	yes	—
16	#16863432	—	2	2	no	—
17	#16863431	—	2	2	no	—
18	#107×D₆₇	C₂²	1	4	yes	—
19	#106×D₆₇	—	2	2	no	—
20	#16863224	C₂²	1	2	yes	complemented
21	#16863223	C₂²	1	4	yes	complemented
22	#16863222	C₂²	1	2	yes	complemented
23	#16863221	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#427×C₆₇	C₂²	1	2	yes	—
26	#367×C₆₇	C₂²	1	2	yes	complemented
27	#366×C₆₇	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₆₇	D₄	1	2	yes	—
32	C₂×C₈×D₆₇	Q₈	1	4	yes	—
33	C₂×C₈×D₆₇	C₂³	1	2	yes	—
34	#14524002	—	2	2	no	—
35	#14524001	—	2	4	no	—
36	#14524000	—	4	2	no	—
37	#14523999	—	4	2	no	—
38	C₄⋊C₄×D₆₇	—	2	4	no	—
39	C₂²⋊C₄×D₆₇	—	4	2	no	—
40	C₄²×D₆₇	—	4	2	no	—
41	#14523982	C₂×C₄	1	8	yes	complemented
42	#14523982	C₂×C₄	1	8	yes	complemented
43	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
44	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
45	#14523977	C₂³	1	4	yes	—
46	#14523976	—	2	4	no	—
47	#14523975	—	2	2	no	—
48	#14523974	—	2	4	no	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	#117×C₆₇	—	2	2	no	—
51	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	C₂³	1	4	yes	—
55	#106×C₆₇	—	2	2	no	—
56	C₂³×D₆₇	D₈	1	1	yes	characteristic
57	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
58	C₂×C₄×D₆₇	—	4	4	no	—
59	C₂×C₄×D₆₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×D₆₇	D₈	1	2	yes	—
61	C₂×C₄×D₆₇	Q₁₆	1	4	yes	—
62	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
63	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
64	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
65	C₈×D₆₇	C₄⋊C₄	1	8	yes	—
66	C₈×D₆₇	C₂²×C₄	1	4	yes	—
67	C₈×D₆₇	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	Dic₆₇⋊C₄	—	4	4	no	—
70	C₂²⋊Dic₆₇	—	4	2	no	—
71	(C₂×D₆₇)⋊C₄	—	4	4	no	—
72	C₄×Dic₆₇	—	4	2	no	—
73	C₄×Dic₆₇	—	4	4	no	—
74	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₂²⋊C₄×C₆₇	—	4	2	no	—
82	C₄²×C₆₇	—	4	2	no	—
83	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
84	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₆₇	—	4	2	no	—
87	C₂²×D₆₇	#107	1	4	yes	—
88	C₂²×D₆₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₆₇	—	4	4	no	—
95	C₂×Dic₆₇	#107	1	4	yes	—
96	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
97	C₄×D₆₇	—	8	8	no	—
98	C₄×D₆₇	#107	1	8	yes	—
99	C₄×D₆₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₆₇	C₂×Q₁₆	1	4	yes	—
101	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
104	C₆₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	C₂×#107	1	2	yes	—
107	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
108	C₂×D₆₇	—	2	4	no	—
109	C₂×D₆₇	—	4	4	no	—
110	C₂×D₆₇	C₂×#107	1	4	yes	—
111	C₄×C₆₇	—	8	4	no	—
112	C₄×C₆₇	C₂×#107	1	2	yes	—
113	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
115	Dic₆₇	—	8	4	no	—
116	Dic₆₇	C₂×#107	1	4	yes	—
117	#33409	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	—	2	2	no	—
119	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
120	C₂×C₆₇	#3414	1	2	yes	—
121	D₆₇	—	2	4	no	—
122	C₂²×M₃₂	—	67	1	no	—
123	C₂×#427	—	67	2	no	—
124	#2159	—	67	4	no	—
125	#2159	—	67	4	no	—
126	#2158	—	67	1	no	—
127	#2158	—	67	2	no	—
128	#2158	D₆₇	1	1	yes	characteristic, complemented
129	C₆₇	#33409	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₈	—	67	1	no	—
131	C₂×#117	—	134	2	no	—
132	C₂×M₃₂	—	67	8	no	—
133	C₂×M₃₂	—	67	2	no	—
134	C₂×M₃₂	—	67	2	no	—
135	C₂²×C₁₆	—	67	1	no	—
136	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
137	#427	—	67	2	no	—
138	#427	—	67	4	no	—
139	#427	—	134	2	no	—
140	#427	—	134	2	no	—
141	#427	C₂×D₆₇	1	2	yes	—
142	C₂×#107	—	67	4	no	—
143	C₂×#106	—	134	2	no	—
144	#367	—	67	2	no	—
145	#367	—	67	4	no	—
146	#367	C₂×D₆₇	1	2	yes	complemented
147	#366	—	67	2	no	—
148	#366	—	67	4	no	—
149	#366	C₂×D₆₇	1	2	yes	complemented
150	C₂³×C₄	—	67	1	no	—
151	C₂²×C₈	—	67	1	no	—
152	C₂²×C₈	—	67	4	no	—
153	C₂²×C₈	—	67	2	no	—
154	C₂²×C₈	—	67	4	no	—
155	C₂²×C₈	—	67	2	no	—
156	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	268	2	no	—
159	#117	—	268	2	no	—
160	#117	—	134	4	no	—
161	#117	—	134	2	no	—
162	C₂×C₄⋊C₄	—	134	4	no	—
163	C₂×C₂²⋊C₄	—	268	2	no	—
164	C₂×C₄²	—	268	2	no	—
165	M₃₂	—	67	8	no	—
166	M₃₂	—	67	8	no	—
167	C₂×C₁₆	—	67	4	no	—
168	C₂×C₁₆	—	67	2	no	—
169	C₂×C₁₆	C₄×D₆₇	1	4	yes	complemented
170	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
172	#107	—	67	4	no	—
173	#107	—	134	4	no	—
174	#107	C₂²×D₆₇	1	4	yes	—
175	#106	—	2	2	no	—
176	#106	—	134	4	no	—
177	#106	—	134	2	no	—
178	C₂⁴	—	67	1	no	—
179	C₂²×C₄	—	268	4	no	—
180	C₂²×C₄	—	67	1	no	—
181	C₂²×C₄	—	67	4	no	—
182	C₂²×C₄	—	67	2	no	—
183	C₂²×C₄	—	67	4	no	—
184	C₂²×C₄	—	67	2	no	—
185	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
186	C₂×C₈	—	67	4	no	—
187	C₂×C₈	—	67	2	no	—
188	C₂×C₈	—	67	8	no	—
189	C₂×C₈	—	67	4	no	—
190	C₂×C₈	—	67	4	no	—
191	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
192	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
193	C₂×C₈	Q₈×D₆₇	1	2	yes	—
194	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	268	4	no	—
197	C₄⋊C₄	—	134	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	268	4	no	—
200	C₂²⋊C₄	—	268	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	268	4	no	—
203	C₄²	—	268	2	no	—
204	C₁₆	—	67	4	no	—
205	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
206	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
207	C₂³	—	268	2	no	—
208	C₂³	—	67	4	no	—
209	C₂³	—	67	2	no	—
210	C₂³	D₈×D₆₇	1	1	yes	characteristic
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	268	4	no	—
213	C₂×C₄	—	536	8	no	—
214	C₂×C₄	—	67	2	no	—
215	C₂×C₄	—	67	4	no	—
216	C₂×C₄	—	67	4	no	—
217	C₂×C₄	—	67	4	no	—
218	C₂×C₄	—	67	8	no	—
219	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
220	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
221	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
222	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
223	C₈	—	67	4	no	—
224	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
225	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
226	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	268	4	no	—
229	C₂²	—	67	4	no	—
230	C₂²	—	134	4	no	—
231	C₂²	#107×D₆₇	1	2	yes	—
232	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	536	4	no	—
235	C₄	—	67	4	no	—
236	C₄	#107×D₆₇	1	2	yes	—
237	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
238	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	134	4	no	—
241	C₂	C₂×#107×D₆₇	1	1	yes	characteristic
242	C₂	#22573342	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2158	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×D₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#33409
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #22561996	C₂	semidirect
4. #22561994	C₂	semidirect
5. #22561993	C₂	semidirect
6. #22561991	C₂	semidirect
7. C₆₇×#2158	C₂	semidirect
10. C₂×#14523982	C₄	semidirect
20. #16863224	C₂²	semidirect
21. #16863223	C₂²	semidirect
22. #16863222	C₂²	semidirect
23. #16863221	C₂²	semidirect
26. #367×C₆₇	C₂²	semidirect
27. #366×C₆₇	C₂²	semidirect
41. #14523982	C₂×C₄	semidirect
42. #14523982	C₂×C₄	semidirect
43. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
51. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
128. #2158	D₆₇	semidirect
129. C₆₇	#33409	semidirect
146. #367	C₂×D₆₇	semidirect
149. #366	C₂×D₆₇	semidirect
169. C₂×C₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523982	2
#427×D₆₇	2
#22561996	2
#22561994	2
#22561993	2
#22561991	2
C₆₇×#2158	2
#33409	67