Subgroups — #30770710

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal, complemented
4	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
7	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863434	—	2	2	no	—
11	#16863434	C₄	1	4	yes	complemented
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523975	—	2	1	no	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	—	2	1	no	—
16	#427×C₆₇	—	2	2	no	—
17	#427×C₆₇	C₄	1	4	yes	complemented
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#106×C₆₇	—	2	1	no	—
20	#14524002	—	2	4	no	—
21	#14524002	—	4	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
24	C₂×C₄×Dic₆₇	—	4	1	no	—
25	C₂×C₆₇⋊C₁₆	—	2	8	no	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₆₇⋊C₁₆	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
29	#14523977	—	2	2	no	—
30	#14523977	C₂×C₄	1	4	yes	—
31	#14523975	—	2	4	no	—
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	D₄	1	2	yes	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	2	yes	—
39	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#117×C₆₇	—	2	4	no	—
42	#117×C₆₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
45	C₂×C₄²×C₆₇	—	4	1	no	—
46	#107×C₆₇	—	2	2	no	—
47	#107×C₆₇	C₂×C₄	1	4	yes	—
48	#106×C₆₇	—	2	4	no	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	2	4	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₂²⋊Dic₆₇	—	4	4	no	—
53	C₄×Dic₆₇	—	4	4	no	—
54	C₆₇⋊C₁₆	—	2	4	no	—
55	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₄²	1	4	yes	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₂²⋊C₄×C₆₇	—	4	4	no	—
81	C₄²×C₆₇	—	4	4	no	—
82	C₂³×C₆₇	—	2	4	no	—
83	C₂³×C₆₇	#102	1	2	yes	—
84	C₂³×C₆₇	#102	1	2	yes	—
85	C₂³×C₆₇	#107	1	2	yes	—
86	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	8	4	no	—
89	C₂×C₄×C₆₇	#95	1	4	yes	—
90	C₂×C₄×C₆₇	#102	1	2	yes	—
91	C₂×C₄×C₆₇	#102	1	2	yes	—
92	C₂×C₄×C₆₇	#103	1	4	yes	—
93	C₂×C₄×C₆₇	#103	1	4	yes	—
94	C₂×C₄×C₆₇	#106	1	4	yes	—
95	C₂×C₄×C₆₇	#107	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×Dic₆₇	—	4	4	no	—
103	C₂×Dic₆₇	—	8	4	no	—
104	C₈×C₆₇	#95	1	4	yes	—
105	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
106	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	4	no	—
111	C₂²×C₆₇	—	2	4	no	—
112	C₂²×C₆₇	#340	1	4	yes	—
113	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
116	C₂²×C₆₇	#508	1	4	yes	—
117	C₄×C₆₇	—	8	4	no	—
118	C₄×C₆₇	#340	1	4	yes	—
119	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
120	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
121	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
123	Dic₆₇	—	8	4	no	—
124	#12556	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	4	no	—
126	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
127	C₂×C₆₇	#2138	1	2	yes	—
128	C₂×C₆₇	#2138	1	2	yes	—
129	C₂×C₆₇	#2158	1	2	yes	—
130	C₂³×C₁₆	—	67	1	no	—
131	C₂×#427	—	67	1	no	—
132	C₂×#427	D₆₇	1	1	yes	characteristic
133	#1380	—	67	4	no	—
134	C₆₇	#12556	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
136	C₂×#117	—	2	1	no	—
137	C₂×#117	—	134	1	no	—
138	C₂²×C₁₆	—	67	4	no	—
139	C₂²×C₁₆	—	67	2	no	—
140	C₂²×C₁₆	—	134	2	no	—
141	C₂²×C₁₆	—	134	2	no	—
142	#427	—	2	2	no	—
143	#427	—	67	4	no	—
144	#427	—	134	2	no	—
145	#427	Dic₆₇	1	4	yes	complemented
146	C₂×#107	—	67	2	no	—
147	C₂×#107	C₂×D₆₇	1	2	yes	—
148	C₂×#106	—	2	1	no	—
149	C₂×#106	—	134	1	no	—
150	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₃₄	1	2	yes	—
152	C₂²×C₈	C₄×D₆₇	1	2	yes	—
153	C₂²×C₈	C₄×D₆₇	1	2	yes	—
154	C₂²×C₈	D₂₆₈	1	2	yes	—
155	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
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	#117	—	2	4	no	—
160	#117	—	4	2	no	—
161	#117	—	268	2	no	—
162	#117	—	134	4	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	134	2	no	—
165	C₂×C₂²⋊C₄	—	4	1	no	—
166	C₂×C₂²⋊C₄	—	268	1	no	—
167	C₂×C₄²	—	4	1	no	—
168	C₂×C₄²	—	268	1	no	—
169	C₂×C₁₆	—	67	4	no	—
170	C₂×C₁₆	—	134	8	no	—
171	C₂×C₁₆	—	134	2	no	—
172	C₂×C₁₆	—	134	2	no	—
173	#107	—	2	2	no	—
174	#107	—	67	4	no	—
175	#107	—	134	2	no	—
176	#107	C₂×Dic₆₇	1	4	yes	—
177	#106	—	2	4	no	—
178	#106	—	134	4	no	—
179	C₂⁴	#11761613	1	1	yes	characteristic
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	268	2	no	—
182	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
184	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
185	C₂²×C₄	#11761613	1	1	yes	characteristic
186	C₂²×C₄	#11761615	1	2	yes	—
187	C₂²×C₄	#11761616	1	2	yes	—
188	C₂²×C₄	#11761618	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
191	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
192	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
193	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
194	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
195	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
197	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
202	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	268	2	no	—
206	C₄⋊C₄	—	134	4	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	268	4	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	268	4	no	—
211	C₁₆	—	134	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	#14523963	1	2	yes	—
214	C₂³	#14523964	1	2	yes	—
215	C₂³	#14523976	1	2	yes	—
216	C₂³	C₂×#11761613	1	1	yes	characteristic
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	268	4	no	—
220	C₂×C₄	—	536	4	no	—
221	C₂×C₄	C₆₇⋊#95	1	4	yes	—
222	C₂×C₄	#14523963	1	2	yes	—
223	C₂×C₄	#14523964	1	2	yes	—
224	C₂×C₄	#14523966	1	4	yes	—
225	C₂×C₄	#14523967	1	4	yes	—
226	C₂×C₄	#14523974	1	4	yes	—
227	C₂×C₄	#14523976	1	2	yes	—
228	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
234	C₈	C₆₇⋊#95	1	4	yes	—
235	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
236	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
237	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
238	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	#16863158	1	4	yes	—
243	C₂²	C₂×#14523963	1	1	yes	characteristic
244	C₂²	C₂×#14523964	1	1	yes	characteristic
245	C₂²	C₂×#14523976	1	1	yes	characteristic
246	C₂²	#16863826	1	4	yes	—
247	C₄	—	8	4	no	—
248	C₄	—	536	4	no	—
249	C₄	#16863158	1	4	yes	—
250	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
251	C₄	C₂×#14523966	1	1	yes	characteristic
252	C₄	C₂×#14523967	1	1	yes	characteristic
253	C₄	C₂×#14523974	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	C₂×#16863158	1	1	yes	characteristic
256	C₂	#22561914	1	2	yes	—
257	C₂	#22561916	1	2	yes	—
258	C₂	#22561991	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11761613
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#12556
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #22558300	C₂	semidirect
8. C₂²×C₆₇⋊C₁₆	C₄	semidirect
11. #16863434	C₄	semidirect
17. #427×C₆₇	C₄	semidirect
134. C₆₇	#12556	semidirect
145. #427	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂×#16863434	2
#22558300	2
C₂×#427×C₆₇	2
#12556	67