Subgroups — #30774055

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863200	C₂	1	2	yes	maximal, complemented
2	C₂×#16863197	C₂	1	2	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14523977	C₂²	1	2	yes	—
8	C₂×#14523975	—	2	1	no	—
9	#16863262	—	2	1	no	—
10	#16863200	—	2	4	no	—
11	#16863197	—	2	4	no	—
12	C₄×C₆₇⋊C₁₆	—	2	2	no	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	—	2	1	no	—
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	—	2	1	no	—
17	#495×C₆₇	—	2	2	no	—
18	#495×C₆₇	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
24	#14523977	—	2	2	no	—
25	#14523977	—	2	2	no	—
26	#14523975	—	4	2	no	—
27	C₆₇⋊#95	—	4	2	no	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₂²×D₄×C₆₇	—	2	1	no	—
30	C₂×Q₁₆×C₆₇	—	2	2	no	—
31	C₂×Q₁₆×C₆₇	—	2	1	no	—
32	C₂×Q₁₆×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×SD₁₆×C₆₇	—	2	4	no	—
35	C₂×SD₁₆×C₆₇	—	4	2	no	—
36	C₂×SD₁₆×C₆₇	—	4	2	no	—
37	C₂×D₈×C₆₇	—	2	2	no	—
38	C₂×D₈×C₆₇	—	2	1	no	—
39	C₂×D₈×C₆₇	—	2	2	no	—
40	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#124×C₆₇	—	2	4	no	—
44	#124×C₆₇	—	4	2	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	2	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₆₇⋊C₁₆	—	2	4	no	—
53	C₂⁴×C₆₇	—	4	1	no	—
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	—	4	1	no	—
56	C₂×Q₈×C₆₇	—	4	1	no	—
57	C₂×Q₈×C₆₇	—	4	1	no	—
58	C₂×Q₈×C₆₇	—	4	1	no	—
59	C₂×Q₈×C₆₇	—	4	1	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	4	1	no	—
62	C₂×D₄×C₆₇	—	4	1	no	—
63	C₂×D₄×C₆₇	—	4	1	no	—
64	C₂×D₄×C₆₇	—	4	1	no	—
65	C₂×D₄×C₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	—	4	1	no	—
67	C₂²×C₄×C₆₇	D₈	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	Q₁₆×C₆₇	—	2	4	no	—
70	Q₁₆×C₆₇	—	4	2	no	—
71	SD₁₆×C₆₇	—	4	8	no	—
72	D₈×C₆₇	—	2	4	no	—
73	D₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	D₈	1	2	yes	—
76	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	#41	1	2	yes	—
81	C₂²⋊C₄×C₆₇	—	4	8	no	—
82	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₆₇	SD₁₆	1	2	yes	—
84	C₂³×C₆₇	—	4	2	no	—
85	C₂³×C₆₇	—	8	1	no	—
86	C₂³×C₆₇	—	8	1	no	—
87	C₂³×C₆₇	—	8	4	no	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₆₇	—	4	2	no	—
90	Q₈×C₆₇	—	4	2	no	—
91	Q₈×C₆₇	—	8	1	no	—
92	Q₈×C₆₇	—	8	1	no	—
93	D₄×C₆₇	—	4	2	no	—
94	D₄×C₆₇	—	4	2	no	—
95	D₄×C₆₇	—	8	1	no	—
96	D₄×C₆₇	—	8	1	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	8	1	no	—
100	C₂×C₄×C₆₇	—	8	1	no	—
101	C₂×C₄×C₆₇	#102	1	4	yes	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×Dic₆₇	—	8	2	no	—
109	C₂×Dic₆₇	—	8	4	no	—
110	C₈×C₆₇	#102	1	4	yes	—
111	C₈×C₆₇	C₄×D₄	1	2	yes	—
112	C₈×C₆₇	#121	1	2	yes	—
113	C₂²×C₆₇	—	8	4	no	—
114	C₂²×C₆₇	—	8	2	no	—
115	C₂²×C₆₇	—	16	2	no	—
116	C₂²×C₆₇	—	16	2	no	—
117	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
118	C₂²×C₆₇	#486	1	1	yes	characteristic
119	C₂²×C₆₇	#493	1	1	yes	characteristic
120	C₂²×C₆₇	#508	1	4	yes	—
121	C₄×C₆₇	—	2	4	no	—
122	C₄×C₆₇	—	8	2	no	—
123	C₄×C₆₇	#390	1	1	yes	characteristic
124	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
125	C₄×C₆₇	#486	1	1	yes	characteristic
126	C₄×C₆₇	#492	1	1	yes	characteristic
127	Dic₆₇	—	16	2	no	—
128	#13204	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	16	2	no	—
130	C₂×C₆₇	#1957	1	1	yes	characteristic
131	C₂×C₆₇	#2138	1	2	yes	—
132	C₂×C₆₇	#2242	1	2	yes	—
133	C₂×C₆₇	#2249	1	2	yes	—
134	C₂×#495	D₆₇	1	1	yes	characteristic
135	C₂×#358	—	67	2	no	—
136	C₂×#357	—	67	2	no	—
137	C₂×C₄×C₁₆	—	67	1	no	—
138	#1846	—	67	1	no	—
139	C₆₇	#13204	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
141	C₂²×SD₁₆	—	2	1	no	—
142	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
143	C₂×#124	—	2	1	no	—
144	C₂²×C₁₆	—	67	2	no	—
145	#495	—	2	2	no	—
146	#495	Dic₆₇	1	4	yes	complemented
147	C₂×#107	—	67	2	no	—
148	C₂×#106	—	134	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#382	—	134	1	no	—
151	#358	—	134	4	no	—
152	#357	—	134	4	no	—
153	C₄×C₁₆	—	134	2	no	—
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×D₄	—	2	1	no	—
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×Q₁₆	—	2	1	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	4	2	no	—
162	C₂×SD₁₆	—	4	2	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
167	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	#124	—	2	4	no	—
170	#124	—	4	2	no	—
171	C₂×C₄⋊C₄	—	134	2	no	—
172	C₂×C₂²⋊C₄	—	4	2	no	—
173	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂×C₁₆	—	67	4	no	—
175	C₂×C₁₆	—	134	2	no	—
176	C₂×C₁₆	—	134	2	no	—
177	#107	—	134	2	no	—
178	#107	—	134	2	no	—
179	#106	—	268	2	no	—
180	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
181	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
182	#95	—	268	2	no	—
183	C₂⁴	—	4	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	268	2	no	—
198	C₂²×C₄	#11761613	1	2	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	8	no	—
203	D₈	—	2	4	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
206	C₂×C₈	#11761613	1	2	yes	—
207	C₂×C₈	#11761616	1	2	yes	—
208	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂×C₈	D₄×D₆₇	1	2	yes	—
210	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂×C₈	#11761627	1	2	yes	—
212	C₄⋊C₄	—	268	2	no	—
213	C₄⋊C₄	—	268	2	no	—
214	C₂²⋊C₄	—	4	8	no	—
215	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
216	C₄²	#11761616	1	2	yes	—
217	C₁₆	—	134	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	8	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	C₂×#11761613	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	Q₈	—	4	2	no	—
225	Q₈	—	8	1	no	—
226	Q₈	—	8	1	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	1	no	—
230	D₄	—	8	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	536	2	no	—
236	C₂×C₄	—	536	4	no	—
237	C₂×C₄	#14523964	1	4	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₂×C₄	#14524030	1	1	yes	characteristic
240	C₂×C₄	#14524039	1	1	yes	characteristic
241	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
242	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
243	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
244	C₈	#14523964	1	4	yes	—
245	C₈	D₄×Dic₆₇	1	2	yes	—
246	C₈	#14524014	1	2	yes	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	16	2	no	—
250	C₂²	—	16	2	no	—
251	C₂²	C₂×#14523964	1	1	yes	characteristic
252	C₂²	#16863742	1	1	yes	characteristic
253	C₂²	#16863772	1	1	yes	characteristic
254	C₂²	#16863826	1	4	yes	—
255	C₄	—	2	4	no	—
256	C₄	—	8	2	no	—
257	C₄	—	1072	2	no	—
258	C₄	#16863299	1	1	yes	characteristic
259	C₄	C₂×#14523964	1	1	yes	characteristic
260	C₄	#16863742	1	1	yes	characteristic
261	C₄	#16863768	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	#22561016	1	1	yes	characteristic
264	C₂	#22561916	1	2	yes	—
265	C₂	#22562440	1	2	yes	—
266	C₂	#22562468	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#495×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₈×C₆₇	C₂²
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#495	D₆₇
hypocenter	C₆₇	#13204
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#16863200	C₂	semidirect
3. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
4. #22560373	C₂	semidirect
18. #495×C₆₇	C₄	semidirect
139. C₆₇	#13204	semidirect
146. #495	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863200	2
C₂×#16863197	2
C₂×C₄×C₆₇⋊C₁₆	2
#22560373	2
C₂×#495×C₆₇	2
#13204	67