Subgroups — #30772330

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863221	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22560238	C₂	1	1	yes	maximal, characteristic
4	#22558300	C₂	1	2	yes	maximal, complemented
5	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14523977	C₂²	1	1	yes	characteristic
8	C₂×#14523977	C₂²	1	1	yes	characteristic
9	#16863256	—	2	1	no	—
10	#16863221	—	2	4	no	—
11	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	complemented
12	#16863159	—	2	1	no	—
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	—	2	4	no	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₆₇	—	2	1	no	—
19	C₂×C₄⋊Dic₆₇	—	2	1	no	—
20	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
24	#14523977	—	2	2	no	—
25	#14523977	—	2	2	no	—
26	C₆₇⋊#95	—	4	1	no	—
27	C₂³×C₄×C₆₇	—	4	1	no	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
39	#107×C₆₇	—	2	4	no	—
40	#107×C₆₇	—	2	2	no	—
41	#107×C₆₇	D₄	1	4	yes	—
42	C₂²×Dic₆₇	—	4	1	no	—
43	C₂²×Dic₆₇	—	4	1	no	—
44	C₄⋊Dic₆₇	—	4	1	no	—
45	C₄⋊Dic₆₇	—	4	1	no	—
46	C₄⋊Dic₆₇	—	4	1	no	—
47	C₄⋊Dic₆₇	—	4	1	no	—
48	C₂²⋊Dic₆₇	—	8	2	no	—
49	C₆₇⋊C₁₆	—	2	4	no	—
50	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	—	2	1	no	—
53	C₂²×C₄×C₆₇	—	2	1	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	4	4	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	—	8	2	no	—
58	C₂²×C₄×C₆₇	—	8	2	no	—
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	2	1	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	4	4	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₂³×C₆₇	—	2	1	no	—
76	C₂³×C₆₇	—	2	1	no	—
77	C₂³×C₆₇	—	2	1	no	—
78	C₂³×C₆₇	—	2	1	no	—
79	C₂³×C₆₇	#107	1	2	yes	—
80	C₂³×C₆₇	D₁₆	1	2	yes	—
81	C₂³×C₆₇	Q₃₂	1	2	yes	—
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	8	8	no	—
91	C₂×C₄×C₆₇	—	8	2	no	—
92	C₂×C₄×C₆₇	—	8	2	no	—
93	C₂×C₄×C₆₇	#107	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
100	C₂×Dic₆₇	—	8	1	no	—
101	C₂×Dic₆₇	—	8	1	no	—
102	C₂×Dic₆₇	—	8	1	no	—
103	C₂×Dic₆₇	—	8	1	no	—
104	C₂×Dic₆₇	—	8	1	no	—
105	C₂×Dic₆₇	—	8	1	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	C₄×D₄	1	2	yes	—
108	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	#366	1	4	yes	complemented
116	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
117	C₂²×C₆₇	#447	1	1	yes	characteristic
118	C₂²×C₆₇	#480	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₆₇	#509	1	1	yes	characteristic
122	C₂²×C₆₇	#510	1	1	yes	characteristic
123	C₄×C₆₇	—	2	2	no	—
124	C₄×C₆₇	—	8	4	no	—
125	C₄×C₆₇	#385	1	1	yes	characteristic
126	C₄×C₆₇	#428	1	1	yes	characteristic
127	C₄×C₆₇	#451	1	1	yes	characteristic
128	C₄×C₆₇	#484	1	1	yes	characteristic
129	Dic₆₇	—	16	1	no	—
130	Dic₆₇	—	16	1	no	—
131	#12919	—	67	1	no	maximal, Sylow
132	C₂×C₆₇	—	2	4	no	—
133	C₂×C₆₇	#1852	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
135	C₂×C₆₇	#2159	1	1	yes	characteristic
136	C₂×C₆₇	#2184	1	1	yes	characteristic
137	C₂×C₆₇	#2188	1	1	yes	characteristic
138	C₂×C₆₇	#2232	1	1	yes	characteristic
139	C₂×C₆₇	#2236	1	1	yes	characteristic
140	C₂²×#107	D₆₇	1	1	yes	characteristic
141	C₂×#366	—	67	2	no	—
142	C₂×C₂²⋊C₁₆	—	67	1	no	—
143	#1808	—	67	1	no	—
144	#1380	—	67	2	no	—
145	C₆₇	#12919	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
147	C₂²×C₄⋊C₄	—	2	1	no	—
148	C₂²×C₁₆	—	67	2	no	—
149	C₂×#107	—	2	1	no	—
150	C₂×#107	—	2	4	no	—
151	C₂×#107	—	67	1	no	—
152	C₂×#107	—	67	1	no	—
153	C₂×#107	C₂×D₆₇	1	2	yes	—
154	#380	—	134	1	no	—
155	#366	—	134	4	no	—
156	C₂²⋊C₁₆	—	67	4	no	—
157	#340	—	134	1	no	—
158	C₂³×C₄	—	4	1	no	—
159	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	Dic₁₃₄	1	2	yes	—
163	C₂²×C₈	C₄×D₆₇	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	2	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₄⋊C₄	—	134	1	no	—
171	C₂×C₄⋊C₄	—	134	1	no	—
172	C₂×C₂²⋊C₄	—	268	1	no	—
173	C₂×C₁₆	—	67	4	no	—
174	C₂×C₁₆	—	134	2	no	—
175	C₂×C₁₆	—	134	2	no	—
176	#107	—	2	4	no	—
177	#107	—	2	2	no	—
178	#107	—	134	2	no	—
179	#107	—	134	2	no	—
180	#107	C₂²⋊D₆₇	1	4	yes	—
181	#95	—	268	1	no	—
182	C₂⁴	#11761613	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	8	2	no	—
190	C₂²×C₄	—	8	2	no	—
191	C₂²×C₄	—	268	1	no	—
192	C₂²×C₄	—	268	1	no	—
193	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
194	C₂²×C₄	#11761613	1	1	yes	characteristic
195	C₂²×C₄	#11761618	1	2	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
204	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
205	C₂×C₈	#11761628	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	268	1	no	—
210	C₄⋊C₄	—	268	1	no	—
211	C₄⋊C₄	—	268	1	no	—
212	C₄⋊C₄	—	268	1	no	—
213	C₂²⋊C₄	—	536	2	no	—
214	C₁₆	—	134	4	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#14523976	1	2	yes	—
220	C₂³	#14523984	1	2	yes	—
221	C₂³	#14523989	1	2	yes	—
222	C₂³	C₂×#11761613	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	536	1	no	—
234	C₂×C₄	—	536	1	no	—
235	C₂×C₄	—	536	1	no	—
236	C₂×C₄	—	536	1	no	—
237	C₂×C₄	—	536	1	no	—
238	C₂×C₄	—	536	1	no	—
239	C₂×C₄	#14523976	1	2	yes	—
240	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
241	C₂×C₄	#14524012	1	1	yes	characteristic
242	C₂×C₄	#14524026	1	1	yes	characteristic
243	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
244	C₂×C₄	#14524069	1	1	yes	characteristic
245	C₂×C₄	#14524074	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
248	C₈	#14524022	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#16863221	1	4	yes	complemented
256	C₂²	C₂×#14523976	1	1	yes	characteristic
257	C₂²	#16863530	1	1	yes	characteristic
258	C₂²	#16863709	1	1	yes	characteristic
259	C₂²	C₂×#14523984	1	1	yes	characteristic
260	C₂²	C₂×#14523989	1	1	yes	characteristic
261	C₂²	#16863832	1	1	yes	characteristic
262	C₂²	#16863837	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	8	4	no	—
265	C₄	—	1072	1	no	—
266	C₄	—	1072	1	no	—
267	C₄	#16863269	1	1	yes	characteristic
268	C₄	#16863436	1	1	yes	characteristic
269	C₄	#16863552	1	1	yes	characteristic
270	C₄	#16863731	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#22560397	1	1	yes	characteristic
273	C₂	C₂×#16863221	1	1	yes	characteristic
274	C₂	#22561995	1	1	yes	characteristic
275	C₂	#22562121	1	1	yes	characteristic
276	C₂	#22562143	1	1	yes	characteristic
277	C₂	#22562389	1	1	yes	characteristic
278	C₂	#22562411	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	#12919
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
1. C₂×#16863221	C₂	semidirect
4. #22558300	C₂	semidirect
11. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
23. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
115. C₂²×C₆₇	#366	semidirect
145. C₆₇	#12919	semidirect
255. C₂²	#16863221	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863221	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
#22560238	2
#22558300	2
C₂²×#107×C₆₇	2
#12919	67