Subgroups — #30771882

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863197	C₂	1	2	yes	maximal
3	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558300	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×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	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	#16863197	—	2	4	no	—
12	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	—	2	1	no	—
14	C₄×D₈×C₆₇	—	2	2	no	—
15	C₄×D₈×C₆₇	C₄	1	4	yes	complemented
16	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#102×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523977	—	2	2	no	—
25	#14523975	—	4	2	no	—
26	C₆₇⋊#95	—	4	1	no	—
27	C₆₇⋊#95	—	4	1	no	—
28	C₂²×D₄×C₆₇	—	2	1	no	—
29	C₂³×C₄×C₆₇	—	4	1	no	—
30	C₂×D₈×C₆₇	—	2	1	no	—
31	C₂×D₈×C₆₇	—	2	1	no	—
32	C₂×D₈×C₆₇	—	2	1	no	—
33	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₆₇	D₄	1	2	yes	—
35	C₂×D₈×C₆₇	Q₈	1	2	yes	—
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₆₇	—	2	4	no	—
39	C₄×D₄×C₆₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	#107×C₆₇	—	2	1	no	—
44	#107×C₆₇	C₂×C₄	1	2	yes	—
45	#102×C₆₇	—	2	4	no	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	D₄	1	2	yes	—
48	C₂²×Dic₆₇	—	4	2	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	2	no	—
51	C₆₇⋊C₁₆	—	4	2	no	—
52	C₂⁴×C₆₇	—	4	1	no	—
53	C₂×D₄×C₆₇	—	2	2	no	—
54	C₂×D₄×C₆₇	—	2	2	no	—
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	—	4	1	no	—
57	C₂×D₄×C₆₇	—	4	1	no	—
58	C₂×D₄×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	8	4	no	—
62	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	D₈×C₆₇	—	2	2	no	—
66	D₈×C₆₇	—	2	2	no	—
67	D₈×C₆₇	—	2	2	no	—
68	D₈×C₆₇	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₆₇	—	2	1	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
77	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	4	1	no	—
80	C₂²⋊C₄×C₆₇	—	4	4	no	—
81	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₆₇	SD₁₆	1	2	yes	—
83	C₂³×C₆₇	—	4	2	no	—
84	C₂³×C₆₇	—	4	2	no	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	—	8	2	no	—
87	C₂³×C₆₇	—	8	2	no	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	D₄×C₆₇	—	2	4	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	4	2	no	—
92	D₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	8	8	no	—
97	C₂×C₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	#102	1	2	yes	—
99	C₂×C₄×C₆₇	#102	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
106	C₂×Dic₆₇	—	8	2	no	—
107	C₂×Dic₆₇	—	8	2	no	—
108	C₂×Dic₆₇	—	8	2	no	—
109	C₈×C₆₇	—	2	2	no	—
110	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₈×C₆₇	#117	1	1	yes	characteristic
112	C₈×C₆₇	#121	1	1	yes	characteristic
113	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂²×C₆₇	—	4	4	no	—
115	C₂²×C₆₇	—	8	2	no	—
116	C₂²×C₆₇	—	8	2	no	—
117	C₂²×C₆₇	—	8	4	no	—
118	C₂²×C₆₇	—	8	4	no	—
119	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
120	C₂²×C₆₇	#459	1	1	yes	characteristic
121	C₂²×C₆₇	#474	1	1	yes	characteristic
122	C₂²×C₆₇	#508	1	2	yes	—
123	C₂²×C₆₇	#509	1	2	yes	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	8	2	no	—
127	C₄×C₆₇	#380	1	1	yes	characteristic
128	C₄×C₆₇	#417	1	1	yes	characteristic
129	C₄×C₆₇	#461	1	1	yes	characteristic
130	C₄×C₆₇	#476	1	1	yes	characteristic
131	Dic₆₇	—	16	2	no	—
132	#12813	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	8	4	no	—
134	C₂×C₆₇	#1802	1	1	yes	characteristic
135	C₂×C₆₇	#2138	1	1	yes	characteristic
136	C₂×C₆₇	#2140	1	1	yes	characteristic
137	C₂×C₆₇	#2210	1	2	yes	—
138	C₂×C₆₇	#2228	1	2	yes	—
139	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
140	C₂×C₄⋊C₁₆	—	67	1	no	—
141	C₂×#357	—	67	2	no	—
142	#1846	—	67	1	no	—
143	#1380	—	67	2	no	—
144	C₆₇	#12813	1	1	yes	characteristic, complemented, Sylow
145	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
146	C₂×C₄×D₄	—	2	1	no	—
147	C₂²×C₁₆	—	67	2	no	—
148	C₄×D₈	—	2	2	no	—
149	C₄×D₈	Dic₆₇	1	4	yes	complemented
150	C₂×#107	—	67	2	no	—
151	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
152	C₂×#106	—	134	1	no	—
153	C₂×#102	—	2	1	no	—
154	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
155	#382	—	134	1	no	—
156	C₄⋊C₁₆	—	134	2	no	—
157	#357	—	134	4	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂³×C₄	—	4	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	Dic₁₃₄	1	2	yes	—
164	C₂×D₈	D₂₆₈	1	2	yes	—
165	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
168	C₄×D₄	—	2	4	no	—
169	C₄×D₄	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	134	2	no	—
172	C₂×C₂²⋊C₄	—	4	1	no	—
173	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂×C₁₆	—	134	2	no	—
175	C₂×C₁₆	—	134	2	no	—
176	C₂×C₁₆	—	134	2	no	—
177	#107	—	2	1	no	—
178	#107	—	134	2	no	—
179	#107	—	134	2	no	—
180	#107	C₂×Dic₆₇	1	2	yes	—
181	#106	—	268	2	no	—
182	#102	—	2	4	no	—
183	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
184	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
185	#95	—	268	1	no	—
186	#95	—	268	1	no	—
187	C₂⁴	—	4	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	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	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	8	4	no	—
197	C₂²×C₄	—	268	2	no	—
198	C₂²×C₄	#11761613	1	1	yes	characteristic
199	C₂²×C₄	#11761613	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	—	2	2	no	—
204	D₈	C₄⋊Dic₆₇	1	4	yes	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
208	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
211	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
212	C₂×C₈	#11761628	1	1	yes	characteristic
213	C₂×C₈	#11761628	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	268	2	no	—
217	C₄⋊C₄	—	268	2	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
220	C₄²	#11761616	1	2	yes	—
221	C₁₆	—	268	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	8	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	C₂×#11761613	1	1	yes	characteristic
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	8	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	536	2	no	—
238	C₂×C₄	—	536	2	no	—
239	C₂×C₄	—	536	2	no	—
240	C₂×C₄	#14523964	1	2	yes	—
241	C₂×C₄	#14523964	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
243	C₂×C₄	#14524017	1	1	yes	characteristic
244	C₂×C₄	#14524022	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
246	C₂×C₄	#14524068	1	1	yes	characteristic
247	C₂×C₄	#14524073	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
250	C₈	#14524002	1	1	yes	characteristic
251	C₈	#14524017	1	1	yes	characteristic
252	C₈	#14524022	1	1	yes	characteristic
253	C₂²	—	4	4	no	—
254	C₂²	—	8	4	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	4	no	—
258	C₂²	C₂×#14523964	1	1	yes	characteristic
259	C₂²	#16863591	1	1	yes	characteristic
260	C₂²	#16863676	1	1	yes	characteristic
261	C₂²	#16863826	1	2	yes	—
262	C₂²	#16863832	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	1072	2	no	—
267	C₄	#16863256	1	1	yes	characteristic
268	C₄	#16863397	1	1	yes	characteristic
269	C₄	#16863601	1	1	yes	characteristic
270	C₄	#16863686	1	1	yes	characteristic
271	C₂	—	8	4	no	—
272	C₂	#22560209	1	1	yes	characteristic
273	C₂	#22561916	1	1	yes	characteristic
274	C₂	#22561925	1	1	yes	characteristic
275	C₂	#22562264	1	2	yes	—
276	C₂	#22562366	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₈×C₆₇	C₂²
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	#12813
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
3. #22560373	C₂	semidirect
4. #22558300	C₂	semidirect
15. C₄×D₈×C₆₇	C₄	semidirect
144. C₆₇	#12813	semidirect
149. C₄×D₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863197	2
#22560373	2
#22558300	2
C₂×C₄×D₈×C₆₇	2
#12813	67