Subgroups — #30771887

Order 17152.

430 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	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863200	C₂	1	2	yes	maximal
3	#22560373	C₂	1	1	yes	maximal, characteristic
4	#22558300	C₂	1	2	yes	maximal
5	C₂×C₄×Q₁₆×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	#16863200	—	2	4	no	—
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
14	C₄×Q₁₆×C₆₇	—	2	2	no	—
15	C₄×Q₁₆×C₆₇	C₄	1	4	yes	complemented
16	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#103×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×Q₈×C₆₇	—	2	1	no	—
29	C₂×Q₁₆×C₆₇	—	2	1	no	—
30	C₂×Q₁₆×C₆₇	—	2	1	no	—
31	C₂×Q₁₆×C₆₇	—	2	1	no	—
32	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₆₇	D₄	1	2	yes	—
34	C₂×Q₁₆×C₆₇	Q₈	1	2	yes	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₆₇	—	2	4	no	—
38	C₄×Q₈×C₆₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
41	C₂×C₄²×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	#103×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	C₂×Q₈×C₆₇	—	2	2	no	—
53	C₂×Q₈×C₆₇	—	2	2	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	C₂²×C₄×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₆₇	—	2	2	no	—
64	Q₁₆×C₆₇	—	2	2	no	—
65	Q₁₆×C₆₇	—	2	2	no	—
66	Q₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	no	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
75	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	4	4	no	—
78	C₄⋊C₄×C₆₇	—	4	1	no	—
79	C₄²×C₆₇	—	4	4	no	—
80	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₆₇	SD₁₆	1	2	yes	—
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₆₇	—	2	4	no	—
84	Q₈×C₆₇	—	4	2	no	—
85	Q₈×C₆₇	—	4	2	no	—
86	Q₈×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	8	2	no	—
94	C₂×C₄×C₆₇	#102	1	2	yes	—
95	C₂×C₄×C₆₇	#102	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
102	C₂×Dic₆₇	—	8	2	no	—
103	C₂×Dic₆₇	—	8	2	no	—
104	C₂×Dic₆₇	—	8	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₆₇	#117	1	1	yes	characteristic
108	C₈×C₆₇	#121	1	1	yes	characteristic
109	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
111	C₂²×C₆₇	#459	1	1	yes	characteristic
112	C₂²×C₆₇	#474	1	1	yes	characteristic
113	C₂²×C₆₇	#508	1	2	yes	—
114	C₂²×C₆₇	#510	1	2	yes	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	4	4	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	#380	1	1	yes	characteristic
120	C₄×C₆₇	#417	1	1	yes	characteristic
121	C₄×C₆₇	#461	1	1	yes	characteristic
122	C₄×C₆₇	#476	1	1	yes	characteristic
123	Dic₆₇	—	16	2	no	—
124	#12814	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	#1802	1	1	yes	characteristic
126	C₂×C₆₇	#2138	1	1	yes	characteristic
127	C₂×C₆₇	#2141	1	1	yes	characteristic
128	C₂×C₆₇	#2211	1	2	yes	—
129	C₂×C₆₇	#2229	1	2	yes	—
130	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
131	C₂×C₄⋊C₁₆	—	67	1	no	—
132	C₂×#358	—	67	2	no	—
133	#1846	—	67	1	no	—
134	#1380	—	67	2	no	—
135	C₆₇	#12814	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
137	C₂×C₄×Q₈	—	2	1	no	—
138	C₂²×C₁₆	—	67	2	no	—
139	C₄×Q₁₆	—	2	2	no	—
140	C₄×Q₁₆	Dic₆₇	1	4	yes	complemented
141	C₂×#107	—	67	2	no	—
142	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
143	C₂×#106	—	134	1	no	—
144	C₂×#103	—	2	1	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	#382	—	134	1	no	—
147	C₄⋊C₁₆	—	134	2	no	—
148	#358	—	134	4	no	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×Q₁₆	Dic₁₃₄	1	2	yes	—
154	C₂×Q₁₆	D₂₆₈	1	2	yes	—
155	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
156	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
158	C₄×Q₈	—	2	4	no	—
159	C₄×Q₈	—	4	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	C₂×C₄⋊C₄	—	134	2	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×C₁₆	—	134	2	no	—
166	C₂×C₁₆	—	134	2	no	—
167	C₂×C₁₆	—	134	2	no	—
168	#107	—	2	1	no	—
169	#107	—	134	2	no	—
170	#107	—	134	2	no	—
171	#107	C₂×Dic₆₇	1	2	yes	—
172	#106	—	268	2	no	—
173	#103	—	2	4	no	—
174	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
175	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
176	#95	—	268	1	no	—
177	#95	—	268	1	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	268	2	no	—
187	C₂²×C₄	#11761613	1	1	yes	characteristic
188	C₂²×C₄	#11761613	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	2	no	—
192	Q₁₆	—	2	2	no	—
193	Q₁₆	C₄⋊Dic₆₇	1	4	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
197	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
200	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
201	C₂×C₈	#11761628	1	1	yes	characteristic
202	C₂×C₈	#11761628	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	268	2	no	—
207	C₄⋊C₄	—	268	2	no	—
208	C₄²	—	4	4	no	—
209	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
210	C₄²	#11761616	1	2	yes	—
211	C₁₆	—	268	2	no	—
212	C₂³	C₂×#11761613	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	536	2	no	—
225	C₂×C₄	—	536	2	no	—
226	C₂×C₄	—	536	2	no	—
227	C₂×C₄	#14523964	1	2	yes	—
228	C₂×C₄	#14523964	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂×C₄	#14524017	1	1	yes	characteristic
231	C₂×C₄	#14524022	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
233	C₂×C₄	#14524068	1	1	yes	characteristic
234	C₂×C₄	#14524073	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
237	C₈	#14524002	1	1	yes	characteristic
238	C₈	#14524017	1	1	yes	characteristic
239	C₈	#14524022	1	1	yes	characteristic
240	C₂²	C₂×#14523964	1	1	yes	characteristic
241	C₂²	#16863591	1	1	yes	characteristic
242	C₂²	#16863676	1	1	yes	characteristic
243	C₂²	#16863826	1	2	yes	—
244	C₂²	#16863837	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	1072	2	no	—
250	C₄	#16863256	1	1	yes	characteristic
251	C₄	#16863397	1	1	yes	characteristic
252	C₄	#16863601	1	1	yes	characteristic
253	C₄	#16863686	1	1	yes	characteristic
254	C₂	#22560209	1	1	yes	characteristic
255	C₂	#22561916	1	1	yes	characteristic
256	C₂	#22561929	1	1	yes	characteristic
257	C₂	#22562271	1	2	yes	—
258	C₂	#22562373	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#12814
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
15. C₄×Q₁₆×C₆₇	C₄	semidirect
135. C₆₇	#12814	semidirect
140. C₄×Q₁₆	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×#16863200	2
#22560373	2
#22558300	2
C₂×C₄×Q₁₆×C₆₇	2
#12814	67