Subgroups — #30754690

Order 17152.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561146	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863151	—	2	4	no	—
10	#16863149	—	2	2	no	—
11	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
12	C₈×C₆₇⋊C₈	—	2	2	no	—
13	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₆₇	—	2	2	no	—
16	C₄×Q₁₆×C₆₇	—	2	2	no	—
17	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#103×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₆₇	—	2	2	no	—
31	C₂×Q₁₆×C₆₇	—	2	2	no	—
32	C₂×Q₁₆×C₆₇	C₈	1	4	yes	complemented
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	—	2	2	no	—
38	C₄×Q₈×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	—	2	2	no	—
44	#103×C₆₇	—	2	4	no	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₆₇⋊C₈	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	4	no	—
53	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	—	2	2	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
65	Q₁₆×C₆₇	—	2	8	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₄⋊C₄×C₆₇	—	4	2	no	—
76	C₄²×C₆₇	—	4	2	no	—
77	C₄²×C₆₇	—	4	2	no	—
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	#41	1	1	yes	characteristic
82	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₆₇	—	2	4	no	—
84	Q₈×C₆₇	—	2	4	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	4	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	4	no	—
93	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₆₇	#135	1	2	yes	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
107	C₆₇⋊C₈	—	2	4	no	—
108	C₆₇⋊C₈	—	2	4	no	—
109	C₆₇⋊C₈	—	8	2	no	—
110	C₂²×C₆₇	#390	1	1	yes	characteristic
111	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
112	C₂²×C₆₇	#436	1	1	yes	characteristic
113	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
114	C₂²×C₆₇	#443	1	2	yes	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	4	4	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₆₇	#408	1	1	yes	characteristic
121	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
122	C₄×C₆₇	#436	1	1	yes	characteristic
123	C₄×C₆₇	#467	1	2	yes	—
124	C₄×C₆₇	#494	1	1	yes	characteristic
125	C₄×C₆₇	#495	1	1	yes	characteristic
126	#9656	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	C₈×Q₁₆	1	2	yes	—
128	C₂×C₆₇	#1591	1	2	yes	—
129	C₂×C₆₇	#1954	1	1	yes	characteristic
130	C₂×C₆₇	#1969	1	1	yes	characteristic
131	C₂×C₆₇	#1974	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
133	#1984	—	67	2	no	—
134	C₂×#335	—	67	1	no	—
135	C₂×Q₈⋊C₈	—	67	2	no	—
136	C₂×C₈²	—	67	1	no	—
137	C₆₇	#9656	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
140	C₄×Q₁₆	—	2	2	no	—
141	C₄×Q₁₆	—	2	2	no	—
142	C₂×#107	Dic₆₇	1	1	yes	characteristic
143	C₂×C₄⋊C₈	—	67	2	no	—
144	C₂×#103	Dic₆₇	1	2	yes	—
145	C₂×C₄×C₈	—	67	1	no	—
146	C₂×C₄×C₈	—	67	1	no	—
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	134	4	no	—
149	#335	—	134	2	no	—
150	Q₈⋊C₈	—	134	4	no	—
151	C₈²	—	134	2	no	—
152	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	C₆₇⋊C₈	1	4	yes	complemented
156	C₂²×C₈	—	67	2	no	—
157	C₂²×C₈	—	134	2	no	—
158	C₂²×C₈	—	134	1	no	—
159	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	C₄⋊C₈	—	134	2	no	—
171	C₄⋊C₈	—	134	2	no	—
172	#103	—	2	4	no	—
173	C₄×C₈	—	67	2	no	—
174	C₄×C₈	—	134	1	no	—
175	C₄×C₈	—	134	1	no	—
176	C₄×C₈	—	134	1	no	—
177	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
178	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	#11761627	1	1	yes	characteristic
190	Q₁₆	—	2	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	268	2	no	—
194	C₂×C₈	—	268	4	no	—
195	C₂×C₈	—	67	4	no	—
196	C₂×C₈	—	134	4	no	—
197	C₂×C₈	—	134	2	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
202	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
203	C₂×C₈	#11761612	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	D₄×D₆₇	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₄²	#11761627	1	1	yes	characteristic
214	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
227	C₂×C₄	#14524014	1	1	yes	characteristic
228	C₂×C₄	#14524014	1	1	yes	characteristic
229	C₂×C₄	#14524030	1	1	yes	characteristic
230	C₂×C₄	#14524039	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
233	C₂×C₄	#14524050	1	1	yes	characteristic
234	C₂×C₄	#14524050	1	1	yes	characteristic
235	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
236	C₂×C₄	#14524066	1	2	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	536	2	no	—
239	C₈	—	134	4	no	—
240	C₈	—	134	4	no	—
241	C₈	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
242	C₂²	#16863299	1	1	yes	characteristic
243	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
244	C₂²	#16863474	1	1	yes	characteristic
245	C₂²	Q₁₆×Dic₆₇	1	2	yes	—
246	C₂²	#16863513	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	—	8	2	no	—
251	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
252	C₄	#16863363	1	1	yes	characteristic
253	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
254	C₄	#16863474	1	1	yes	characteristic
255	C₄	#16863634	1	2	yes	—
256	C₄	#16863774	1	1	yes	characteristic
257	C₄	#16863781	1	1	yes	characteristic
258	C₂	Q₁₆×C₆₇⋊C₈	1	2	yes	—
259	C₂	#22559106	1	2	yes	—
260	C₂	#22561004	1	1	yes	characteristic
261	C₂	#22561071	1	1	yes	characteristic
262	C₂	#22561098	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#9656
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₆₇	C₈	semidirect
53. C₂×C₆₇⋊C₈	Q₁₆	semidirect
137. C₆₇	#9656	semidirect
155. C₂×Q₁₆	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22561146	2
C₂×#16863149	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
C₂×C₈×C₆₇⋊C₈	2
C₂×C₄×Q₁₆×C₆₇	2
#9656	67