Subgroups — #30749100

Order 17152.

346 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560347	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
3	#22558899	C₂	1	2	yes	maximal
4	#22558083	C₂	1	2	yes	maximal
5	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863161	—	2	1	no	—
9	#16863161	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863145	—	2	2	no	—
13	#16863142	—	2	2	no	—
14	#533×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	—	2	1	no	—
16	#446×C₆₇	C₄	1	2	yes	—
17	C₈×Q₈×C₆₇	—	2	1	no	—
18	C₈×Q₈×C₆₇	C₄	1	2	yes	—
19	#424×C₆₇	—	2	1	no	—
20	#423×C₆₇	C₂²	1	1	yes	characteristic
21	#405×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
34	#125×C₆₇	—	2	1	no	—
35	C₂²⋊Q₈×C₆₇	—	2	1	no	—
36	C₂²⋊Q₈×C₆₇	Q₈	1	2	yes	—
37	C₄×Q₈×C₆₇	—	2	1	no	—
38	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
39	#117×C₆₇	—	2	1	no	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	1	no	—
44	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
45	#97×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	—	2	2	no	—
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	#11761612	—	4	2	no	—
52	#11761612	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	8	2	no	—
56	C₂×Q₈×C₆₇	—	2	1	no	—
57	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
61	M₁₆×C₆₇	—	4	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	#41	1	1	yes	characteristic
81	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₆₇	—	4	2	no	—
83	Q₈×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#135	1	2	yes	—
95	C₂×C₄×C₆₇	#137	1	2	yes	—
96	C₈×C₆₇	—	2	1	no	—
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	—	4	2	no	—
101	C₆₇⋊C₈	—	8	2	no	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	#380	1	1	yes	characteristic
106	C₂²×C₆₇	#479	1	2	yes	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	4	2	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	#416	1	1	yes	characteristic
113	C₄×C₆₇	#419	1	1	yes	characteristic
114	C₄×C₆₇	#464	1	2	yes	—
115	C₄×C₆₇	#471	1	2	yes	—
116	#8687	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	#1805	1	1	yes	characteristic
119	C₂×C₆₇	#1813	1	1	yes	characteristic
120	C₂×C₆₇	#1815	1	1	yes	characteristic
121	#2960	D₆₇	1	1	yes	characteristic
122	#1837	—	67	1	no	—
123	C₂×#334	—	67	1	no	—
124	#1550	—	67	2	no	—
125	#1292	—	67	2	no	—
126	C₆₇	#8687	1	1	yes	characteristic, complemented, Sylow
127	#533	C₂×D₆₇	1	1	yes	characteristic
128	#446	—	2	1	no	—
129	#446	Dic₆₇	1	2	yes	—
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	Dic₆₇	1	2	yes	—
132	#424	—	2	1	no	—
133	#423	—	67	2	no	—
134	#423	C₂×D₆₇	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	67	2	no	—
136	#405	—	2	1	no	—
137	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	#341	—	134	1	no	—
139	#341	—	134	1	no	—
140	#334	—	134	1	no	—
141	#334	—	134	1	no	—
142	#333	—	134	2	no	—
143	#332	—	134	2	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	134	2	no	—
146	C₂²×C₈	—	134	2	no	—
147	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
149	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
150	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
151	#125	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	Dic₁₃₄	1	2	yes	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
156	#117	—	2	1	no	—
157	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	134	2	no	—
162	C₄⋊C₈	—	134	2	no	—
163	C₄⋊C₈	—	134	2	no	—
164	C₄⋊C₈	—	134	2	no	—
165	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
169	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
174	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
176	C₂²×C₄	#11761628	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	M₁₆	—	268	2	no	—
179	M₁₆	—	268	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	268	2	no	—
184	C₂×C₈	—	268	2	no	—
185	C₂×C₈	—	536	2	no	—
186	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
201	C₄²	#11761628	1	1	yes	characteristic
202	C₂³	#14524022	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
211	C₂×C₄	#14524002	1	1	yes	characteristic
212	C₂×C₄	#14524017	1	1	yes	characteristic
213	C₂×C₄	#14524017	1	1	yes	characteristic
214	C₂×C₄	#14524022	1	1	yes	characteristic
215	C₂×C₄	#14524063	1	2	yes	—
216	C₂×C₄	#14524074	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	536	2	no	—
223	C₈	—	536	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#16863256	1	1	yes	characteristic
227	C₂²	#16863703	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	#16863392	1	1	yes	characteristic
234	C₄	#16863405	1	1	yes	characteristic
235	C₄	#16863618	1	2	yes	—
236	C₄	#16863659	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#22560224	1	1	yes	characteristic
239	C₂	#22560259	1	1	yes	characteristic
240	C₂	#22560268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#8687
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
126. C₆₇	#8687	semidirect

Maximal subgroups

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

subgroup	index
#22560347	2
C₂×#16863147	2
#22558899	2
#22558083	2
C₆₇×#2960	2
#8687	67