Subgroups — #30748629

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	2	yes	maximal
2	C₂×#16863145	C₂	1	2	yes	maximal
3	#22558092	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
8	#16863145	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
11	#391×C₆₇	C₄	1	2	yes	—
12	#389×C₆₇	—	2	2	no	—
13	#384×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
20	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
21	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₄⋊Q₈×C₆₇	—	2	2	no	—
23	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	—
25	C₄×Q₈×C₆₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	#95×C₆₇	—	2	4	no	—
37	C₂×C₆₇⋊C₈	—	4	4	no	—
38	C₂×C₆₇⋊C₈	—	4	4	no	—
39	C₂×C₆₇⋊C₈	—	4	4	no	—
40	C₂×Q₈×C₆₇	—	2	2	no	—
41	C₂×Q₈×C₆₇	—	4	2	no	—
42	C₂×Q₈×C₆₇	—	4	2	no	—
43	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
51	C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₄⋊C₄×C₆₇	—	4	4	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₄²×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	—	4	1	no	—
61	C₄²×C₆₇	—	4	4	no	—
62	C₄²×C₆₇	SD₁₆	1	4	yes	—
63	C₄²×C₆₇	Q₁₆	1	4	yes	—
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₆₇	#41	1	2	yes	—
67	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₆₇	—	4	4	no	—
69	Q₈×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	1	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	#100	1	2	yes	—
82	C₂×C₄×C₆₇	#103	1	4	yes	—
83	C₂×C₄×C₆₇	#103	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₆₇	#137	1	2	yes	—
92	C₆₇⋊C₈	—	8	4	no	—
93	C₂²×C₆₇	#380	1	1	yes	characteristic
94	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
95	C₂²×C₆₇	#413	1	1	yes	characteristic
96	C₂²×C₆₇	C₂×#103	1	2	yes	—
97	C₂²×C₆₇	#417	1	2	yes	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	4	no	—
101	C₄×C₆₇	—	4	4	no	—
102	C₄×C₆₇	—	4	1	no	—
103	C₄×C₆₇	—	8	2	no	—
104	C₄×C₆₇	#333	1	4	yes	complemented
105	C₄×C₆₇	#460	1	2	yes	—
106	C₄×C₆₇	#462	1	2	yes	—
107	C₄×C₆₇	#475	1	2	yes	—
108	C₄×C₆₇	#477	1	2	yes	—
109	#8587	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	C₂×#333	1	1	yes	characteristic
111	C₂×C₆₇	#1550	1	2	yes	—
112	C₂×C₆₇	#1560	1	1	yes	characteristic
113	C₂×C₆₇	#1800	1	1	yes	characteristic
114	C₂×C₆₇	#1803	1	2	yes	—
115	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
116	#1842	—	67	2	no	—
117	C₂×#333	—	67	2	no	—
118	#1296	—	67	2	no	—
119	C₆₇	#8587	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
121	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
122	C₂×C₄⋊C₈	—	67	4	no	—
123	C₂×C₄⋊C₈	—	134	2	no	—
124	C₂×C₄⋊C₈	—	134	2	no	—
125	#391	Dic₆₇	1	2	yes	—
126	#389	—	2	2	no	—
127	#384	C₂×D₆₇	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	2	2	no	—
129	C₄³	C₂×D₆₇	1	1	yes	characteristic
130	#333	—	134	4	no	—
131	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
132	C₂²×C₈	—	134	4	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	Dic₁₃₄	1	2	yes	—
135	C₄⋊Q₈	D₂₆₈	1	2	yes	—
136	C₄×Q₈	—	4	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
145	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
146	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
147	C₄⋊C₈	—	268	4	no	—
148	C₄⋊C₈	—	268	4	no	—
149	C₄⋊C₈	—	134	4	no	—
150	C₄⋊C₈	—	134	4	no	—
151	#95	—	2	4	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	C₄⋊Dic₆₇	1	4	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
163	C₂×C₈	—	268	4	no	—
164	C₂×C₈	—	268	4	no	—
165	C₂×C₈	—	268	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	#11761615	1	4	yes	—
178	C₄²	#11761618	1	4	yes	—
179	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
181	C₄²	#11761628	1	2	yes	—
182	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#14523960	1	2	yes	—
197	C₂×C₄	#14523967	1	4	yes	—
198	C₂×C₄	#14523967	1	4	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
200	C₂×C₄	#14524002	1	1	yes	characteristic
201	C₂×C₄	#14524017	1	2	yes	—
202	C₂×C₄	#14524022	1	2	yes	—
203	C₂×C₄	C₂×#11761615	1	2	yes	—
204	C₂×C₄	C₂×#11761618	1	2	yes	—
205	C₂×C₄	#14524068	1	2	yes	—
206	C₂×C₄	#14524073	1	2	yes	—
207	C₈	—	536	4	no	—
208	C₂²	#16863256	1	1	yes	characteristic
209	C₂²	C₂×#14523960	1	1	yes	characteristic
210	C₂²	#16863380	1	1	yes	characteristic
211	C₂²	C₂×#14523967	1	2	yes	—
212	C₂²	#16863397	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	#16863145	1	4	yes	complemented
220	C₄	#16863596	1	2	yes	—
221	C₄	#16863606	1	2	yes	—
222	C₄	#16863681	1	2	yes	—
223	C₄	#16863691	1	2	yes	—
224	C₂	C₂×#16863145	1	1	yes	characteristic
225	C₂	#22558899	1	2	yes	—
226	C₂	#22558944	1	1	yes	characteristic
227	C₂	#22560201	1	1	yes	characteristic
228	C₂	#22560214	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#8587
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
104. C₄×C₆₇	#333	semidirect
119. C₆₇	#8587	semidirect
219. C₄	#16863145	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₂×#16863145	2
#22558092	2
C₄×C₄⋊Q₈×C₆₇	2
#8587	67