Subgroups — #30774347

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863212	C₂	1	2	yes	maximal
3	#22561929	C₂	1	2	yes	maximal
4	#22560088	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	C₂²	1	1	yes	characteristic
8	#16863434	C₂²	1	1	yes	characteristic
9	#16863351	—	2	1	no	—
10	#16863217	—	2	2	no	—
11	#16863212	—	2	4	no	—
12	#16863200	—	2	4	no	—
13	#16863156	—	2	1	no	—
14	C₂²×Q₁₆×C₆₇	—	2	1	no	—
15	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
17	#494×C₆₇	—	2	2	no	—
18	#494×C₆₇	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	1	no	—
21	#14524002	—	2	1	no	—
22	#14523983	—	2	2	no	—
23	#14523983	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	#14523979	—	2	1	no	—
26	#14523979	D₄	1	2	yes	—
27	#14523977	D₄	1	2	yes	—
28	#14523975	—	2	1	no	—
29	#14523953	—	4	1	no	—
30	C₈×Dic₆₇	—	4	1	no	—
31	C₂²×Q₈×C₆₇	—	2	2	no	—
32	C₂×Q₁₆×C₆₇	—	2	4	no	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	2	no	—
35	C₂×Q₁₆×C₆₇	—	2	2	no	—
36	C₂×Q₁₆×C₆₇	—	2	2	no	—
37	C₂×Q₁₆×C₆₇	—	2	2	no	—
38	C₂×Q₁₆×C₆₇	—	4	1	no	—
39	C₂×Q₁₆×C₆₇	—	4	1	no	—
40	C₂×Q₁₆×C₆₇	—	4	2	no	—
41	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₇	—	2	4	no	—
45	C₄⋊Q₈×C₆₇	—	4	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	#11761612	—	2	2	no	—
51	#11761612	—	4	1	no	—
52	C₂×C₆₇⋊C₈	—	4	1	no	—
53	C₄⋊Dic₆₇	—	2	2	no	—
54	C₂²⋊Dic₆₇	—	4	1	no	—
55	C₄×Dic₆₇	—	4	1	no	—
56	C₆₇⋊C₁₆	—	4	2	no	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	4	2	no	—
60	C₂×Q₈×C₆₇	—	4	2	no	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂×Q₈×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₆₇	—	2	4	no	—
67	Q₁₆×C₆₇	—	2	4	no	—
68	Q₁₆×C₆₇	—	4	4	no	—
69	Q₁₆×C₆₇	—	4	4	no	—
70	Q₁₆×C₆₇	—	4	2	no	—
71	Q₁₆×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
74	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
75	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₆₇	—	4	8	no	—
83	C₄²×C₆₇	—	2	1	no	—
84	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
85	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
86	Q₈×C₆₇	—	4	4	no	—
87	Q₈×C₆₇	—	4	4	no	—
88	Q₈×C₆₇	—	8	2	no	—
89	Q₈×C₆₇	—	8	2	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	4	4	no	—
93	C₂×C₄×C₆₇	—	8	2	no	—
94	C₂×C₄×C₆₇	—	8	2	no	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×D₈	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×Dic₆₇	—	4	2	no	—
102	C₈×C₆₇	#102	1	2	yes	—
103	C₈×C₆₇	#102	1	2	yes	—
104	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₈×C₆₇	#124	1	1	yes	characteristic
107	C₈×C₆₇	#127	1	1	yes	characteristic
108	C₆₇⋊C₈	—	4	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	#418	1	1	yes	characteristic
111	C₂²×C₆₇	#459	1	1	yes	characteristic
112	C₂²×C₆₇	#460	1	1	yes	characteristic
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	—	8	4	no	—
115	C₄×C₆₇	#390	1	1	yes	characteristic
116	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
117	C₄×C₆₇	#459	1	1	yes	characteristic
118	C₄×C₆₇	#460	1	1	yes	characteristic
119	Dic₆₇	—	8	2	no	—
120	#13262	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	1	no	—
122	C₂×C₆₇	—	2	1	no	—
123	C₂×C₆₇	#1982	1	1	yes	characteristic
124	C₂×C₆₇	#2214	1	2	yes	—
125	C₂×#494	D₆₇	1	1	yes	characteristic
126	C₂×#364	—	67	1	no	—
127	C₂×#362	—	67	2	no	—
128	#2141	—	67	2	no	—
129	#1772	—	67	1	no	—
130	C₆₇	#13262	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₁₆	—	2	1	no	—
132	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
133	C₂×C₄⋊Q₈	—	2	1	no	—
134	C₂×M₃₂	—	67	2	no	—
135	#494	—	2	2	no	—
136	#494	Dic₆₇	1	4	yes	complemented
137	C₂×#108	—	67	1	no	—
138	#427	—	67	1	no	—
139	#405	—	134	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#364	—	134	2	no	—
142	#362	—	134	4	no	—
143	#358	—	134	4	no	—
144	#339	—	134	1	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂×Q₁₆	—	2	4	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	4	1	no	—
153	C₂×Q₁₆	—	4	1	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
156	C₂×M₁₆	—	134	1	no	—
157	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	4	no	—
160	C₄⋊Q₈	—	4	2	no	—
161	#117	—	134	1	no	—
162	C₂×C₄⋊C₄	—	4	2	no	—
163	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
164	M₃₂	—	134	2	no	—
165	M₃₂	—	134	2	no	—
166	C₂×C₁₆	—	134	2	no	—
167	#108	—	67	2	no	—
168	#108	—	134	1	no	—
169	#107	—	67	2	no	—
170	#106	—	134	1	no	—
171	#97	—	268	1	no	—
172	C₄×C₈	—	268	1	no	—
173	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
174	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	Q₁₆	—	2	4	no	—
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	4	no	—
187	Q₁₆	—	4	4	no	—
188	Q₁₆	—	4	2	no	—
189	Q₁₆	—	4	2	no	—
190	M₁₆	—	268	1	no	—
191	M₁₆	—	134	2	no	—
192	C₂×C₈	—	268	1	no	—
193	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
194	C₂×C₈	#11761613	1	1	yes	characteristic
195	C₂×C₈	#11761613	1	1	yes	characteristic
196	C₂×C₈	#11761616	1	2	yes	—
197	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
199	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₈	#11761627	1	1	yes	characteristic
202	C₂×C₈	#11761627	1	1	yes	characteristic
203	C₄⋊C₄	—	4	8	no	—
204	C₄⋊C₄	—	134	2	no	—
205	C₂²⋊C₄	—	268	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	268	1	no	—
208	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
209	C₁₆	—	268	2	no	—
210	C₂³	#14524069	1	1	yes	characteristic
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	Q₈	—	8	2	no	—
214	Q₈	—	8	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	268	2	no	—
221	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
222	C₂×C₄	#14524014	1	1	yes	characteristic
223	C₂×C₄	#14524014	1	1	yes	characteristic
224	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
225	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
226	C₂×C₄	#14524069	1	1	yes	characteristic
227	C₈	—	268	2	no	—
228	C₈	#14523964	1	2	yes	—
229	C₈	#14523964	1	2	yes	—
230	C₈	D₄×Dic₆₇	1	1	yes	characteristic
231	C₈	D₄×Dic₆₇	1	1	yes	characteristic
232	C₈	#14524030	1	1	yes	characteristic
233	C₈	#14524039	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	#16863401	1	1	yes	characteristic
236	C₂²	#16863588	1	1	yes	characteristic
237	C₂²	#16863593	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	8	4	no	—
240	C₄	—	536	2	no	—
241	C₄	#16863299	1	1	yes	characteristic
242	C₄	C₂×#14523964	1	1	yes	characteristic
243	C₄	#16863588	1	1	yes	characteristic
244	C₄	#16863593	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#22561137	1	1	yes	characteristic
248	C₂	#22562287	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863588
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle	C₂²×C₆₇	#459
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#13262
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
18. #494×C₆₇	C₄	semidirect
130. C₆₇	#13262	semidirect
136. #494	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
C₂×#16863212	2
#22561929	2
#22560088	2
C₂×#494×C₆₇	2
#13262	67