Subgroups — C₂×#22559803 #30843477

Order 17152.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863792	C₂	1	2	yes	maximal
3	C₂×#16863748	C₂	1	2	yes	maximal
4	#22559803	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524043	C₂²	1	2	yes	—
8	#16863801	—	2	2	no	—
9	#16863801	C₂²	1	4	yes	—
10	#16863792	—	2	4	no	—
11	#16863792	C₂²	1	8	yes	—
12	#16863748	—	2	4	no	—
13	#16863748	C₂²	1	8	yes	—
14	C₂×#14523977	—	2	2	no	—
15	C₂×#14523974	—	2	4	no	—
16	C₂×#14523966	—	2	4	no	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×C₆₇⋊C₈	C₂²	1	4	yes	—
20	C₂²×Q₁₆×C₆₇	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#494×C₆₇	—	2	2	no	—
23	#494×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₃₄	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
27	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
28	#14524043	—	2	2	no	—
29	#14524043	—	2	2	no	—
30	#14524043	—	2	2	no	—
31	#14524043	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	#14523977	—	2	8	no	—
35	#14523974	—	2	16	no	—
36	#14523966	—	2	16	no	—
37	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
38	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
39	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₂²×Q₈×C₆₇	—	2	2	no	—
42	C₂×Q₁₆×C₆₇	—	2	8	no	—
43	C₂×Q₁₆×C₆₇	—	4	8	no	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₆₇	D₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂×Dic₁₃₄	—	2	4	no	—
55	C₂×Dic₁₃₄	—	4	8	no	—
56	C₂×Dic₁₃₄	—	4	2	no	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	Q₁₆	1	8	yes	complemented
60	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
61	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
62	C₄⋊Dic₆₇	—	2	4	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₄⋊Dic₆₇	—	4	2	no	—
66	C₂×Q₈×C₆₇	—	2	4	no	—
67	C₂×Q₈×C₆₇	—	4	2	no	—
68	C₂×Q₈×C₆₇	—	4	8	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₆₇	—	4	16	no	—
74	C₂×C₈×C₆₇	SD₁₆	1	8	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
76	C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂³×C₆₇	#127	1	1	yes	characteristic
85	Q₈×C₆₇	—	4	8	no	—
86	Q₈×C₆₇	—	8	4	no	—
87	C₂×C₄×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	8	4	no	—
89	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
93	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₆₇	#135	1	4	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₆₇	—	4	4	no	—
99	C₂×Dic₆₇	—	8	4	no	—
100	Dic₁₃₄	—	4	8	no	—
101	Dic₁₃₄	—	8	4	no	—
102	C₈×C₆₇	C₂×SD₁₆	1	8	yes	—
103	C₆₇⋊C₈	—	2	4	no	—
104	C₆₇⋊C₈	C₂×Q₁₆	1	8	yes	complemented
105	C₂²×C₆₇	#492	1	2	yes	—
106	C₂²×C₆₇	#494	1	2	yes	—
107	C₂²×C₆₇	#495	1	2	yes	—
108	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
109	C₄×C₆₇	—	8	4	no	—
110	C₄×C₆₇	#492	1	2	yes	—
111	C₄×C₆₇	#494	1	2	yes	—
112	C₄×C₆₇	#495	1	2	yes	—
113	C₄×C₆₇	C₂²×SD₁₆	1	2	yes	—
114	C₄×C₆₇	C₂²×Q₁₆	1	2	yes	—
115	C₄×C₆₇	C₂×#135	1	2	yes	—
116	Dic₆₇	—	8	4	no	—
117	C₂×#1707	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	#1707	1	4	yes	complemented
119	C₂×C₆₇	C₂×#492	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
122	C₂×#500	—	67	1	no	—
123	C₂×#498	—	67	2	no	—
124	C₂×#494	D₆₇	1	1	yes	characteristic
125	C₂×#487	—	67	2	no	—
126	#1707	—	67	8	no	—
127	C₂×C₈²	—	67	1	no	—
128	C₆₇	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	2	2	no	—
130	C₂×C₄⋊Q₈	—	67	2	no	—
131	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
132	#500	—	67	4	no	—
133	#500	—	134	2	no	—
134	#498	—	67	8	no	—
135	#498	—	134	4	no	—
136	#494	—	2	2	no	—
137	#494	C₂×D₆₇	1	4	yes	—
138	#487	—	67	8	no	—
139	#487	—	134	4	no	—
140	C₂×#107	—	134	2	no	—
141	C₂×#106	—	134	4	no	—
142	C₂×#103	—	134	4	no	—
143	C₂×C₄×C₈	—	67	1	no	—
144	C₂×C₄×C₈	—	67	1	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	C₈²	—	67	4	no	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	134	2	no	—
149	C₂×Q₁₆	—	2	8	no	—
150	C₂×Q₁₆	—	4	8	no	—
151	C₂²×C₈	—	67	2	no	—
152	C₂²×C₈	—	67	2	no	—
153	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	67	4	no	—
158	C₄⋊Q₈	—	134	2	no	—
159	C₄⋊Q₈	—	134	2	no	—
160	C₄⋊Q₈	—	134	2	no	—
161	C₄⋊Q₈	C₂²×D₆₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	134	2	no	—
165	C₂×C₄⋊C₄	—	134	2	no	—
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	#107	—	134	8	no	—
168	#106	—	134	16	no	—
169	#103	—	134	16	no	—
170	C₄×C₈	—	67	2	no	—
171	C₄×C₈	—	67	2	no	—
172	C₄×C₈	—	67	2	no	—
173	C₄×C₈	—	67	2	no	—
174	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
175	C₄×C₈	C₂²×D₆₇	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	8	no	—
181	C₂×Q₈	—	134	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	268	2	no	—
184	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
187	Q₁₆	—	4	16	no	—
188	C₂×C₈	—	67	8	no	—
189	C₂×C₈	—	67	4	no	—
190	C₂×C₈	—	67	4	no	—
191	C₂×C₈	—	134	4	no	—
192	C₂×C₈	#11761616	1	8	yes	—
193	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	268	2	no	—
199	C₄⋊C₄	—	268	2	no	—
200	C₄⋊C₄	—	134	4	no	—
201	C₄⋊C₄	—	134	4	no	—
202	C₄²	D₄×D₆₇	1	1	yes	characteristic
203	C₄²	D₄×D₆₇	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₄²	C₂³×D₆₇	1	1	yes	characteristic
206	C₂³	#14524039	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	268	8	no	—
210	Q₈	—	536	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	268	4	no	—
214	C₂×C₄	—	536	4	no	—
215	C₂×C₄	#14524039	1	1	yes	characteristic
216	C₂×C₄	#14524039	1	1	yes	characteristic
217	C₂×C₄	#14524039	1	1	yes	characteristic
218	C₂×C₄	C₂×#11761616	1	4	yes	—
219	C₂×C₄	Q₁₆×D₆₇	1	4	yes	—
220	C₂×C₄	#14524066	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
224	C₈	—	67	8	no	—
225	C₈	—	134	4	no	—
226	C₈	C₂×#11761616	1	8	yes	—
227	C₂²	#16863768	1	2	yes	—
228	C₂²	#16863774	1	2	yes	—
229	C₂²	#16863781	1	2	yes	—
230	C₂²	C₂×#14524039	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	536	4	no	—
233	C₄	#16863768	1	2	yes	—
234	C₄	#16863774	1	2	yes	—
235	C₄	#16863781	1	2	yes	—
236	C₄	C₂²×#11761616	1	2	yes	—
237	C₄	C₂×Q₁₆×D₆₇	1	2	yes	—
238	C₄	C₂×#14524066	1	2	yes	—
239	C₂	#22559803	1	4	yes	direct product factor
240	C₂	C₂×#16863768	1	1	yes	characteristic
241	C₂	C₂×#16863774	1	1	yes	characteristic
242	C₂	C₂×#16863781	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	C₂×#1707
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
4. #22559803	C₂	direct (N × H)
59. C₂×C₆₇⋊C₈	Q₁₆	semidirect
104. C₆₇⋊C₈	C₂×Q₁₆	semidirect
118. C₂×C₆₇	#1707	semidirect
128. C₆₇	C₂×#1707	semidirect
239. C₂	#22559803	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863801	2
C₂×#16863792	2
C₂×#16863748	2
#22559803	2
C₂×C₈×C₆₇⋊C₈	2
C₂×#494×C₆₇	2
C₂×#1707	67

Subgroups — C2×#22559803 #30843477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559803 #30843477