Subgroups — C₂×#22559703 #30838437

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863681	C₂	1	2	yes	maximal
3	#22559703	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863143	C₂	1	2	yes	maximal
6	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	—	2	1	no	—
8	C₂²×Dic₂₆₈	—	2	1	no	—
9	C₂×#14524043	C₂²	1	2	yes	—
10	#16863776	—	2	2	no	—
11	#16863776	C₂²	1	4	yes	—
12	#16863681	—	2	4	no	—
13	#16863681	C₂²	1	8	yes	—
14	C₂×#14523974	—	2	2	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14523967	—	2	2	no	—
17	#16863147	C₂²	1	4	yes	—
18	#16863143	C₂²	1	8	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#446×C₆₇	—	2	2	no	—
21	#446×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₆₇	—	2	1	no	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₃₄	—	2	2	no	—
27	C₂×Dic₂₆₈	—	2	4	no	—
28	C₂×Dic₂₆₈	—	2	4	no	—
29	C₂×Dic₂₆₈	—	4	4	no	—
30	C₂×Dic₂₆₈	—	4	4	no	—
31	C₂²×C₆₇⋊C₈	—	2	2	no	—
32	#14524043	—	2	2	no	—
33	#14524043	—	2	2	no	—
34	#14524043	—	2	2	no	—
35	#14524043	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	2	no	—
38	#14523974	—	2	8	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
41	#14523967	—	2	8	no	—
42	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	—	2	1	no	—
47	C₄×Q₈×C₆₇	—	2	1	no	—
48	C₄×Q₈×C₆₇	D₄	1	2	yes	—
49	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	—	2	4	no	—
55	C₄⋊C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
57	#97×C₆₇	—	2	4	no	—
58	C₄×C₈×C₆₇	D₄	1	2	yes	—
59	C₄×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂×Dic₁₃₄	—	2	4	no	—
61	C₂×Dic₁₃₄	—	4	4	no	—
62	C₂×Dic₁₃₄	—	4	4	no	—
63	C₂×Dic₁₃₄	—	4	2	no	—
64	C₂²×Dic₆₇	—	4	2	no	—
65	Dic₂₆₈	—	4	8	no	—
66	Dic₂₆₈	—	4	8	no	—
67	C₂×C₆₇⋊C₈	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	4	4	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	4	2	no	—
72	C₄⋊Dic₆₇	—	4	2	no	—
73	C₂×Q₈×C₆₇	—	2	1	no	—
74	C₂×Q₈×C₆₇	—	2	4	no	—
75	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₆₇	#41	1	1	yes	characteristic
96	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₂³×C₆₇	#121	1	1	yes	characteristic
98	Q₈×C₆₇	—	2	4	no	—
99	Q₈×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₆₇	#137	1	4	yes	—
111	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₆₇	—	4	4	no	—
115	C₂×Dic₆₇	—	8	4	no	—
116	Dic₁₃₄	—	4	4	no	—
117	Dic₁₃₄	—	4	4	no	—
118	Dic₁₃₄	—	8	4	no	—
119	C₈×C₆₇	—	2	2	no	—
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
122	C₆₇⋊C₈	—	4	4	no	—
123	C₂²×C₆₇	#460	1	2	yes	—
124	C₂²×C₆₇	#470	1	2	yes	—
125	C₂²×C₆₇	#473	1	2	yes	—
126	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	4	2	no	—
130	C₄×C₆₇	—	4	2	no	—
131	C₄×C₆₇	#460	1	2	yes	—
132	C₄×C₆₇	#470	1	2	yes	—
133	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
134	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#137	1	2	yes	—
136	Dic₆₇	—	8	4	no	—
137	C₂×#1690	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	#1690	1	4	yes	complemented
139	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
140	C₂×C₆₇	C₂×#470	1	1	yes	characteristic
141	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
142	C₂×#494	—	67	1	no	—
143	C₂×#475	—	67	2	no	—
144	C₂×#446	D₆₇	1	1	yes	characteristic
145	#1690	—	67	8	no	—
146	C₂×#334	—	67	1	no	—
147	C₂×#332	—	67	2	no	—
148	C₆₇	C₂×#1690	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₁₆	—	134	1	no	—
150	C₂²×Q₁₆	—	134	1	no	—
151	C₂×C₄⋊Q₈	—	67	2	no	—
152	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
153	#494	—	67	4	no	—
154	#494	—	134	2	no	—
155	#475	—	67	8	no	—
156	#475	—	134	4	no	—
157	#446	—	2	2	no	—
158	#446	C₂×D₆₇	1	4	yes	—
159	C₂×#106	—	134	2	no	—
160	C₂×C₄⋊C₈	—	2	1	no	—
161	C₂×C₄⋊C₈	—	67	2	no	—
162	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
163	C₂×#103	—	134	2	no	—
164	C₂×#97	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
166	#334	—	67	4	no	—
167	#332	—	67	8	no	—
168	C₂²×Q₈	—	134	2	no	—
169	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
170	C₂×Q₁₆	—	268	4	no	—
171	C₂×Q₁₆	—	268	4	no	—
172	C₂×Q₁₆	—	134	4	no	—
173	C₂×Q₁₆	—	134	4	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	134	2	no	—
176	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
178	C₄⋊Q₈	—	67	4	no	—
179	C₄⋊Q₈	—	134	2	no	—
180	C₄⋊Q₈	—	134	2	no	—
181	C₄⋊Q₈	—	134	2	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	D₂₆₈	1	2	yes	—
185	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	134	2	no	—
188	C₂×C₄⋊C₄	—	134	2	no	—
189	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
192	#106	—	134	8	no	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	67	4	no	—
196	C₄⋊C₈	—	134	2	no	—
197	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
198	#103	—	134	8	no	—
199	#97	—	2	4	no	—
200	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
201	C₄×C₈	C₂²×D₆₇	1	2	yes	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	268	2	no	—
205	C₂×Q₈	—	268	4	no	—
206	C₂×Q₈	—	268	4	no	—
207	C₂×Q₈	—	134	4	no	—
208	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	268	2	no	—
212	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
214	C₂²×C₄	#11761628	1	1	yes	characteristic
215	Q₁₆	—	268	8	no	—
216	Q₁₆	—	268	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	268	4	no	—
221	C₂×C₈	—	134	4	no	—
222	C₂×C₈	#11761615	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	268	2	no	—
229	C₄⋊C₄	—	268	2	no	—
230	C₄⋊C₄	—	134	4	no	—
231	C₄⋊C₄	—	134	4	no	—
232	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
237	C₄²	#11761628	1	1	yes	characteristic
238	C₄²	C₂³×D₆₇	1	1	yes	characteristic
239	C₂³	#14524017	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	268	4	no	—
243	Q₈	—	268	4	no	—
244	Q₈	—	536	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	268	4	no	—
251	C₂×C₄	—	536	4	no	—
252	C₂×C₄	#14524017	1	1	yes	characteristic
253	C₂×C₄	#14524017	1	1	yes	characteristic
254	C₂×C₄	#14524017	1	1	yes	characteristic
255	C₂×C₄	C₂×#11761615	1	2	yes	—
256	C₂×C₄	#14524068	1	2	yes	—
257	C₂×C₄	#14524076	1	4	yes	—
258	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
260	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	4	2	no	—
263	C₈	—	268	4	no	—
264	C₈	C₂×#11761615	1	4	yes	—
265	C₂²	#16863596	1	2	yes	—
266	C₂²	#16863654	1	2	yes	—
267	C₂²	#16863671	1	2	yes	—
268	C₂²	C₂×#14524017	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	536	4	no	—
274	C₄	#16863596	1	2	yes	—
275	C₄	#16863654	1	2	yes	—
276	C₄	C₂²×#11761615	1	1	yes	characteristic
277	C₄	C₂×#14524068	1	1	yes	characteristic
278	C₄	C₂×#14524076	1	2	yes	—
279	C₂	#22559703	1	4	yes	direct product factor
280	C₂	C₂×#16863596	1	1	yes	characteristic
281	C₂	C₂×#16863654	1	1	yes	characteristic
282	C₂	C₂×#16863671	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#446×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#446	D₆₇
hypocenter	C₆₇	C₂×#1690
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #22559703	C₂	direct (N × H)
138. C₂×C₆₇	#1690	semidirect
148. C₆₇	C₂×#1690	semidirect
279. C₂	#22559703	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×#16863776	2
C₂×#16863681	2
#22559703	2
C₂×#16863147	2
C₂×#16863143	2
C₂×#446×C₆₇	2
C₂×#1690	67

Subgroups — C2×#22559703 #30838437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559703 #30838437