Subgroups — C₂×#22558998 #30824916

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558998	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863149	C₂	1	2	yes	maximal, complemented
5	C₂×#16863147	C₂	1	2	yes	maximal, complemented
6	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	—	2	1	no	—
8	#16863449	—	2	2	no	—
9	#16863449	C₂²	1	4	yes	—
10	#16863420	—	2	2	no	—
11	#16863420	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
17	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
18	#16863149	C₂²	1	8	yes	complemented
19	#16863147	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#431×C₆₇	C₄	1	4	yes	—
22	#431×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
24	C₂×C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#11761612	—	2	4	no	—
27	C₂×#11761612	—	4	4	no	—
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
36	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
37	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
38	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
39	C₄×C₆₇⋊C₈	—	2	4	no	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
53	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₆₇	C₂³	1	4	yes	—
56	#11761612	—	4	8	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	4	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₆₇	—	2	8	no	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
82	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂×D₄	1	2	yes	—
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	2	no	—
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	#108	1	4	yes	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₇	#131	1	4	yes	—
115	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₆₇	#137	1	2	yes	—
117	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₆₇	—	2	8	no	—
121	C₆₇⋊C₈	—	4	2	no	—
122	C₆₇⋊C₈	—	4	4	no	—
123	C₆₇⋊C₈	—	4	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	#424	1	2	yes	—
129	C₂²×C₆₇	#428	1	2	yes	—
130	C₂²×C₆₇	C₂×#108	1	2	yes	—
131	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	#424	1	2	yes	—
137	C₄×C₆₇	#428	1	2	yes	—
138	C₄×C₆₇	C₂×#108	1	4	yes	—
139	C₄×C₆₇	C₂×#131	1	2	yes	—
140	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
142	C₂×#1574	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	4	2	no	—
144	C₂×C₆₇	#1574	1	4	yes	complemented
145	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
147	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
148	C₂×#431	—	67	1	no	—
149	C₂×#431	D₆₇	1	1	yes	characteristic
150	C₂×#423	—	67	1	no	—
151	#1574	—	67	8	no	—
152	C₂×#335	—	67	2	no	—
153	C₂×#334	—	67	2	no	—
154	C₆₇	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	134	1	no	—
156	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
157	#431	—	67	4	no	—
158	#431	—	134	2	no	—
159	#431	Dic₆₇	1	4	yes	—
160	#431	C₂×D₆₇	1	4	yes	—
161	#423	—	67	4	no	—
162	#423	—	134	2	no	—
163	C₂×C₄⋊C₈	—	67	1	no	—
164	C₂×C₄⋊C₈	—	67	2	no	—
165	C₂×C₄⋊C₈	—	67	1	no	—
166	C₂×C₄⋊C₈	—	134	1	no	—
167	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
168	C₂×C₂²⋊C₈	—	134	1	no	—
169	C₂×C₂²⋊C₈	Dic₆₇	1	2	yes	—
170	C₂×C₄×C₈	—	134	1	no	—
171	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
172	#335	—	67	8	no	—
173	#334	—	67	8	no	—
174	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
175	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
177	C₂×M₁₆	—	268	4	no	—
178	C₂×M₁₆	—	134	4	no	—
179	C₂²×C₈	—	134	2	no	—
180	C₂²×C₈	—	134	1	no	—
181	C₂²×C₈	—	134	1	no	—
182	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
183	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
184	C₂×C₄²	Dic₁₃₄	1	2	yes	—
185	C₂×C₄²	D₂₆₈	1	2	yes	—
186	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
189	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
190	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
191	C₄⋊C₈	—	67	2	no	—
192	C₄⋊C₈	—	67	4	no	—
193	C₄⋊C₈	—	67	2	no	—
194	C₄⋊C₈	—	134	2	no	—
195	C₄⋊C₈	—	134	1	no	—
196	C₄⋊C₈	—	134	1	no	—
197	C₄⋊C₈	—	134	4	no	—
198	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
199	C₂²⋊C₈	—	134	4	no	—
200	C₂²⋊C₈	C₂×Dic₆₇	1	8	yes	—
201	C₄×C₈	—	134	4	no	—
202	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
203	C₄×C₈	C₂²×D₆₇	1	4	yes	—
204	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
218	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
221	M₁₆	—	268	8	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	268	4	no	—
224	C₂×C₈	—	268	2	no	—
225	C₂×C₈	—	268	2	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	—	134	2	no	—
228	C₂×C₈	—	134	4	no	—
229	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
230	C₄²	C₄⋊Dic₆₇	1	2	yes	—
231	C₄²	C₄⋊Dic₆₇	1	2	yes	—
232	C₄²	C₄⋊Dic₆₇	1	2	yes	—
233	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
234	C₄²	C₂²×Dic₆₇	1	2	yes	—
235	C₄²	C₂×D₂₆₈	1	2	yes	—
236	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
237	C₄²	C₂×Dic₁₃₄	1	2	yes	—
238	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
239	C₄²	C₂³×D₆₇	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
244	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	#14523979	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
262	C₂×C₄	#14524050	1	4	yes	—
263	C₂×C₄	#14524071	1	2	yes	—
264	C₂×C₄	#14524076	1	2	yes	—
265	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	268	4	no	—
270	C₈	—	268	2	no	—
271	C₈	—	268	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	#16863423	1	2	yes	—
277	C₂²	#16863438	1	2	yes	—
278	C₂²	C₂×#14523979	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#16863423	1	2	yes	—
285	C₄	#16863438	1	2	yes	—
286	C₄	C₂×#14523979	1	4	yes	—
287	C₄	C₂×#14524050	1	2	yes	—
288	C₄	C₂×#14524071	1	1	yes	characteristic
289	C₄	C₂×#14524076	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#22558998	1	4	yes	direct product factor
292	C₂	C₂×#16863423	1	1	yes	characteristic
293	C₂	C₂×#16863438	1	1	yes	characteristic
294	C₂	C₂²×#14523979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	C₂×#1574
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22558998	C₂	direct (N × H)
4. C₂×#16863149	C₂	semidirect
5. C₂×#16863147	C₂	semidirect
18. #16863149	C₂²	semidirect
19. #16863147	C₂²	semidirect
144. C₂×C₆₇	#1574	semidirect
154. C₆₇	C₂×#1574	semidirect
291. C₂	#22558998	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×#16863449	2
C₂×#16863420	2
#22558998	2
C₂×#16863149	2
C₂×#16863147	2
C₂×#431×C₆₇	2
C₂×#1574	67

Subgroups — C2×#22558998 #30824916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558998 #30824916