Subgroups — C₂×#22560109 #30853889

Order 17152.

1318 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×Dic₆₇	C₂	1	2	yes	maximal
4	#22560109	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863158	C₂	1	2	yes	maximal
6	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₆₇	—	2	2	no	—
10	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×#14523977	C₄	1	8	yes	complemented
12	C₂×#14523977	C₂²	1	4	yes	—
13	C₂×#14523953	C₄	1	8	yes	complemented
14	C₂×#14523953	C₂²	1	4	yes	—
15	C₄⋊C₄×Dic₆₇	—	2	4	no	—
16	C₄⋊C₄×Dic₆₇	C₂²	1	8	yes	—
17	C₂×C₆₇⋊#95	—	2	2	no	—
18	#16863158	C₂²	1	8	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#106×C₆₇	C₄	1	8	yes	complemented
22	C₂×#106×C₆₇	C₂²	1	4	yes	—
23	C₂³×Dic₆₇	—	2	2	no	—
24	C₂³×Dic₆₇	D₄	1	2	yes	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	4	no	—
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
33	C₂×C₄×Dic₆₇	—	2	8	no	—
34	C₂×C₄×Dic₆₇	—	4	8	no	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₆₇	Q₈	1	4	yes	—
38	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
39	#14523977	C₂×C₄	1	16	yes	complemented
40	#14523953	C₂×C₄	1	16	yes	complemented
41	C₆₇⋊#95	—	2	8	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
52	#106×C₆₇	C₂×C₄	1	16	yes	complemented
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	8	no	—
56	C₂²×Dic₆₇	—	2	4	no	—
57	C₂²×Dic₆₇	—	4	4	no	—
58	C₂²×Dic₆₇	—	4	4	no	—
59	C₂²×Dic₆₇	C₄⋊C₄	1	8	yes	—
60	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
61	C₂×C₆₇⋊C₈	—	2	8	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	2	2	no	—
64	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
65	C₄⋊Dic₆₇	—	2	8	no	—
66	C₄⋊Dic₆₇	—	2	4	no	—
67	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
68	C₄×Dic₆₇	—	4	16	no	—
69	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	—
70	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
71	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
72	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	2	8	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	—	4	4	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₄²	1	16	yes	complemented
86	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₆₇	—	2	8	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
92	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₆₇	C₄×D₄	1	2	yes	—
94	C₂³×C₆₇	C₄×D₄	1	2	yes	—
95	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
96	C₂³×C₆₇	#137	1	4	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	—	2	8	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	4	16	no	—
102	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	#117	1	4	yes	—
105	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₆₇	—	2	4	no	—
116	C₂×Dic₆₇	—	2	4	no	—
117	C₂×Dic₆₇	—	2	16	no	—
118	C₂×Dic₆₇	—	2	8	no	—
119	C₂×Dic₆₇	—	4	4	no	—
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	—	4	16	no	—
122	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
123	C₈×C₆₇	C₂×C₄²	1	8	yes	complemented
124	C₆₇⋊C₈	—	2	4	no	—
125	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
126	C₂²×C₆₇	#428	1	8	yes	—
127	C₂²×C₆₇	#440	1	8	yes	—
128	C₂²×C₆₇	#441	1	4	yes	—
129	C₂²×C₆₇	#442	1	4	yes	—
130	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₂²×C₆₇	C₂×#137	1	2	yes	—
135	C₄×C₆₇	—	4	8	no	—
136	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
137	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
139	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	Dic₆₇	—	2	8	no	—
142	Dic₆₇	—	4	8	no	—
143	C₂×#1776	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	#1776	1	8	yes	complemented
145	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#428	1	2	yes	—
147	C₂×C₆₇	C₂×#440	1	2	yes	—
148	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
149	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
150	C₂²×#107	—	67	1	no	—
151	C₂²×#106	D₆₇	1	1	yes	characteristic
152	C₂²×#97	—	67	1	no	—
153	C₂×C₄×C₄⋊C₄	—	67	2	no	—
154	#1776	—	67	8	no	—
155	C₂×#340	—	67	2	no	—
156	C₆₇	C₂×#1776	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₈	—	67	1	no	—
158	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂²×C₄⋊C₄	—	67	2	no	—
160	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
161	C₂²×C₄²	—	67	1	no	—
162	C₂²×C₄²	—	134	2	no	—
163	C₂×#107	—	67	8	no	—
164	C₂×#107	—	67	4	no	—
165	C₂×#106	Dic₆₇	1	8	yes	complemented
166	C₂×#106	C₂×D₆₇	1	4	yes	—
167	C₂×#97	—	67	8	no	—
168	C₂×#97	—	67	4	no	—
169	C₄×C₄⋊C₄	—	67	8	no	—
170	C₄×C₄⋊C₄	—	134	4	no	—
171	C₂×#95	—	134	2	no	—
172	#340	—	67	8	no	—
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	67	2	no	—
175	C₂³×C₄	—	134	2	no	—
176	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
177	C₂²×C₈	—	67	4	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	—	134	2	no	—
180	C₂²×C₈	—	134	2	no	—
181	C₂²×C₈	C₄×D₆₇	1	4	yes	—
182	C₂²×C₈	C₄×D₆₇	1	4	yes	—
183	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
184	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	67	8	no	—
188	C₂×C₄⋊C₄	—	67	4	no	—
189	C₂×C₄⋊C₄	—	134	2	no	—
190	C₂×C₄⋊C₄	—	134	4	no	—
191	C₂×C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
192	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
193	C₂×C₄²	—	268	8	no	—
194	C₂×C₄²	—	67	4	no	—
195	C₂×C₄²	—	67	2	no	—
196	C₂×C₄²	—	67	4	no	—
197	C₂×C₄²	—	67	2	no	—
198	C₂×C₄²	—	134	8	no	—
199	#107	—	67	16	no	—
200	#106	C₂×Dic₆₇	1	16	yes	complemented
201	#97	—	67	16	no	—
202	#95	—	134	8	no	—
203	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	268	4	no	—
209	C₂²×C₄	—	268	4	no	—
210	C₂²×C₄	—	67	8	no	—
211	C₂²×C₄	—	67	4	no	—
212	C₂²×C₄	—	134	8	no	—
213	C₂²×C₄	—	134	4	no	—
214	C₂²×C₄	—	134	4	no	—
215	C₂²×C₄	—	134	4	no	—
216	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
219	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
221	C₂²×C₄	#11761627	1	2	yes	—
222	C₂²×C₄	#11761629	1	2	yes	—
223	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
224	C₂×C₈	—	67	4	no	—
225	C₂×C₈	—	134	8	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	—	134	2	no	—
228	C₂×C₈	C₄×Dic₆₇	1	16	yes	complemented
229	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
230	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
231	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	67	8	no	—
235	C₄⋊C₄	—	134	8	no	—
236	C₄⋊C₄	—	134	4	no	—
237	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
238	C₄²	—	268	16	no	—
239	C₄²	—	67	8	no	—
240	C₄²	—	67	4	no	—
241	C₄²	—	67	4	no	—
242	C₂³	C₄⋊C₄×D₆₇	1	2	yes	—
243	C₂³	#14524005	1	2	yes	—
244	C₂³	D₄×Dic₆₇	1	2	yes	—
245	C₂³	#14524070	1	4	yes	—
246	C₂³	#14524072	1	4	yes	—
247	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	268	16	no	—
253	C₂×C₄	—	268	4	no	—
254	C₂×C₄	—	268	4	no	—
255	C₂×C₄	—	67	8	no	—
256	C₂×C₄	—	134	8	no	—
257	C₂×C₄	—	134	4	no	—
258	C₂×C₄	—	134	4	no	—
259	C₂×C₄	—	134	16	no	—
260	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
261	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
262	C₂×C₄	#14524001	1	4	yes	—
263	C₂×C₄	#14524005	1	2	yes	—
264	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
265	C₂×C₄	#14524008	1	4	yes	—
266	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
267	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
272	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
273	C₈	—	134	4	no	—
274	C₈	C₂×C₄×Dic₆₇	1	8	yes	complemented
275	C₂²	C₄⋊C₄×Dic₆₇	1	4	yes	—
276	C₂²	#16863435	1	8	yes	—
277	C₂²	#16863497	1	8	yes	—
278	C₂²	#16863502	1	4	yes	—
279	C₂²	#16863507	1	4	yes	—
280	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
281	C₂²	C₂×#14524005	1	1	yes	characteristic
282	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
283	C₂²	C₂×#14524070	1	2	yes	—
284	C₂²	C₂×#14524072	1	2	yes	—
285	C₄	—	4	8	no	—
286	C₄	—	268	8	no	—
287	C₄	—	134	8	no	—
288	C₄	C₄⋊C₄×Dic₆₇	1	4	yes	—
289	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
290	C₄	C₂×#14524001	1	1	yes	characteristic
291	C₄	C₂×#14524008	1	1	yes	characteristic
292	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
293	C₂	#22560109	1	8	yes	direct product factor
294	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
295	C₂	C₂×#16863435	1	2	yes	—
296	C₂	C₂×#16863497	1	2	yes	—
297	C₂	C₂×#16863502	1	1	yes	characteristic
298	C₂	C₂×#16863507	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#106×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#106	D₆₇
hypocenter	C₆₇	C₂×#1776
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
4. #22560109	C₂	direct (N × H)
11. C₂×#14523977	C₄	semidirect
13. C₂×#14523953	C₄	semidirect
21. C₂×#106×C₆₇	C₄	semidirect
39. #14523977	C₂×C₄	semidirect
40. #14523953	C₂×C₄	semidirect
52. #106×C₆₇	C₂×C₄	semidirect
85. C₂×C₈×C₆₇	C₄²	semidirect
123. C₈×C₆₇	C₂×C₄²	semidirect
144. C₂×C₆₇	#1776	semidirect
156. C₆₇	C₂×#1776	semidirect
165. C₂×#106	Dic₆₇	semidirect
200. #106	C₂×Dic₆₇	semidirect
228. C₂×C₈	C₄×Dic₆₇	semidirect
274. C₈	C₂×C₄×Dic₆₇	semidirect
293. C₂	#22560109	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523977	2
C₂²×#14523953	2
C₂×C₄⋊C₄×Dic₆₇	2
#22560109	2
C₂×#16863158	2
C₂²×#106×C₆₇	2
C₂×#1776	67

Subgroups — C2×#22560109 #30853889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560109 #30853889