Subgroups — C₂×#22561714 #30884939

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863439	C₂	1	2	yes	maximal
2	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
3	#22561714	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863164	C₂	1	2	yes	maximal
5	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
6	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	#16863439	—	2	4	no	—
10	#16863439	C₂²	1	8	yes	—
11	C₂×#14523977	—	2	2	no	—
12	C₂×#14523977	—	2	2	no	—
13	C₂×#14523977	C₂²	1	4	yes	—
14	C₂×#14523976	—	2	2	no	—
15	C₂×#14523974	—	2	2	no	—
16	#16863164	C₂²	1	8	yes	—
17	#16863161	C₂²	1	4	yes	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	#424×C₆₇	—	2	2	no	—
22	#424×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
24	C₂³×Dic₆₇	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	4	no	—
27	C₂×#11761612	Q₈	1	8	yes	—
28	C₂×#11761612	C₂³	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
36	#14523977	—	2	8	no	—
37	#14523976	—	2	8	no	—
38	#14523974	—	2	8	no	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	1	no	—
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	D₄	1	2	yes	—
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	#117×C₆₇	—	2	1	no	—
49	#117×C₆₇	—	2	1	no	—
50	#117×C₆₇	D₄	1	2	yes	—
51	#117×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄²×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	—	2	8	no	—
57	C₂²×Dic₆₇	—	2	4	no	—
58	C₂²×Dic₆₇	—	4	2	no	—
59	C₂²×Dic₆₇	—	4	2	no	—
60	C₂²×Dic₆₇	—	8	4	no	—
61	#11761612	—	2	8	no	—
62	#11761612	—	2	4	no	—
63	#11761612	C₂×Q₈	1	8	yes	—
64	C₂×C₆₇⋊C₈	—	2	8	no	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₄⋊Dic₆₇	—	2	4	no	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	4	2	no	—
69	C₄⋊Dic₆₇	—	4	2	no	—
70	C₂⁴×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₆₇	#41	1	4	yes	—
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
89	C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂²⋊C₄×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	#121	1	1	yes	characteristic
100	C₂³×C₆₇	#121	1	1	yes	characteristic
101	C₂³×C₆₇	#125	1	4	yes	—
102	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₆₇	#123	1	2	yes	—
112	C₂×C₄×C₆₇	#123	1	2	yes	—
113	C₂×C₄×C₆₇	#125	1	4	yes	—
114	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	—	8	4	no	—
122	C₂×Dic₆₇	—	8	4	no	—
123	C₂×Dic₆₇	—	8	4	no	—
124	C₈×C₆₇	—	2	4	no	—
125	C₈×C₆₇	—	4	2	no	—
126	C₆₇⋊C₈	—	2	8	no	—
127	C₂²×C₆₇	—	2	8	no	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	#398	1	2	yes	—
133	C₂²×C₆₇	#398	1	2	yes	—
134	C₂²×C₆₇	#472	1	2	yes	—
135	C₂²×C₆₇	#473	1	2	yes	—
136	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#125	1	2	yes	—
138	C₄×C₆₇	—	4	2	no	—
139	C₄×C₆₇	—	4	2	no	—
140	C₄×C₆₇	#398	1	2	yes	—
141	C₄×C₆₇	#398	1	2	yes	—
142	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
143	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
144	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	Dic₆₇	—	8	4	no	—
146	C₂×#2089	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	#2089	1	4	yes	complemented
150	C₂×C₆₇	C₂×#398	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
153	C₂×#428	—	67	2	no	—
154	C₂²×#107	—	67	1	no	—
155	C₂×#424	D₆₇	1	1	yes	characteristic
156	#2089	—	67	8	no	—
157	C₂×#343	—	67	2	no	—
158	C₂×#341	—	67	1	no	—
159	C₆₇	C₂×#2089	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	67	2	no	—
161	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
162	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
163	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	67	2	no	—
165	#428	—	67	8	no	—
166	#428	—	134	4	no	—
167	C₂×#107	—	67	4	no	—
168	C₂×#107	—	134	2	no	—
169	C₂×#107	—	134	2	no	—
170	C₂×#107	—	134	2	no	—
171	C₂×#106	—	134	2	no	—
172	#424	—	2	2	no	—
173	#424	C₂×D₆₇	1	4	yes	—
174	C₂×C₄⋊C₈	—	2	2	no	—
175	#343	—	67	8	no	—
176	#341	—	67	4	no	—
177	C₂³×C₄	—	134	2	no	—
178	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	67	8	no	—
182	C₂×M₁₆	—	67	4	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	4	no	—
185	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	134	2	no	—
189	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
191	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	D₂₆₈	1	2	yes	—
195	#117	C₂²×D₆₇	1	2	yes	—
196	C₂×C₄⋊C₄	—	67	4	no	—
197	C₂×C₄⋊C₄	—	134	2	no	—
198	C₂×C₄⋊C₄	—	134	2	no	—
199	C₂×C₄⋊C₄	—	134	2	no	—
200	C₂×C₄⋊C₄	—	134	2	no	—
201	C₂×C₄⋊C₄	—	134	2	no	—
202	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
203	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₄²	—	2	1	no	—
206	#107	—	134	8	no	—
207	#107	—	134	8	no	—
208	#106	—	134	8	no	—
209	C₄⋊C₈	—	2	8	no	—
210	C₂⁴	#11761628	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	268	2	no	—
214	C₂²×C₄	—	268	2	no	—
215	C₂²×C₄	—	536	4	no	—
216	C₂²×C₄	—	134	4	no	—
217	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
221	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
222	C₂²×C₄	#11761627	1	4	yes	—
223	C₂²×C₄	#11761628	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	67	8	no	—
228	M₁₆	—	134	8	no	—
229	M₁₆	—	134	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	134	8	no	—
236	C₂×C₈	—	134	4	no	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	268	2	no	—
241	C₄⋊C₄	—	268	2	no	—
242	C₄⋊C₄	—	134	4	no	—
243	C₄⋊C₄	—	134	4	no	—
244	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
245	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	#14524017	1	1	yes	characteristic
253	C₂³	#14524017	1	1	yes	characteristic
254	C₂³	#14524033	1	4	yes	—
255	C₂³	C₂×#11761628	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	268	4	no	—
261	C₂×C₄	—	536	4	no	—
262	C₂×C₄	—	536	4	no	—
263	C₂×C₄	—	536	4	no	—
264	C₂×C₄	#14524017	1	1	yes	characteristic
265	C₂×C₄	#14524017	1	1	yes	characteristic
266	C₂×C₄	#14524023	1	4	yes	—
267	C₂×C₄	#14524023	1	4	yes	—
268	C₂×C₄	#14524024	1	2	yes	—
269	C₂×C₄	#14524024	1	2	yes	—
270	C₂×C₄	#14524033	1	4	yes	—
271	C₂×C₄	#14524041	1	2	yes	—
272	C₂×C₄	#14524041	1	2	yes	—
273	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
276	C₂×C₄	C₂×#11761627	1	2	yes	—
277	C₈	—	2	4	no	—
278	C₈	—	4	2	no	—
279	C₈	—	134	8	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#16863326	1	2	yes	—
286	C₂²	#16863326	1	2	yes	—
287	C₂²	#16863666	1	2	yes	—
288	C₂²	#16863671	1	2	yes	—
289	C₂²	C₂×#14524017	1	1	yes	characteristic
290	C₂²	C₂×#14524033	1	2	yes	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	536	4	no	—
294	C₄	#16863326	1	2	yes	—
295	C₄	#16863326	1	2	yes	—
296	C₄	C₂×#14524023	1	2	yes	—
297	C₄	C₂×#14524024	1	1	yes	characteristic
298	C₄	C₂×#14524041	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#22561714	1	4	yes	direct product factor
302	C₂	C₂×#16863326	1	1	yes	characteristic
303	C₂	C₂×#16863666	1	1	yes	characteristic
304	C₂	C₂×#16863671	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#424×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
hypercenter	C₂×#424	D₆₇
hypocenter	C₆₇	C₂×#2089
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
3. #22561714	C₂	direct (N × H)
149. C₂×C₆₇	#2089	semidirect
159. C₆₇	C₂×#2089	semidirect
301. C₂	#22561714	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×#16863439	2
C₂²×#14523977	2
#22561714	2
C₂×#16863164	2
C₂×#16863161	2
C₂×#424×C₆₇	2
C₂×#2089	67

Subgroups — C2×#22561714 #30884939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561714 #30884939