Subgroups — C₂×#22569229 #31720333

Order 17152.

2230 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569229	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×#16863474	C₂	1	6	yes	maximal, complemented
4	C₂×#16863453	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×#11761612	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	6	yes	—
8	Q₈×C₆₇⋊C₈	C₂²	1	16	yes	complemented
9	#16863474	C₂²	1	48	yes	complemented
10	#16863453	C₂²	1	24	yes	complemented
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
12	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
13	C₄×#11761612	C₂²	1	24	yes	complemented
14	C₂×#14523954	C₂²	1	3	yes	—
15	C₂×#14523954	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
17	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
18	C₄×#41×C₆₇	C₄	1	8	yes	—
19	C₄×#41×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₄	1	3	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
24	C₂×#117×C₆₇	C₄	1	3	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
26	C₂×#11761612	—	2	24	no	—
27	C₂×#11761612	C₂³	1	24	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
30	C₄⋊(C₆₇⋊C₈)	C₂³	1	24	yes	—
31	C₂²⋊(C₆₇⋊C₈)	C₂³	1	24	yes	—
32	#14523954	C₂³	1	4	yes	—
33	#14523954	C₂³	1	12	yes	—
34	C₄×C₆₇⋊C₈	C₂³	1	24	yes	—
35	C₂×#41×C₆₇	—	2	8	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	C₂×C₄	1	3	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
42	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
43	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
45	C₄×D₄×C₆₇	C₂×C₄	1	24	yes	—
46	C₄×D₄×C₆₇	C₂×C₄	1	12	yes	—
47	C₄×D₄×C₆₇	C₂³	1	12	yes	—
48	#117×C₆₇	C₂×C₄	1	24	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
51	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
58	#11761612	—	2	48	no	—
59	C₂×C₆₇⋊C₈	—	2	12	no	—
60	C₂×C₆₇⋊C₈	#41	1	8	yes	complemented
61	C₂×C₆₇⋊C₈	C₂⁴	1	12	yes	—
62	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
63	C₂⁴×C₆₇	C₂²×C₄	1	3	yes	—
64	#41×C₆₇	—	2	32	no	—
65	C₂×Q₈×C₆₇	M₁₆	1	8	yes	complemented
66	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂×D₄×C₆₇	—	2	12	no	—
68	C₂×D₄×C₆₇	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₆₇	—	2	12	no	—
70	C₂²×C₄×C₆₇	—	2	12	no	—
71	C₂²×C₄×C₆₇	—	4	6	no	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
83	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
84	C₂²⋊C₄×C₆₇	C₂²×C₄	1	24	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
87	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
88	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
89	C₄²×C₆₇	C₂⁴	1	2	yes	—
90	C₄²×C₆₇	C₂⁴	1	6	yes	—
91	C₂³×C₆₇	—	2	6	no	—
92	C₂³×C₆₇	—	2	6	no	—
93	C₂³×C₆₇	—	2	6	no	—
94	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
95	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₆₇	C₂×M₁₆	1	16	yes	complemented
97	D₄×C₆₇	—	2	24	no	—
98	C₂×C₄×C₆₇	—	2	6	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	6	no	—
102	C₂×C₄×C₆₇	—	2	12	no	—
103	C₂×C₄×C₆₇	—	2	24	no	—
104	C₂×C₄×C₆₇	—	2	12	no	—
105	C₂×C₄×C₆₇	—	4	12	no	—
106	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
113	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
114	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
115	C₆₇⋊C₈	—	2	12	no	—
116	C₆₇⋊C₈	C₂×#41	1	8	yes	complemented
117	C₂²×C₆₇	—	2	6	no	—
118	C₂²×C₆₇	—	2	6	no	—
119	C₂²×C₆₇	—	2	6	no	—
120	C₂²×C₆₇	—	4	12	no	—
121	C₂²×C₆₇	C₄×#41	1	2	yes	—
122	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
123	C₂²×C₆₇	#568	1	2	yes	—
124	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
125	C₄×C₆₇	—	2	6	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	2	6	no	—
129	C₄×C₆₇	C₄×#41	1	2	yes	—
130	C₄×C₆₇	C₂²×M₁₆	1	12	yes	—
131	C₄×C₆₇	C₂²×#41	1	2	yes	—
132	C₂×#2971	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	4	6	no	—
134	C₂×C₆₇	#2971	1	4	yes	complemented
135	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
136	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
137	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
138	C₂×C₄×#41	D₆₇	1	1	yes	characteristic
139	#2971	—	67	16	no	—
140	C₂×C₈×Q₈	—	67	2	no	—
141	C₂×#436	—	67	6	no	—
142	C₂×#432	—	67	3	no	—
143	C₂×C₄×M₁₆	—	67	3	no	—
144	C₆₇	C₂×#2971	1	1	yes	characteristic, complemented, Sylow
145	C₂²×#41	Dic₆₇	1	1	yes	characteristic
146	C₂²×M₁₆	—	67	6	no	—
147	C₄×#41	Dic₆₇	1	8	yes	—
148	C₄×#41	C₂×D₆₇	1	8	yes	—
149	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
150	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
151	C₂×C₄×D₄	Dic₆₇	1	3	yes	—
152	C₂×C₄×D₄	C₂×D₆₇	1	3	yes	—
153	C₂×#117	Dic₆₇	1	3	yes	—
154	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
155	C₈×Q₈	—	67	16	no	—
156	#436	—	67	48	no	—
157	#432	—	67	24	no	—
158	C₂×C₄⋊C₈	—	67	6	no	—
159	C₂×C₂²⋊C₈	—	67	6	no	—
160	C₄×M₁₆	—	67	24	no	—
161	C₂×#97	—	67	1	no	—
162	C₂×#97	—	67	3	no	—
163	C₂×C₄×C₈	—	67	6	no	—
164	C₂×#41	—	2	8	no	—
165	C₂×#41	C₂×Dic₆₇	1	8	yes	—
166	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂²×D₄	C₂×Dic₆₇	1	3	yes	—
168	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
169	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
170	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
171	C₂×M₁₆	—	67	24	no	—
172	C₂×M₁₆	—	134	24	no	—
173	C₂²×C₈	—	67	2	no	—
174	C₂²×C₈	—	67	6	no	—
175	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
176	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
177	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
178	C₄×D₄	C₂×Dic₆₇	1	24	yes	—
179	C₄×D₄	C₂×Dic₆₇	1	12	yes	—
180	C₄×D₄	C₂²×D₆₇	1	12	yes	—
181	#117	C₂×Dic₆₇	1	24	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
184	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	6	yes	—
185	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
186	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
187	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
188	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
189	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
190	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
191	C₄⋊C₈	—	67	24	no	—
192	C₂²⋊C₈	—	67	24	no	—
193	#97	—	67	4	no	—
194	#97	—	67	12	no	—
195	C₄×C₈	—	67	24	no	—
196	C₂⁴	C₂²×Dic₆₇	1	3	yes	—
197	#41	—	2	32	no	—
198	C₂×Q₈	#11761612	1	8	yes	complemented
199	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
200	C₂×D₄	—	2	12	no	—
201	C₂×D₄	C₂²×Dic₆₇	1	12	yes	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	6	no	—
205	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
208	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
209	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
210	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
211	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
212	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
213	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
214	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
215	M₁₆	—	134	48	no	—
216	C₂×C₈	—	67	8	no	—
217	C₂×C₈	—	67	4	no	—
218	C₂×C₈	—	67	12	no	—
219	C₂×C₈	—	134	12	no	—
220	C₄⋊C₄	C₂²×Dic₆₇	1	12	yes	—
221	C₄⋊C₄	C₂²×Dic₆₇	1	12	yes	—
222	C₂²⋊C₄	C₂²×Dic₆₇	1	24	yes	—
223	C₄²	C₂²×Dic₆₇	1	2	yes	—
224	C₄²	C₂²×Dic₆₇	1	12	yes	—
225	C₄²	C₂²×Dic₆₇	1	12	yes	—
226	C₄²	C₂²×Dic₆₇	1	6	yes	—
227	C₄²	C₂³×D₆₇	1	6	yes	—
228	C₄²	C₂³×D₆₇	1	2	yes	—
229	C₂³	—	2	6	no	—
230	C₂³	—	2	6	no	—
231	C₂³	—	2	6	no	—
232	C₂³	C₂³×Dic₆₇	1	6	yes	—
233	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
234	Q₈	C₂×#11761612	1	16	yes	complemented
235	D₄	—	2	24	no	—
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	6	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	24	no	—
242	C₂×C₄	—	2	12	no	—
243	C₂×C₄	—	4	12	no	—
244	C₂×C₄	C₂×#11761612	1	12	yes	—
245	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
246	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
247	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
248	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
249	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
251	C₂×C₄	#41×D₆₇	1	4	yes	—
252	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
253	C₈	—	67	8	no	—
254	C₈	—	134	12	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	2	6	no	—
257	C₂²	—	2	6	no	—
258	C₂²	—	4	12	no	—
259	C₂²	#41×Dic₆₇	1	2	yes	—
260	C₂²	C₂²×#11761612	1	2	yes	—
261	C₂²	#16864163	1	2	yes	—
262	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	6	no	—
266	C₄	—	2	6	no	—
267	C₄	#41×Dic₆₇	1	2	yes	—
268	C₄	C₂²×#11761612	1	12	yes	—
269	C₄	C₂×#41×D₆₇	1	2	yes	—
270	C₂	—	4	6	no	—
271	C₂	#22569229	1	4	yes	direct product factor
272	C₂	C₂×#41×Dic₆₇	1	1	yes	characteristic
273	C₂	C₂³×#11761612	1	1	yes	characteristic
274	C₂	C₂×#16864163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×C₄×#41×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×#41×C₆₇	C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×C₄×#41	D₆₇
hypocenter	C₆₇	C₂×#2971
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22569229	C₂	direct (N × H)
2. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
3. C₂×#16863474	C₂	semidirect
4. C₂×#16863453	C₂	semidirect
5. C₂×C₄×#11761612	C₂	semidirect
8. Q₈×C₆₇⋊C₈	C₂²	semidirect
9. #16863474	C₂²	semidirect
10. #16863453	C₂²	semidirect
13. C₄×#11761612	C₂²	semidirect
60. C₂×C₆₇⋊C₈	#41	semidirect
65. C₂×Q₈×C₆₇	M₁₆	semidirect
96. Q₈×C₆₇	C₂×M₁₆	semidirect
116. C₆₇⋊C₈	C₂×#41	semidirect
134. C₂×C₆₇	#2971	semidirect
144. C₆₇	C₂×#2971	semidirect
198. C₂×Q₈	#11761612	semidirect
234. Q₈	C₂×#11761612	semidirect
271. C₂	#22569229	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569229	2
C₂×Q₈×C₆₇⋊C₈	2
C₂×#16863474	2
C₂×#16863453	2
C₂×C₄×#11761612	2
C₂×C₄×#41×C₆₇	2
C₂×#2971	67

Subgroups — C2×#22569229 #31720333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569229 #31720333