Subgroups — C₂×#22562021 #31008953

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562021	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16863217	C₂	1	2	yes	maximal, complemented
3	C₂×#16863178	C₂	1	2	yes	maximal
4	C₂×#16863174	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	4	yes	—
7	#16863217	C₂²	1	8	yes	complemented
8	#16863178	C₂²	1	8	yes	—
9	#16863174	C₂²	1	8	yes	complemented
10	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
13	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
14	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
15	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#14523983	—	2	4	no	—
20	#14523983	C₂³	1	8	yes	—
21	C₂×C₆₇⋊C₁₆	—	2	4	no	—
22	C₂³×C₄×C₆₇	—	2	1	no	—
23	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	—	2	1	no	—
36	C₂×C₄²×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
40	#97×C₆₇	C₂×C₄	1	4	yes	—
41	#97×C₆₇	C₂×C₄	1	2	yes	—
42	#97×C₆₇	C₂³	1	2	yes	—
43	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	C₂³	1	2	yes	—
46	C₆₇⋊C₁₆	—	2	8	no	—
47	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	2	1	no	—
50	C₂²×C₄×C₆₇	—	2	1	no	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	—	2	1	no	—
53	C₂²×C₄×C₆₇	—	2	4	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	4	4	no	—
56	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	M₁₆×C₆₇	—	2	4	no	—
61	M₁₆×C₆₇	—	2	8	no	—
62	M₁₆×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	2	1	no	—
65	C₂×C₈×C₆₇	—	2	1	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
68	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₆₇	#41	1	2	yes	—
74	C₂×C₈×C₆₇	#41	1	2	yes	—
75	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₄²×C₆₇	—	2	1	no	—
77	C₄²×C₆₇	—	2	4	no	—
78	C₄²×C₆₇	—	2	1	no	—
79	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₆₇	—	2	1	no	—
82	C₂³×C₆₇	—	2	1	no	—
83	C₂³×C₆₇	—	4	2	no	—
84	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	8	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	#117	1	2	yes	—
107	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
108	C₈×C₆₇	C₂×#41	1	2	yes	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	—	4	2	no	—
114	C₂²×C₆₇	#432	1	1	yes	characteristic
115	C₂²×C₆₇	#432	1	1	yes	characteristic
116	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	4	2	no	—
120	C₄×C₆₇	#432	1	1	yes	characteristic
121	C₄×C₆₇	#432	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
123	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×#2165	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	1	no	—
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	4	2	no	—
128	C₂×C₆₇	#2165	1	2	yes	complemented
129	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
130	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
131	#2165	—	67	8	no	—
132	C₂×#364	—	67	2	no	—
133	C₂×#349	—	67	2	no	—
134	C₂×#347	—	67	2	no	—
135	C₆₇	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
136	C₂²×M₁₆	Dic₆₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
138	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
139	C₂×M₃₂	—	67	4	no	—
140	C₄×M₁₆	Dic₆₇	1	4	yes	—
141	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
142	C₂×#97	Dic₆₇	1	1	yes	characteristic
143	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
144	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	#364	—	67	8	no	—
147	#349	—	67	8	no	—
148	#347	—	67	8	no	—
149	C₂³×C₄	—	2	1	no	—
150	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
156	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
157	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
158	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
167	M₃₂	—	67	8	no	—
168	M₃₂	—	134	4	no	—
169	C₂×C₁₆	—	134	4	no	—
170	#97	C₂×Dic₆₇	1	4	yes	—
171	#97	C₂×Dic₆₇	1	2	yes	—
172	#97	C₂²×D₆₇	1	2	yes	—
173	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
174	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
175	C₄×C₈	C₂²×D₆₇	1	2	yes	—
176	C₂⁴	#11761612	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	#11761612	1	1	yes	characteristic
186	C₂²×C₄	#11761612	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	8	no	—
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	#11761612	1	2	yes	—
197	C₂×C₈	#11761612	1	2	yes	—
198	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
199	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
200	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
202	C₂×C₈	#11761628	1	2	yes	—
203	C₂×C₈	#11761628	1	2	yes	—
204	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
205	C₄²	—	2	1	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂²×Dic₆₇	1	2	yes	—
209	C₄²	C₂²×Dic₆₇	1	2	yes	—
210	C₁₆	—	134	8	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂×#11761612	1	1	yes	characteristic
215	C₂³	C₂×#11761612	1	2	yes	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	#14524002	1	1	yes	characteristic
228	C₂×C₄	#14524002	1	1	yes	characteristic
229	C₂×C₄	C₂×#11761612	1	2	yes	—
230	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
231	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
233	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	#14524002	1	2	yes	—
237	C₈	C₂×#11761612	1	4	yes	—
238	C₈	C₂×#11761628	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#16863453	1	1	yes	characteristic
245	C₂²	#16863453	1	1	yes	characteristic
246	C₂²	C₂²×#11761612	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	#16863453	1	1	yes	characteristic
251	C₄	#16863453	1	1	yes	characteristic
252	C₄	C₂×#14524002	1	1	yes	characteristic
253	C₄	C₂²×#11761612	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#22562021	1	2	yes	direct product factor
258	C₂	C₂×#16863453	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₂×#2165
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #22562021	C₂	direct (N × H)
2. C₂×#16863217	C₂	semidirect
4. C₂×#16863174	C₂	semidirect
7. #16863217	C₂²	semidirect
9. #16863174	C₂²	semidirect
128. C₂×C₆₇	#2165	semidirect
135. C₆₇	C₂×#2165	semidirect
257. C₂	#22562021	direct (N × H)

Maximal subgroups

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

subgroup	index
#22562021	2
C₂×#16863217	2
C₂×#16863178	2
C₂×#16863174	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#2165	67

Subgroups — C2×#22562021 #31008953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562021 #31008953