Subgroups — C₂×#22560115 #30853932

Order 17152.

1046 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	2	yes	maximal
2	C₂×C₄×#11761612	C₂	1	2	yes	maximal
3	#22560115	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863161	C₂	1	2	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	4	yes	—
7	C₂×#14523979	C₄	1	16	yes	complemented
8	C₂×#14523979	C₂²	1	8	yes	—
9	C₄×#11761612	—	2	4	no	—
10	C₄×#11761612	C₂²	1	8	yes	—
11	C₂×#14523954	—	2	2	no	—
12	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
13	#16863161	C₂²	1	8	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₇	C₄	1	4	yes	—
18	C₂×#97×C₆₇	C₄	1	4	yes	—
19	C₂×#97×C₆₇	C₂²	1	4	yes	—
20	C₂×#11761612	—	2	4	no	—
21	C₂×#11761612	—	2	8	no	—
22	C₂×#11761612	C₂×C₄	1	16	yes	—
23	C₂×#11761612	C₂³	1	8	yes	—
24	C₂²×C₆₇⋊C₈	—	2	4	no	—
25	#14523979	C₂×C₄	1	32	yes	complemented
26	#14523954	—	2	8	no	—
27	C₄×C₆₇⋊C₈	—	2	8	no	—
28	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	4	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
44	#97×C₆₇	C₂×C₄	1	16	yes	—
45	#11761612	—	2	16	no	—
46	#11761612	—	2	8	no	—
47	#11761612	C₂²×C₄	1	16	yes	—
48	C₂×C₆₇⋊C₈	—	2	16	no	—
49	C₂×C₆₇⋊C₈	—	2	8	no	—
50	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	—	4	4	no	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	#41	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	—	2	4	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	C₄²	1	8	yes	—
67	C₂×C₈×C₆₇	C₄²	1	8	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
71	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
72	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
74	C₄²×C₆₇	C₂×D₄	1	4	yes	—
75	C₂³×C₆₇	—	2	4	no	—
76	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
79	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	8	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	—	4	4	no	—
83	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#117	1	2	yes	—
89	C₂×C₄×C₆₇	#117	1	2	yes	—
90	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
97	C₈×C₆₇	—	2	4	no	—
98	C₈×C₆₇	C₂×C₄²	1	8	yes	—
99	C₆₇⋊C₈	—	2	16	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	8	no	—
103	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
104	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
105	C₂²×C₆₇	#430	1	4	yes	—
106	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
108	C₄×C₆₇	—	2	8	no	—
109	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
110	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
113	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
114	C₂×#1778	—	67	1	no	maximal, Sylow
115	C₂×C₆₇	—	2	2	no	—
116	C₂×C₆₇	—	2	2	no	—
117	C₂×C₆₇	#1778	1	4	yes	complemented
118	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#430	1	2	yes	—
120	C₂²×#108	—	67	2	no	—
121	C₂×C₄×M₁₆	—	67	2	no	—
122	C₂²×#97	D₆₇	1	1	yes	characteristic
123	#1778	—	67	8	no	—
124	C₂×#341	—	67	2	no	—
125	C₆₇	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
126	C₂²×M₁₆	—	67	4	no	—
127	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
128	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
130	C₂×#108	—	67	16	no	—
131	C₂×#108	—	67	8	no	—
132	C₄×M₁₆	—	67	8	no	—
133	C₄×M₁₆	—	134	4	no	—
134	C₂×#97	—	134	2	no	—
135	C₂×#97	Dic₆₇	1	4	yes	—
136	C₂×#97	Dic₆₇	1	4	yes	—
137	C₂×#97	C₂×D₆₇	1	4	yes	—
138	C₂×C₄×C₈	—	134	2	no	—
139	#341	—	67	8	no	—
140	C₂³×C₄	Dic₁₃₄	1	2	yes	—
141	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
142	C₂×M₁₆	—	67	16	no	—
143	C₂×M₁₆	—	67	8	no	—
144	C₂×M₁₆	—	134	4	no	—
145	C₂×M₁₆	—	134	8	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	134	4	no	—
148	C₂²×C₈	C₄×D₆₇	1	8	yes	—
149	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
150	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
151	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
152	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
153	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
154	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
155	C₂×C₄²	Dic₁₃₄	1	2	yes	—
156	C₂×C₄²	D₂₆₈	1	2	yes	—
157	C₂×C₄²	D₂₆₈	1	2	yes	—
158	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
159	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
160	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
161	#108	—	67	32	no	—
162	#97	—	134	8	no	—
163	#97	C₂×Dic₆₇	1	16	yes	—
164	C₄×C₈	—	134	8	no	—
165	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
168	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
169	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
170	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
174	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
175	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
176	C₂²×C₄	#11761628	1	4	yes	—
177	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
178	M₁₆	—	67	16	no	—
179	M₁₆	—	134	16	no	—
180	M₁₆	—	134	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	134	16	no	—
185	C₂×C₈	—	134	8	no	—
186	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
187	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
188	C₂×C₈	C₂×C₄×D₆₇	1	8	yes	—
189	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
190	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
191	C₄²	C₄⋊Dic₆₇	1	4	yes	—
192	C₄²	C₄⋊Dic₆₇	1	4	yes	—
193	C₄²	C₂²×Dic₆₇	1	4	yes	—
194	C₄²	C₂×D₂₆₈	1	4	yes	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
198	C₂³	C₄×Dic₁₃₄	1	4	yes	—
199	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
208	C₂×C₄	#14524002	1	2	yes	—
209	C₂×C₄	#14524002	1	2	yes	—
210	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
211	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
212	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
213	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
214	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
216	C₂×C₄	C₂×#11761628	1	2	yes	—
217	C₈	—	2	4	no	—
218	C₈	—	134	16	no	—
219	C₈	C₂×C₄×Dic₆₇	1	8	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄×C₄⋊Dic₆₇	1	2	yes	—
224	C₂²	C₄×C₄⋊Dic₆₇	1	2	yes	—
225	C₂²	#16863445	1	4	yes	—
226	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
227	C₂²	C₂×C₄×Dic₁₃₄	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
230	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
231	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
232	C₄	C₂×#14524002	1	1	yes	characteristic
233	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	#22560115	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
238	C₂	C₂×#16863445	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#1778
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22560115	C₂	direct (N × H)
7. C₂×#14523979	C₄	semidirect
25. #14523979	C₂×C₄	semidirect
117. C₂×C₆₇	#1778	semidirect
125. C₆₇	C₂×#1778	semidirect
236. C₂	#22560115	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×#14523979	2
C₂×C₄×#11761612	2
#22560115	2
C₂×#16863161	2
C₂²×#97×C₆₇	2
C₂×#1778	67

Subgroups — C2×#22560115 #30853932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560115 #30853932