Subgroups — C₂×#22560326 #30855974

Order 17152.

1246 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
2	#22560326	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863151	C₂	1	4	yes	maximal
4	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
8	#16863151	C₂²	1	16	yes	—
9	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₄³×C₆₇	C₄	1	4	yes	—
15	C₄³×C₆₇	C₂²	1	4	yes	—
16	C₂²×C₆₇⋊C₈	—	2	8	no	—
17	C₂²×C₆₇⋊C₈	—	2	16	no	—
18	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
19	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	—	2	8	no	—
26	C₂×C₄²×C₆₇	—	2	4	no	—
27	C₂×C₄²×C₆₇	—	2	8	no	—
28	C₂×C₄²×C₆₇	C₈	1	8	yes	complemented
29	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₆₇	D₄	1	4	yes	—
35	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂×C₆₇⋊C₈	—	2	8	no	—
39	C₂×C₆₇⋊C₈	—	2	16	no	—
40	C₂×C₆₇⋊C₈	—	4	16	no	—
41	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	—	2	4	no	—
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
52	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₆₇	#41	1	8	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₄²×C₆₇	—	2	8	no	—
63	C₄²×C₆₇	—	2	16	no	—
64	C₄²×C₆₇	—	2	8	no	—
65	C₄²×C₆₇	—	2	8	no	—
66	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
67	C₄²×C₆₇	C₂×C₈	1	16	yes	complemented
68	C₄²×C₆₇	C₂×D₄	1	4	yes	—
69	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₆₇	#117	1	4	yes	—
72	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
73	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
74	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	8	no	—
82	C₂×C₄×C₆₇	—	2	16	no	—
83	C₂×C₄×C₆₇	—	2	16	no	—
84	C₂×C₄×C₆₇	C₄⋊C₈	1	16	yes	—
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₆₇	#117	1	4	yes	—
89	C₂×C₄×C₆₇	#124	1	8	yes	—
90	C₂×C₄×C₆₇	#125	1	8	yes	—
91	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
92	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
94	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
98	C₆₇⋊C₈	—	4	8	no	—
99	C₂²×C₆₇	#382	1	4	yes	—
100	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
101	C₂²×C₆₇	#423	1	4	yes	—
102	C₂²×C₆₇	#431	1	8	yes	—
103	C₂²×C₆₇	#432	1	8	yes	—
104	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#117	1	2	yes	—
106	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
107	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
108	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
109	C₄×C₆₇	—	2	4	no	—
110	C₄×C₆₇	—	2	8	no	—
111	C₄×C₆₇	—	2	8	no	—
112	C₄×C₆₇	#382	1	4	yes	—
113	C₄×C₆₇	C₂×C₄⋊C₈	1	8	yes	—
114	C₄×C₆₇	C₂×#124	1	2	yes	—
115	C₄×C₆₇	C₂×#125	1	2	yes	—
116	C₂×#1831	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	#1831	1	8	yes	complemented
118	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
119	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#431	1	2	yes	—
122	C₂×C₆₇	C₂×#432	1	2	yes	—
123	C₂²×C₄⋊C₈	—	67	2	no	—
124	C₂×C₄³	D₆₇	1	1	yes	characteristic
125	#1831	—	67	8	no	—
126	C₂×#336	—	67	4	no	—
127	C₆₇	C₂×#1831	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₈	—	67	4	no	—
129	C₂²×C₄²	Dic₆₇	1	2	yes	—
130	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
131	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
132	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
133	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	67	8	no	—
135	C₂×C₄⋊C₈	—	134	8	no	—
136	C₄³	Dic₆₇	1	4	yes	—
137	C₄³	C₂×D₆₇	1	4	yes	—
138	#336	—	67	16	no	—
139	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
140	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
141	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
142	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
144	C₂²×C₈	—	67	8	no	—
145	C₂²×C₈	—	134	8	no	—
146	C₂²×C₈	—	134	16	no	—
147	C₂×C₄²	—	2	8	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	—	2	8	no	—
150	C₂×C₄²	C₆₇⋊C₈	1	8	yes	complemented
151	C₂×C₄²	Dic₁₃₄	1	4	yes	—
152	C₂×C₄²	D₂₆₈	1	4	yes	—
153	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
154	C₂×C₄²	C₂×Dic₆₇	1	8	yes	—
155	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
156	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
157	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
158	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
159	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
160	C₄⋊C₈	—	134	16	no	—
161	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
168	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
169	C₂²×C₄	C₂×C₆₇⋊C₈	1	8	yes	—
170	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
171	C₂²×C₄	#11761612	1	8	yes	—
172	C₂²×C₄	#11761612	1	4	yes	—
173	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
175	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
178	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
179	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
180	C₂²×C₄	#11761628	1	8	yes	—
181	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
182	C₂×C₈	—	268	16	no	—
183	C₂×C₈	—	134	8	no	—
184	C₂×C₈	—	134	16	no	—
185	C₄²	—	2	8	no	—
186	C₄²	—	2	16	no	—
187	C₄²	—	2	8	no	—
188	C₄²	—	2	8	no	—
189	C₄²	C₄⋊Dic₆₇	1	8	yes	—
190	C₄²	C₂×C₆₇⋊C₈	1	16	yes	complemented
191	C₄²	C₂×D₂₆₈	1	4	yes	—
192	C₄²	C₂×Dic₁₃₄	1	4	yes	—
193	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
194	C₂³	#14524002	1	4	yes	—
195	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
196	C₂³	C₂×#11761612	1	2	yes	—
197	C₂³	C₂×#11761612	1	4	yes	—
198	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	16	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
211	C₂×C₄	#14524002	1	4	yes	—
212	C₂×C₄	#14524031	1	8	yes	—
213	C₂×C₄	#14524035	1	8	yes	—
214	C₂×C₄	C₂²×C₆₇⋊C₈	1	8	yes	—
215	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
216	C₂×C₄	C₂×#11761612	1	8	yes	—
217	C₂×C₄	C₂×#11761612	1	4	yes	—
218	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
219	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761628	1	4	yes	—
221	C₈	—	268	8	no	—
222	C₂²	#16863262	1	4	yes	—
223	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	4	yes	—
224	C₂²	#16863420	1	4	yes	—
225	C₂²	#16863449	1	8	yes	—
226	C₂²	#16863453	1	8	yes	—
227	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
228	C₂²	C₂×#14524002	1	2	yes	—
229	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
230	C₂²	C₂²×#11761612	1	1	yes	characteristic
231	C₂²	C₂²×#11761612	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	8	no	—
234	C₄	—	2	8	no	—
235	C₄	#16863262	1	4	yes	—
236	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	8	yes	—
237	C₄	C₂×#14524031	1	2	yes	—
238	C₄	C₂×#14524035	1	2	yes	—
239	C₂	#22560326	1	8	yes	direct product factor
240	C₂	C₂×#16863262	1	1	yes	characteristic
241	C₂	C₂²×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
242	C₂	C₂×#16863420	1	1	yes	characteristic
243	C₂	C₂×#16863449	1	2	yes	—
244	C₂	C₂×#16863453	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄³×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	C₂×#1831
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22560326	C₂	direct (N × H)
28. C₂×C₄²×C₆₇	C₈	semidirect
67. C₄²×C₆₇	C₂×C₈	semidirect
117. C₂×C₆₇	#1831	semidirect
127. C₆₇	C₂×#1831	semidirect
150. C₂×C₄²	C₆₇⋊C₈	semidirect
190. C₄²	C₂×C₆₇⋊C₈	semidirect
239. C₂	#22560326	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×C₄⋊(C₆₇⋊C₈)	2
#22560326	2
C₂×#16863151	2
C₂×C₄³×C₆₇	2
C₂×#1831	67

Subgroups — C2×#22560326 #30855974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560326 #30855974