Subgroups — C₂×#22558291 #30770684

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	2	yes	maximal
2	#22558291	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	2	no	—
5	C₂×#14523983	—	2	4	no	—
6	C₂×#14523983	C₄	1	8	yes	complemented
7	C₂×#14523983	C₂²	1	4	yes	—
8	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
9	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	#14523983	—	2	8	no	—
16	#14523983	—	2	4	no	—
17	#14523983	C₂×C₄	1	8	yes	complemented
18	C₂×C₆₇⋊C₁₆	—	2	8	no	—
19	C₂×C₆₇⋊C₁₆	—	2	4	no	—
20	C₂³×C₄×C₆₇	—	2	1	no	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₈	1	2	yes	—
25	C₂²×C₈×C₆₇	C₈	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	4	yes	—
32	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	—	2	2	no	—
36	C₂×C₄²×C₆₇	—	2	1	no	—
37	C₂×C₄²×C₆₇	C₈	1	4	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	—	2	4	no	—
40	C₄×C₈×C₆₇	—	2	4	no	—
41	C₆₇⋊C₁₆	—	2	8	no	—
42	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	8	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	C₄²	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	C₂×C₈	1	4	yes	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	#97	1	2	yes	—
78	C₂³×C₆₇	C₂²⋊C₈	1	2	yes	—
79	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
80	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
81	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₇	—	2	16	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	#95	1	2	yes	—
86	C₂×C₄×C₆₇	#95	1	2	yes	—
87	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₆₇	#97	1	2	yes	—
90	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₈×C₆₇	—	2	4	no	—
104	C₈×C₆₇	#95	1	2	yes	—
105	C₈×C₆₇	#95	1	2	yes	—
106	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂²×C₆₇	—	2	4	no	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	#336	1	2	yes	—
113	C₂²×C₆₇	#336	1	2	yes	—
114	C₂²×C₆₇	#364	1	4	yes	—
115	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
116	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₆₇	—	2	8	no	—
119	C₄×C₆₇	#336	1	2	yes	—
120	C₄×C₆₇	#336	1	2	yes	—
121	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
122	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
123	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂×#1376	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	2	2	no	—
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	#1376	1	4	yes	complemented
130	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
131	C₂×C₆₇	C₂×#364	1	2	yes	—
132	C₂²×M₃₂	—	67	2	no	—
133	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
134	#1376	—	67	4	no	—
135	C₆₇	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
137	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
139	C₂×M₃₂	—	67	8	no	—
140	C₂×M₃₂	—	67	4	no	—
141	C₂×M₃₂	—	134	2	no	—
142	C₂×M₃₂	—	134	4	no	—
143	C₂²×C₁₆	—	134	2	no	—
144	C₂×C₄×C₈	—	2	4	no	—
145	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
147	C₂³×C₄	—	2	1	no	—
148	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	C₆₇⋊C₈	1	2	yes	—
152	C₂²×C₈	C₆₇⋊C₈	1	2	yes	—
153	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
154	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
155	C₂²×C₈	C₄×D₆₇	1	4	yes	—
156	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
157	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
159	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
161	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₆₇⋊C₈	1	4	yes	—
165	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
166	M₃₂	—	67	8	no	—
167	M₃₂	—	134	8	no	—
168	M₃₂	—	134	4	no	—
169	C₂×C₁₆	—	134	8	no	—
170	C₂×C₁₆	—	134	4	no	—
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	—	2	4	no	—
173	C₂⁴	#11761612	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
179	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
184	C₂²×C₄	#11761612	1	1	yes	characteristic
185	C₂²×C₄	#11761612	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
192	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
193	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
194	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
195	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
196	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
197	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
200	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
202	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
203	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
207	C₁₆	—	134	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	#14523954	1	2	yes	—
210	C₂³	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
211	C₂³	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
212	C₂³	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
213	C₂³	C₂×#11761612	1	1	yes	characteristic
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₆₇⋊#95	1	2	yes	—
218	C₂×C₄	C₆₇⋊#95	1	2	yes	—
219	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
220	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
221	C₂×C₄	#14523954	1	2	yes	—
222	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
223	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
224	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
225	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
226	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
227	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
228	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
234	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	C₆₇⋊#95	1	2	yes	—
237	C₈	C₆₇⋊#95	1	2	yes	—
238	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
239	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	#16863151	1	2	yes	—
245	C₂²	#16863151	1	2	yes	—
246	C₂²	#16863217	1	4	yes	—
247	C₂²	C₂×#14523954	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
249	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	#16863151	1	2	yes	—
252	C₄	#16863151	1	2	yes	—
253	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
254	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
255	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
256	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	2	2	no	—
260	C₂	#22558291	1	4	yes	direct product factor
261	C₂	C₂×#16863151	1	1	yes	characteristic
262	C₂	C₂×#16863217	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₂₆₈
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₄⋊C₈
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#1376
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22558291	C₂	direct (N × H)
6. C₂×#14523983	C₄	semidirect
17. #14523983	C₂×C₄	semidirect
129. C₂×C₆₇	#1376	semidirect
135. C₆₇	C₂×#1376	semidirect
260. C₂	#22558291	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×#14523983	2
#22558291	2
C₂²×C₄×C₈×C₆₇	2
C₂×#1376	67

Subgroups — C2×#22558291 #30770684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558291 #30770684