Subgroups — C₂×#22558293 #30770690

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#22558293	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	4	no	—
5	C₂×#14523983	—	2	2	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×#97×C₆₇	—	2	2	no	—
13	C₂×#97×C₆₇	—	2	2	no	—
14	C₂×#97×C₆₇	C₄	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	#14523983	—	2	8	no	—
17	#14523983	—	2	4	no	—
18	#14523983	C₂×C₄	1	8	yes	complemented
19	C₂×C₆₇⋊C₁₆	—	2	8	no	—
20	C₂×C₆₇⋊C₁₆	—	2	4	no	—
21	C₂³×C₄×C₆₇	C₈	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	—	2	1	no	—
24	C₂²×C₈×C₆₇	—	2	1	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₈	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₂²×C₈×C₆₇	D₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	2	yes	—
33	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	—	2	1	no	—
36	C₂×C₄²×C₆₇	—	2	1	no	—
37	C₂×C₄²×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	—	2	1	no	—
39	C₂×C₄²×C₆₇	C₈	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	#97×C₆₇	—	2	4	no	—
42	#97×C₆₇	—	2	4	no	—
43	C₆₇⋊C₁₆	—	2	8	no	—
44	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₄×C₆₇	—	4	4	no	—
48	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
56	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₄²	1	2	yes	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	—	2	4	no	—
80	C₂³×C₆₇	C₄×C₈	1	2	yes	—
81	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
82	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
84	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	4	4	no	—
87	C₂×C₄×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	#95	1	2	yes	—
90	C₂×C₄×C₆₇	#95	1	2	yes	—
91	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₆₇	#97	1	2	yes	—
93	C₂×C₄×C₆₇	#97	1	2	yes	—
94	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
100	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₆₇	—	2	4	no	—
108	C₈×C₆₇	#95	1	4	yes	—
109	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	2	4	no	—
115	C₂²×C₆₇	#336	1	2	yes	—
116	C₂²×C₆₇	#336	1	2	yes	—
117	C₂²×C₆₇	#349	1	4	yes	—
118	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₄×C₆₇	—	4	4	no	—
122	C₄×C₆₇	#336	1	2	yes	—
123	C₄×C₆₇	#336	1	2	yes	—
124	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
125	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
126	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×#1377	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	2	1	no	—
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	#1377	1	4	yes	complemented
133	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#349	1	2	yes	—
135	C₂²×M₃₂	—	67	2	no	—
136	C₂²×#97	D₆₇	1	1	yes	characteristic
137	#1377	—	67	4	no	—
138	C₆₇	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
142	C₂×M₃₂	—	67	8	no	—
143	C₂×M₃₂	—	67	4	no	—
144	C₂×M₃₂	—	134	4	no	—
145	C₂×M₃₂	—	134	2	no	—
146	C₂²×C₁₆	—	134	2	no	—
147	C₂×#97	—	2	2	no	—
148	C₂×#97	—	2	2	no	—
149	C₂×#97	Dic₆₇	1	2	yes	—
150	C₂×#97	C₂×D₆₇	1	2	yes	—
151	C₂³×C₄	C₆₇⋊C₈	1	2	yes	—
152	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	C₆₇⋊C₈	1	4	yes	—
157	C₂²×C₈	Dic₁₃₄	1	2	yes	—
158	C₂²×C₈	C₄×D₆₇	1	4	yes	—
159	C₂²×C₈	D₂₆₈	1	2	yes	—
160	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
161	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
164	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₆₇⋊C₈	1	2	yes	—
170	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
171	M₃₂	—	67	8	no	—
172	M₃₂	—	134	8	no	—
173	M₃₂	—	134	4	no	—
174	C₂×C₁₆	—	134	8	no	—
175	C₂×C₁₆	—	134	4	no	—
176	#97	—	2	4	no	—
177	#97	—	2	4	no	—
178	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
186	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
187	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
189	C₂²×C₄	#11761612	1	2	yes	—
190	C₂²×C₄	#11761612	1	1	yes	characteristic
191	C₂²×C₄	#11761612	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
198	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
199	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
200	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
201	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
202	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
203	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
204	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
205	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
206	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
207	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
208	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₁₆	—	134	8	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₄×C₆₇⋊C₈	1	2	yes	—
216	C₂³	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
217	C₂³	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
218	C₂³	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
219	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₆₇⋊#95	1	2	yes	—
225	C₂×C₄	C₆₇⋊#95	1	2	yes	—
226	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
227	C₂×C₄	#14523954	1	2	yes	—
228	C₂×C₄	#14523954	1	2	yes	—
229	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
230	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
231	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
233	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
234	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
235	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
236	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
241	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	C₆₇⋊#95	1	4	yes	—
244	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
245	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	#16863151	1	2	yes	—
251	C₂²	#16863151	1	2	yes	—
252	C₂²	#16863178	1	4	yes	—
253	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
254	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
255	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
256	C₄	—	4	4	no	—
257	C₄	#16863151	1	2	yes	—
258	C₄	#16863151	1	2	yes	—
259	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
260	C₄	C₂×#14523954	1	1	yes	characteristic
261	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
262	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	2	no	—
265	C₂	—	2	1	no	—
266	C₂	#22558293	1	4	yes	direct product factor
267	C₂	C₂×#16863151	1	1	yes	characteristic
268	C₂	C₂×#16863178	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂²⋊C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#1377
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22558293	C₂	direct (N × H)
6. C₂×#14523983	C₄	semidirect
18. #14523983	C₂×C₄	semidirect
132. C₂×C₆₇	#1377	semidirect
138. C₆₇	C₂×#1377	semidirect
266. C₂	#22558293	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
#22558293	2
C₂²×#97×C₆₇	2
C₂×#1377	67

Subgroups — C2×#22558293 #30770690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558293 #30770690