Subgroups — C₂×C₆₇×#2215 #30712848

Order 17152.

806 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×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#466×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2215	C₂	1	8	yes	maximal, direct product factor
5	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#357×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#121×C₆₇	—	2	2	no	—
10	C₂×D₁₆×C₆₇	—	2	4	no	—
11	C₂×D₁₆×C₆₇	C₂²	1	4	yes	complemented
12	C₂²×C₁₆×C₆₇	—	2	4	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	complemented
16	#466×C₆₇	—	2	4	no	—
17	#466×C₆₇	C₂²	1	8	yes	complemented
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#102×C₆₇	—	2	2	no	—
20	#366×C₆₇	C₂²	1	4	yes	complemented
21	#357×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×D₄×C₆₇	—	2	2	no	—
23	C₂²×D₄×C₆₇	—	4	2	no	—
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂×D₈×C₆₇	—	2	4	no	—
26	C₂×D₈×C₆₇	—	2	2	no	—
27	C₂×D₈×C₆₇	—	2	4	no	—
28	C₂×D₈×C₆₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	D₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	#121×C₆₇	—	2	8	no	—
36	#121×C₆₇	—	4	4	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
39	D₁₆×C₆₇	—	2	8	no	—
40	C₂×C₁₆×C₆₇	—	2	2	no	—
41	C₂×C₁₆×C₆₇	—	2	8	no	—
42	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
45	#107×C₆₇	—	2	2	no	—
46	#107×C₆₇	C₂³	1	4	yes	—
47	#102×C₆₇	—	2	8	no	—
48	C₂⁴×C₆₇	—	4	2	no	—
49	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
50	C₂×D₄×C₆₇	—	2	4	no	—
51	C₂×D₄×C₆₇	—	4	8	no	—
52	C₂×D₄×C₆₇	—	4	4	no	—
53	C₂×D₄×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	4	4	no	—
55	C₂×D₄×C₆₇	—	8	8	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	D₈	1	2	yes	—
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	D₈×C₆₇	—	2	4	no	—
64	D₈×C₆₇	—	2	4	no	—
65	D₈×C₆₇	—	4	4	no	—
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	#41	1	2	yes	—
74	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₂²⋊C₄×C₆₇	—	4	8	no	—
78	C₁₆×C₆₇	—	2	2	no	—
79	C₁₆×C₆₇	C₂×D₄	1	4	yes	complemented
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	4	4	no	—
83	C₂³×C₆₇	—	8	4	no	—
84	C₂³×C₆₇	—	8	4	no	—
85	C₂³×C₆₇	—	8	4	no	—
86	C₂³×C₆₇	D₁₆	1	4	yes	complemented
87	C₂³×C₆₇	C₂×D₈	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	D₄×C₆₇	—	4	4	no	—
90	D₄×C₆₇	—	4	4	no	—
91	D₄×C₆₇	—	8	16	no	—
92	D₄×C₆₇	—	8	4	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	8	4	no	—
99	C₂×C₄×C₆₇	#121	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#135	1	2	yes	—
103	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	#121	1	2	yes	—
106	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	8	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	8	4	no	—
112	C₂²×C₆₇	—	8	4	no	—
113	C₂²×C₆₇	—	8	4	no	—
114	C₂²×C₆₇	—	16	8	no	—
115	C₂²×C₆₇	#466	1	2	yes	—
116	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
117	C₂²×C₆₇	C₂×D₁₆	1	4	yes	complemented
118	C₂²×C₆₇	#508	1	2	yes	—
119	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	8	4	no	—
122	C₄×C₆₇	#466	1	2	yes	—
123	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
124	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
125	C₂×#2215	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	—	2	4	no	—
127	C₂×C₆₇	—	16	4	no	—
128	C₂×C₆₇	#2215	1	4	yes	direct product factor
129	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
130	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
131	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
132	C₂²×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
133	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
134	C₂×#466	C₂×C₆₇	1	2	yes	complemented
135	#2215	C₂×C₆₇	1	8	yes	direct product factor
136	C₂×#366	C₂×C₆₇	1	1	yes	characteristic, complemented
137	C₂×#357	C₂×C₆₇	1	2	yes	complemented
138	C₆₇	C₂×#2215	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
140	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
141	C₂×#121	—	2	2	no	—
142	C₂×D₁₆	—	2	4	no	—
143	C₂×D₁₆	C₂²×C₆₇	1	4	yes	complemented
144	C₂²×C₁₆	—	2	4	no	—
145	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
146	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	complemented
148	#466	—	2	4	no	—
149	#466	C₂²×C₆₇	1	8	yes	complemented
150	C₂×#107	C₂²×C₆₇	1	2	yes	—
151	C₂×#102	—	2	2	no	—
152	#366	C₂²×C₆₇	1	4	yes	complemented
153	#357	C₂²×C₆₇	1	8	yes	complemented
154	C₂²×D₄	—	2	2	no	—
155	C₂²×D₄	—	4	2	no	—
156	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
157	C₂×D₈	—	2	4	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	C₂³×C₆₇	1	4	yes	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₆₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
166	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
167	#121	—	2	8	no	—
168	#121	—	4	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	4	2	no	—
171	D₁₆	—	2	8	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	8	no	—
174	C₂×C₁₆	D₄×C₆₇	1	4	yes	complemented
175	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
176	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
177	#107	—	2	2	no	—
178	#107	C₂³×C₆₇	1	4	yes	—
179	#102	—	2	8	no	—
180	C₂⁴	—	4	2	no	—
181	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	4	8	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	8	8	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
192	C₂²×C₄	D₈×C₆₇	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
195	D₈	—	2	4	no	—
196	D₈	—	2	4	no	—
197	D₈	—	4	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
205	C₂×C₈	#41×C₆₇	1	2	yes	—
206	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	C₂×D₄×C₆₇	1	4	yes	complemented
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	4	no	—
216	C₂³	—	8	4	no	—
217	C₂³	—	8	4	no	—
218	C₂³	D₁₆×C₆₇	1	4	yes	complemented
219	C₂³	C₂×D₈×C₆₇	1	2	yes	—
220	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
221	D₄	—	4	4	no	—
222	D₄	—	4	4	no	—
223	D₄	—	8	16	no	—
224	D₄	—	8	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	#121×C₆₇	1	2	yes	—
232	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
234	C₂×C₄	#135×C₆₇	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#121×C₆₇	1	2	yes	—
238	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
239	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	16	8	no	—
247	C₂²	#466×C₆₇	1	2	yes	—
248	C₂²	C₂×D₁₆×C₆₇	1	2	yes	—
249	C₂²	C₂×D₁₆×C₆₇	1	4	yes	complemented
250	C₂²	#508×C₆₇	1	2	yes	—
251	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	#466×C₆₇	1	2	yes	—
255	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
256	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	16	4	no	—
259	C₂	C₆₇×#2215	1	4	yes	direct product factor
260	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
261	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
262	C₂	C₂×#508×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆×C₆₇	C₂	semidirect
2. C₂³×C₁₆×C₆₇	C₂	semidirect
3. C₂×#466×C₆₇	C₂	semidirect
4. C₆₇×#2215	C₂	direct (N × H)
5. C₂×#366×C₆₇	C₂	semidirect
6. C₂×#357×C₆₇	C₂	semidirect
11. C₂×D₁₆×C₆₇	C₂²	semidirect
15. C₂²×C₁₆×C₆₇	C₂²	semidirect
17. #466×C₆₇	C₂²	semidirect
20. #366×C₆₇	C₂²	semidirect
21. #357×C₆₇	C₂²	semidirect
42. C₂×C₁₆×C₆₇	D₄	semidirect
79. C₁₆×C₆₇	C₂×D₄	semidirect
86. C₂³×C₆₇	D₁₆	semidirect
117. C₂²×C₆₇	C₂×D₁₆	semidirect
125. C₂×#2215	C₆₇	direct (N × H)
128. C₂×C₆₇	#2215	direct (N × H)
132. C₂²×D₁₆	C₂×C₆₇	semidirect
133. C₂³×C₁₆	C₂×C₆₇	semidirect
134. C₂×#466	C₂×C₆₇	semidirect
135. #2215	C₂×C₆₇	direct (N × H)
136. C₂×#366	C₂×C₆₇	semidirect
137. C₂×#357	C₂×C₆₇	semidirect
138. C₆₇	C₂×#2215	direct (N × H)
143. C₂×D₁₆	C₂²×C₆₇	semidirect
147. C₂²×C₁₆	C₂²×C₆₇	semidirect
149. #466	C₂²×C₆₇	semidirect
152. #366	C₂²×C₆₇	semidirect
153. #357	C₂²×C₆₇	semidirect
174. C₂×C₁₆	D₄×C₆₇	semidirect
211. C₁₆	C₂×D₄×C₆₇	semidirect
218. C₂³	D₁₆×C₆₇	semidirect
249. C₂²	C₂×D₁₆×C₆₇	semidirect
259. C₂	C₆₇×#2215	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×D₁₆×C₆₇	2
C₂³×C₁₆×C₆₇	2
C₂×#466×C₆₇	2
C₆₇×#2215	2
C₂×#366×C₆₇	2
C₂×#357×C₆₇	2
C₂×#2215	67

Subgroups — C2×C67×#2215 #30712848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2215 #30712848