Subgroups — C₂×C₆₇×#2226 #30712872

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2226	C₂	1	8	yes	maximal, direct product factor
4	C₂×#367×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#357×C₆₇	C₂	1	2	yes	maximal
7	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
9	C₂×C₄×D₄×C₆₇	—	2	1	no	—
10	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
11	#500×C₆₇	—	2	2	no	—
12	#500×C₆₇	C₂²	1	4	yes	complemented
13	C₄×D₈×C₆₇	—	2	2	no	—
14	C₄×D₈×C₆₇	C₂²	1	4	yes	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#102×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#367×C₆₇	C₂²	1	8	yes	complemented
21	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
22	#357×C₆₇	C₂²	1	8	yes	—
23	C₂²×Q₈×C₆₇	—	4	1	no	—
24	C₂²×D₄×C₆₇	—	2	1	no	—
25	C₂³×C₄×C₆₇	—	4	1	no	—
26	C₂×D₈×C₆₇	—	2	1	no	—
27	C₂×D₈×C₆₇	—	2	2	no	—
28	C₂×D₈×C₆₇	Q₈	1	4	yes	complemented
29	C₂×D₈×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	—	2	4	no	—
33	C₄⋊Q₈×C₆₇	—	4	2	no	—
34	C₄×D₄×C₆₇	—	2	4	no	—
35	C₄×D₄×C₆₇	—	4	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
39	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₇	—	2	4	no	—
42	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
43	#107×C₆₇	—	2	2	no	—
44	#107×C₆₇	—	2	4	no	—
45	#107×C₆₇	—	2	1	no	—
46	#107×C₆₇	C₂³	1	4	yes	—
47	#107×C₆₇	C₂³	1	2	yes	—
48	#102×C₆₇	—	2	4	no	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₄×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂⁴×C₆₇	—	4	1	no	—
52	C₂×Q₈×C₆₇	—	4	4	no	—
53	C₂×Q₈×C₆₇	—	8	4	no	—
54	C₂×D₄×C₆₇	—	2	4	no	—
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	—	4	1	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	8	4	no	—
62	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	D₈×C₆₇	—	2	4	no	—
66	D₈×C₆₇	—	2	2	no	—
67	D₈×C₆₇	C₂×Q₈	1	4	yes	complemented
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₆₇	#41	1	2	yes	—
73	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	—	4	4	no	—
79	C₂²⋊C₄×C₆₇	—	4	4	no	—
80	C₄²×C₆₇	D₈	1	2	yes	—
81	C₄²×C₆₇	C₂×D₄	1	2	yes	—
82	C₁₆×C₆₇	—	2	4	no	—
83	C₂³×C₆₇	—	4	2	no	—
84	C₂³×C₆₇	—	4	4	no	—
85	C₂³×C₆₇	—	8	2	no	—
86	C₂³×C₆₇	—	8	2	no	—
87	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
88	Q₈×C₆₇	—	8	8	no	—
89	D₄×C₆₇	—	2	4	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	4	4	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	—	8	2	no	—
96	C₂×C₄×C₆₇	—	8	4	no	—
97	C₂×C₄×C₆₇	—	8	8	no	—
98	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
99	C₂×C₄×C₆₇	C₂²⋊Q₈	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	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₆₇	#137	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₆₇	—	4	4	no	—
110	C₂²×C₆₇	—	8	2	no	—
111	C₂²×C₆₇	—	8	2	no	—
112	C₂²×C₆₇	—	8	8	no	—
113	C₂²×C₆₇	#474	1	2	yes	—
114	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
115	C₂²×C₆₇	#509	1	2	yes	—
116	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	—	8	4	no	—
120	C₄×C₆₇	#474	1	2	yes	—
121	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
122	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
123	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
124	C₂×#2226	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₆₇	—	8	4	no	—
126	C₂×C₆₇	#2226	1	4	yes	direct product factor
127	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
128	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
130	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
131	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
132	#2226	C₂×C₆₇	1	8	yes	direct product factor
133	C₂×#367	C₂×C₆₇	1	2	yes	complemented
134	C₂×C₄⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
135	C₂×#357	C₂×C₆₇	1	2	yes	—
136	C₆₇	C₂×#2226	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	—	2	1	no	—
139	C₂×C₄×D₄	—	2	1	no	—
140	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
141	#500	—	2	2	no	—
142	#500	C₂²×C₆₇	1	4	yes	complemented
143	C₄×D₈	—	2	2	no	—
144	C₄×D₈	C₂²×C₆₇	1	4	yes	—
145	C₂×#107	—	2	1	no	—
146	C₂×#107	C₂²×C₆₇	1	2	yes	—
147	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
148	C₂×#102	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
150	#367	C₂²×C₆₇	1	8	yes	complemented
151	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	complemented
152	#357	C₂²×C₆₇	1	8	yes	—
153	C₂²×Q₈	—	4	1	no	—
154	C₂²×D₄	—	2	1	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	Q₈×C₆₇	1	4	yes	complemented
159	C₂×D₈	C₂³×C₆₇	1	2	yes	—
160	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
173	#107	—	2	4	no	—
174	#107	—	2	1	no	—
175	#107	—	2	2	no	—
176	#107	C₂³×C₆₇	1	4	yes	—
177	#107	C₂³×C₆₇	1	2	yes	—
178	#102	—	2	4	no	—
179	C₄×C₈	D₄×C₆₇	1	2	yes	—
180	C₄×C₈	C₂³×C₆₇	1	2	yes	—
181	C₂⁴	—	4	1	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	8	4	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	8	4	no	—
192	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
193	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
195	D₈	—	2	4	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂×Q₈×C₆₇	1	4	yes	complemented
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
202	C₂×C₈	#41×C₆₇	1	2	yes	—
203	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	D₈×C₆₇	1	2	yes	—
211	C₄²	C₂×D₄×C₆₇	1	2	yes	—
212	C₁₆	—	2	4	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	8	2	no	—
216	C₂³	—	8	2	no	—
217	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
218	Q₈	—	8	8	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	—	8	8	no	—
228	C₂×C₄	SD₃₂×C₆₇	1	4	yes	—
229	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
230	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
231	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
233	C₂×C₄	#137×C₆₇	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₂²⋊Q₈×C₆₇	1	2	yes	—
237	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
238	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
239	C₂²	—	4	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	8	no	—
243	C₂²	#474×C₆₇	1	2	yes	—
244	C₂²	C₂×SD₃₂×C₆₇	1	2	yes	—
245	C₂²	#509×C₆₇	1	2	yes	—
246	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#474×C₆₇	1	2	yes	—
251	C₄	C₂×SD₃₂×C₆₇	1	4	yes	—
252	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
253	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
254	C₂	—	8	4	no	—
255	C₂	C₆₇×#2226	1	4	yes	direct product factor
256	C₂	C₂×#474×C₆₇	1	1	yes	characteristic
257	C₂	C₂²×SD₃₂×C₆₇	1	1	yes	characteristic
258	C₂	C₂×#509×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₆₇	C₂	semidirect
3. C₆₇×#2226	C₂	direct (N × H)
4. C₂×#367×C₆₇	C₂	semidirect
5. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
12. #500×C₆₇	C₂²	semidirect
20. #367×C₆₇	C₂²	semidirect
21. C₄⋊C₁₆×C₆₇	C₂²	semidirect
28. C₂×D₈×C₆₇	Q₈	semidirect
67. D₈×C₆₇	C₂×Q₈	semidirect
124. C₂×#2226	C₆₇	direct (N × H)
126. C₂×C₆₇	#2226	direct (N × H)
130. C₂×#500	C₂×C₆₇	semidirect
132. #2226	C₂×C₆₇	direct (N × H)
133. C₂×#367	C₂×C₆₇	semidirect
134. C₂×C₄⋊C₁₆	C₂×C₆₇	semidirect
136. C₆₇	C₂×#2226	direct (N × H)
142. #500	C₂²×C₆₇	semidirect
150. #367	C₂²×C₆₇	semidirect
151. C₄⋊C₁₆	C₂²×C₆₇	semidirect
158. C₂×D₈	Q₈×C₆₇	semidirect
197. D₈	C₂×Q₈×C₆₇	semidirect
255. C₂	C₆₇×#2226	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#500×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2226	2
C₂×#367×C₆₇	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#357×C₆₇	2
C₂×#2226	67

Subgroups — C2×C67×#2226 #30712872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2226 #30712872