Subgroups — C₂×C₆₇×#2884 #30739320

Order 17152.

4818 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#519×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2884	C₂	1	16	yes	maximal, direct product factor
4	C₂×#410×C₆₇	C₂	1	12	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
6	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
7	C₂×2₋¹⁺⁴×C₆₇	C₄	1	8	yes	complemented
8	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	8	yes	—
9	C₂²×#41×C₆₇	C₄	1	4	yes	complemented
10	C₂²×#41×C₆₇	C₂²	1	6	yes	—
11	C₂³×Q₈×C₆₇	C₄	1	4	yes	complemented
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#519×C₆₇	—	2	8	no	—
14	#519×C₆₇	C₂²	1	16	yes	complemented
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×#117×C₆₇	C₂²	1	12	yes	—
17	#410×C₆₇	C₂²	1	96	yes	complemented
18	C₂×#99×C₆₇	C₂²	1	16	yes	—
19	2₋¹⁺⁴×C₆₇	—	2	64	no	—
20	2₋¹⁺⁴×C₆₇	—	2	64	no	—
21	C₂×#41×C₆₇	—	2	16	no	—
22	C₂×#41×C₆₇	—	2	16	no	—
23	C₂×#41×C₆₇	—	2	24	no	—
24	C₂×#41×C₆₇	—	2	24	no	—
25	C₂×#41×C₆₇	C₂×C₄	1	24	yes	—
26	C₂×#41×C₆₇	C₂×C₄	1	32	yes	complemented
27	C₂×#41×C₆₇	C₂³	1	24	yes	—
28	C₂²×Q₈×C₆₇	—	2	16	no	—
29	C₂²×Q₈×C₆₇	—	2	16	no	—
30	C₂²×Q₈×C₆₇	C₂×C₄	1	6	yes	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	32	yes	complemented
34	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
35	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
37	C₂²×D₄×C₆₇	C₂×C₄	1	6	yes	—
38	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
41	C₄×Q₈×C₆₇	—	2	64	no	—
42	#117×C₆₇	—	2	24	no	—
43	#117×C₆₇	C₂³	1	48	yes	—
44	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
45	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
46	C₂×C₄²×C₆₇	—	2	12	no	—
47	#99×C₆₇	C₂³	1	64	yes	—
48	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
50	#41×C₆₇	—	2	128	no	—
51	#41×C₆₇	—	2	96	no	—
52	#41×C₆₇	—	2	96	no	—
53	C₂×Q₈×C₆₇	—	2	128	no	—
54	C₂×Q₈×C₆₇	—	2	12	no	—
55	C₂×Q₈×C₆₇	—	2	12	no	—
56	C₂×Q₈×C₆₇	C₂²⋊C₄	1	16	yes	—
57	C₂×Q₈×C₆₇	C₂²⋊C₄	1	16	yes	—
58	C₂×Q₈×C₆₇	C₂²⋊C₄	1	32	yes	—
59	C₂×Q₈×C₆₇	C₂²×C₄	1	24	yes	—
60	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
61	C₂×Q₈×C₆₇	C₂×D₄	1	32	yes	—
62	C₂×D₄×C₆₇	—	2	8	no	—
63	C₂×D₄×C₆₇	—	2	8	no	—
64	C₂×D₄×C₆₇	—	2	12	no	—
65	C₂×D₄×C₆₇	—	2	12	no	—
66	C₂×D₄×C₆₇	C₂²×C₄	1	24	yes	—
67	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
68	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₆₇	—	2	24	no	—
70	C₂²×C₄×C₆₇	—	2	24	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	48	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
79	C₄⋊C₄×C₆₇	—	2	48	no	—
80	C₂²⋊C₄×C₆₇	—	2	8	no	—
81	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
82	C₄²×C₆₇	—	2	48	no	—
83	C₂³×C₆₇	—	2	4	no	—
84	C₂³×C₆₇	—	2	4	no	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	—	4	8	no	—
87	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
88	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
89	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
91	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
92	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
93	Q₈×C₆₇	—	2	48	no	—
94	Q₈×C₆₇	C₂×C₂²⋊C₄	1	64	yes	—
95	D₄×C₆₇	—	2	16	no	—
96	D₄×C₆₇	—	2	16	no	—
97	D₄×C₆₇	—	2	48	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	96	no	—
101	C₂×C₄×C₆₇	—	2	16	no	—
102	C₂×C₄×C₆₇	—	4	16	no	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	24	yes	—
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	8	no	—
112	C₂²×C₆₇	—	4	1	no	—
113	C₂²×C₆₇	—	4	2	no	—
114	C₂²×C₆₇	—	4	1	no	—
115	C₂²×C₆₇	—	4	8	no	—
116	C₂²×C₆₇	—	4	8	no	—
117	C₂²×C₆₇	—	4	8	no	—
118	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
120	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
122	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
123	C₄×C₆₇	—	2	8	no	—
124	C₄×C₆₇	—	4	16	no	—
125	C₄×C₆₇	C₂²×C₂²⋊C₄	1	24	yes	—
126	C₂×#2884	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₆₇	—	2	2	no	—
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	—	2	1	no	—
130	C₂×C₆₇	—	2	2	no	—
131	C₂×C₆₇	—	4	8	no	—
132	C₂×C₆₇	#2884	1	2	yes	direct product factor
133	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic
135	C₂×#519	C₂×C₆₇	1	2	yes	complemented
136	#2884	C₂×C₆₇	1	16	yes	direct product factor
137	C₂×#410	C₂×C₆₇	1	12	yes	complemented
138	C₆₇	C₂×#2884	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×2₋¹⁺⁴	—	2	8	no	—
140	C₂×2₋¹⁺⁴	—	2	8	no	—
141	C₂×2₋¹⁺⁴	C₄×C₆₇	1	8	yes	complemented
142	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	8	yes	—
143	C₂²×#41	C₄×C₆₇	1	4	yes	complemented
144	C₂²×#41	C₂²×C₆₇	1	6	yes	—
145	C₂³×Q₈	C₄×C₆₇	1	4	yes	complemented
146	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
147	#519	—	2	8	no	—
148	#519	C₂²×C₆₇	1	16	yes	complemented
149	C₂×C₄×Q₈	—	2	8	no	—
150	C₂×#117	C₂²×C₆₇	1	12	yes	—
151	#410	C₂²×C₆₇	1	96	yes	complemented
152	C₂×#99	C₂²×C₆₇	1	16	yes	—
153	2₋¹⁺⁴	—	2	64	no	—
154	2₋¹⁺⁴	—	2	64	no	—
155	C₂×#41	—	2	16	no	—
156	C₂×#41	—	2	16	no	—
157	C₂×#41	—	2	24	no	—
158	C₂×#41	—	2	24	no	—
159	C₂×#41	C₂×C₄×C₆₇	1	24	yes	—
160	C₂×#41	C₂×C₄×C₆₇	1	32	yes	complemented
161	C₂×#41	C₂³×C₆₇	1	24	yes	—
162	C₂²×Q₈	—	2	16	no	—
163	C₂²×Q₈	—	2	16	no	—
164	C₂²×Q₈	C₂×C₄×C₆₇	1	6	yes	—
165	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
166	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
167	C₂²×Q₈	C₂×C₄×C₆₇	1	32	yes	complemented
168	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
169	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
170	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
171	C₂²×D₄	C₂×C₄×C₆₇	1	6	yes	—
172	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
173	C₂³×C₄	C₂×C₄×C₆₇	1	12	yes	—
174	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
175	C₄×Q₈	—	2	64	no	—
176	#117	—	2	24	no	—
177	#117	C₂³×C₆₇	1	48	yes	—
178	C₂×C₄⋊C₄	—	2	12	no	—
179	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
180	C₂×C₄²	—	2	12	no	—
181	#99	C₂³×C₆₇	1	64	yes	—
182	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
183	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
184	#41	—	2	128	no	—
185	#41	—	2	96	no	—
186	#41	—	2	96	no	—
187	C₂×Q₈	—	2	128	no	—
188	C₂×Q₈	—	2	12	no	—
189	C₂×Q₈	—	2	12	no	—
190	C₂×Q₈	C₂²⋊C₄×C₆₇	1	16	yes	—
191	C₂×Q₈	C₂²⋊C₄×C₆₇	1	16	yes	—
192	C₂×Q₈	C₂²⋊C₄×C₆₇	1	32	yes	—
193	C₂×Q₈	C₂²×C₄×C₆₇	1	24	yes	—
194	C₂×Q₈	C₂²×C₄×C₆₇	1	16	yes	—
195	C₂×Q₈	C₂×D₄×C₆₇	1	32	yes	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	12	no	—
199	C₂×D₄	—	2	12	no	—
200	C₂×D₄	C₂²×C₄×C₆₇	1	24	yes	—
201	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
202	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
203	C₂²×C₄	—	2	24	no	—
204	C₂²×C₄	—	2	24	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₆₇	1	48	yes	—
208	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
209	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
210	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
211	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
212	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
213	C₄⋊C₄	—	2	48	no	—
214	C₂²⋊C₄	—	2	8	no	—
215	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
216	C₄²	—	2	48	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	8	no	—
221	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
222	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
223	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
224	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
225	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
226	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
227	Q₈	—	2	48	no	—
228	Q₈	C₂×C₂²⋊C₄×C₆₇	1	64	yes	—
229	D₄	—	2	16	no	—
230	D₄	—	2	16	no	—
231	D₄	—	2	48	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	96	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	4	16	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	12	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	12	yes	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	24	yes	—
240	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
241	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
242	C₂×C₄	C₂²×D₄×C₆₇	1	24	yes	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	8	no	—
252	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
253	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
256	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	4	16	no	—
259	C₄	C₂²×C₂²⋊C₄×C₆₇	1	24	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	2	no	—
264	C₂	—	4	8	no	—
265	C₂	C₆₇×#2884	1	2	yes	direct product factor
266	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×2₋¹⁺⁴×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#519×C₆₇	C₂	semidirect
3. C₆₇×#2884	C₂	direct (N × H)
4. C₂×#410×C₆₇	C₂	semidirect
7. C₂×2₋¹⁺⁴×C₆₇	C₄	semidirect
9. C₂²×#41×C₆₇	C₄	semidirect
11. C₂³×Q₈×C₆₇	C₄	semidirect
14. #519×C₆₇	C₂²	semidirect
17. #410×C₆₇	C₂²	semidirect
26. C₂×#41×C₆₇	C₂×C₄	semidirect
33. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
126. C₂×#2884	C₆₇	direct (N × H)
132. C₂×C₆₇	#2884	direct (N × H)
135. C₂×#519	C₂×C₆₇	semidirect
136. #2884	C₂×C₆₇	direct (N × H)
137. C₂×#410	C₂×C₆₇	semidirect
138. C₆₇	C₂×#2884	direct (N × H)
141. C₂×2₋¹⁺⁴	C₄×C₆₇	semidirect
143. C₂²×#41	C₄×C₆₇	semidirect
145. C₂³×Q₈	C₄×C₆₇	semidirect
148. #519	C₂²×C₆₇	semidirect
151. #410	C₂²×C₆₇	semidirect
160. C₂×#41	C₂×C₄×C₆₇	semidirect
167. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
265. C₂	C₆₇×#2884	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂×#519×C₆₇	2
C₆₇×#2884	2
C₂×#410×C₆₇	2
C₂×#2884	67

Subgroups — C2×C67×#2884 #30739320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2884 #30739320