Subgroups — C₂×C₆₇×#1960 #30700448

Order 17152.

1598 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×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#103×C₆₇	C₂	1	4	yes	maximal
3	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1960	C₂	1	8	yes	maximal, direct product factor
6	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#103×C₆₇	—	2	16	no	—
16	C₂×#103×C₆₇	C₂²	1	16	yes	—
17	C₂×#97×C₆₇	—	2	4	no	—
18	C₂×#97×C₆₇	C₂²	1	4	yes	—
19	#382×C₆₇	—	2	2	no	—
20	#382×C₆₇	C₂²	1	4	yes	—
21	C₂×#95×C₆₇	—	2	2	no	—
22	C₂²×Q₈×C₆₇	—	2	4	no	—
23	C₂²×Q₈×C₆₇	—	2	2	no	—
24	C₂²×Q₈×C₆₇	—	2	8	no	—
25	C₂²×Q₈×C₆₇	—	2	8	no	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
28	C₂³×C₄×C₆₇	—	2	2	no	—
29	C₂³×C₄×C₆₇	—	2	2	no	—
30	C₂³×C₄×C₆₇	D₄	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	—	2	4	no	—
33	C₂²×C₈×C₆₇	—	2	4	no	—
34	C₂²×C₈×C₆₇	D₄	1	8	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
36	C₄⋊Q₈×C₆₇	—	2	16	no	—
37	C₄⋊Q₈×C₆₇	—	2	8	no	—
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
39	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₇	D₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	#103×C₆₇	—	2	32	no	—
49	#97×C₆₇	—	2	8	no	—
50	#95×C₆₇	—	2	8	no	—
51	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈×C₆₇	—	2	8	no	—
53	C₂×Q₈×C₆₇	—	2	4	no	—
54	C₂×Q₈×C₆₇	—	2	32	no	—
55	C₂×Q₈×C₆₇	—	2	8	no	—
56	C₂×Q₈×C₆₇	—	2	8	no	—
57	C₂×Q₈×C₆₇	—	4	16	no	—
58	C₂×Q₈×C₆₇	—	4	4	no	—
59	C₂×Q₈×C₆₇	—	4	4	no	—
60	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₆₇	—	2	8	no	—
62	C₂²×C₄×C₆₇	—	2	8	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	4	4	no	—
66	C₂²×C₄×C₆₇	—	4	4	no	—
67	C₂²×C₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	—	4	8	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₆₇	#41	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	16	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
81	C₄⋊C₄×C₆₇	—	2	32	no	—
82	C₄⋊C₄×C₆₇	—	2	8	no	—
83	C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
86	C₄²×C₆₇	C₂×D₄	1	4	yes	—
87	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂³×C₆₇	#124	1	2	yes	—
89	C₂³×C₆₇	#127	1	2	yes	—
90	C₂³×C₆₇	#137	1	8	yes	—
91	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₆₇	—	2	16	no	—
93	Q₈×C₆₇	—	4	8	no	—
94	Q₈×C₆₇	—	4	16	no	—
95	C₂×C₄×C₆₇	—	2	8	no	—
96	C₂×C₄×C₆₇	—	2	16	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	4	16	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	8	no	—
103	C₂×C₄×C₆₇	—	8	8	no	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₆₇	#121	1	8	yes	—
108	C₂×C₄×C₆₇	#124	1	2	yes	—
109	C₂×C₄×C₆₇	#127	1	2	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
114	C₈×C₆₇	—	2	8	no	—
115	C₂²×C₆₇	#390	1	4	yes	—
116	C₂²×C₆₇	#419	1	8	yes	—
117	C₂²×C₆₇	#491	1	8	yes	—
118	C₂²×C₆₇	#497	1	8	yes	—
119	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
121	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
122	C₂²×C₆₇	C₂×#137	1	4	yes	—
123	C₄×C₆₇	—	2	8	no	—
124	C₄×C₆₇	—	4	8	no	—
125	C₄×C₆₇	—	8	4	no	—
126	C₄×C₆₇	#390	1	4	yes	—
127	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
128	C₄×C₆₇	C₂×#121	1	2	yes	—
129	C₂×#1960	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₆₇	#1960	1	8	yes	direct product factor
131	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
132	C₂×C₆₇	C₂×#419	1	2	yes	—
133	C₂×C₆₇	C₂×#491	1	2	yes	—
134	C₂×C₆₇	C₂×#497	1	2	yes	—
135	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
136	C₂²×#103	C₂×C₆₇	1	4	yes	—
137	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
138	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
139	#1960	C₂×C₆₇	1	8	yes	direct product factor
140	C₆₇	C₂×#1960	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
142	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
143	C₂×C₄⋊Q₈	—	2	2	no	—
144	C₂×C₄⋊Q₈	—	2	4	no	—
145	C₂×C₄⋊Q₈	C₄×C₆₇	1	8	yes	complemented
146	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
147	C₂²×C₄⋊C₄	—	2	2	no	—
148	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
149	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
150	C₂×#103	—	2	16	no	—
151	C₂×#103	C₂²×C₆₇	1	16	yes	—
152	C₂×#97	—	2	4	no	—
153	C₂×#97	C₂²×C₆₇	1	4	yes	—
154	#382	—	2	2	no	—
155	#382	C₂²×C₆₇	1	4	yes	—
156	C₂×#95	—	2	2	no	—
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	—	2	8	no	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
162	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	D₄×C₆₇	1	2	yes	—
166	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	D₄×C₆₇	1	8	yes	—
170	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
171	C₄⋊Q₈	—	2	16	no	—
172	C₄⋊Q₈	—	2	8	no	—
173	C₄⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
174	C₂×C₄⋊C₄	—	2	16	no	—
175	C₂×C₄⋊C₄	—	2	8	no	—
176	C₂×C₄⋊C₄	—	2	8	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
179	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
180	C₂×C₄²	D₄×C₆₇	1	2	yes	—
181	C₂×C₄²	D₄×C₆₇	1	4	yes	—
182	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
183	#103	—	2	32	no	—
184	#97	—	2	8	no	—
185	#95	—	2	8	no	—
186	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
187	C₂×Q₈	—	2	8	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	32	no	—
190	C₂×Q₈	—	2	8	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	4	16	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
205	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
210	C₂²×C₄	#41×C₆₇	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
212	C₂×C₈	—	2	16	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
216	C₄⋊C₄	—	2	32	no	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
220	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
221	C₄²	C₂×D₄×C₆₇	1	4	yes	—
222	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
223	C₂³	#124×C₆₇	1	2	yes	—
224	C₂³	#127×C₆₇	1	2	yes	—
225	C₂³	#137×C₆₇	1	8	yes	—
226	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
227	Q₈	—	2	16	no	—
228	Q₈	—	4	8	no	—
229	Q₈	—	4	16	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	16	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	8	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
241	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
242	C₂×C₄	#121×C₆₇	1	8	yes	—
243	C₂×C₄	#124×C₆₇	1	2	yes	—
244	C₂×C₄	#127×C₆₇	1	2	yes	—
245	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
247	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₂²	#390×C₆₇	1	4	yes	—
251	C₂²	#419×C₆₇	1	8	yes	—
252	C₂²	#491×C₆₇	1	8	yes	—
253	C₂²	#497×C₆₇	1	8	yes	—
254	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
256	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
257	C₂²	C₂×#137×C₆₇	1	4	yes	—
258	C₄	—	2	8	no	—
259	C₄	—	4	8	no	—
260	C₄	—	8	4	no	—
261	C₄	#390×C₆₇	1	4	yes	—
262	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
263	C₄	C₂×#121×C₆₇	1	2	yes	—
264	C₂	C₆₇×#1960	1	8	yes	direct product factor
265	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
266	C₂	C₂×#419×C₆₇	1	2	yes	—
267	C₂	C₂×#491×C₆₇	1	2	yes	—
268	C₂	C₂×#497×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1960	C₂	direct (N × H)
10. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
38. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
129. C₂×#1960	C₆₇	direct (N × H)
130. C₂×C₆₇	#1960	direct (N × H)
139. #1960	C₂×C₆₇	direct (N × H)
140. C₆₇	C₂×#1960	direct (N × H)
145. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
173. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
264. C₂	C₆₇×#1960	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×C₄⋊Q₈×C₆₇	2
C₂²×#103×C₆₇	2
C₂²×#97×C₆₇	2
C₂×#382×C₆₇	2
C₆₇×#1960	2
C₂×#1960	67

Subgroups — C2×C67×#1960 #30700448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1960 #30700448