Subgroups — C₆₇×#20755 #30700006

Order 17152.

922 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1832	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1831	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1729	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×#117×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#423×C₆₇	—	2	4	no	—
12	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
19	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
20	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
21	#376×C₆₇	C₂²	1	2	yes	—
22	C₄³×C₆₇	C₂²	1	2	yes	—
23	#336×C₆₇	C₂²	1	16	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	—	2	4	no	—
28	C₂×M₁₆×C₆₇	—	2	8	no	—
29	C₂²×C₈×C₆₇	—	2	4	no	—
30	C₂²×C₈×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
33	#117×C₆₇	—	2	4	no	—
34	#117×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	4	yes	—
42	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	—	2	4	no	—
47	C₄⋊C₈×C₆₇	—	2	4	no	—
48	C₄⋊C₈×C₆₇	—	2	4	no	—
49	C₄⋊C₈×C₆₇	—	2	4	no	—
50	#95×C₆₇	C₂×C₄	1	4	yes	—
51	#95×C₆₇	C₂×C₄	1	4	yes	—
52	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	—	2	4	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₆₇	#41	1	4	yes	—
66	C₂²×C₄×C₆₇	#41	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₆₇	—	4	8	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	8	no	—
72	C₂×C₈×C₆₇	—	2	8	no	—
73	C₂×C₈×C₆₇	—	4	8	no	—
74	C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₂²⋊C₄×C₆₇	—	2	8	no	—
76	C₄²×C₆₇	—	2	8	no	—
77	C₄²×C₆₇	—	2	4	no	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	—	4	4	no	—
80	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
81	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
82	C₄²×C₆₇	C₂×D₄	1	2	yes	—
83	C₄²×C₆₇	C₂×D₄	1	2	yes	—
84	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
85	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	2	no	—
88	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₆₇	#117	1	4	yes	—
90	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	8	no	—
95	C₂×C₄×C₆₇	—	2	8	no	—
96	C₂×C₄×C₆₇	—	2	8	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	#117	1	4	yes	—
105	C₂×C₄×C₆₇	#124	1	4	yes	—
106	C₂×C₄×C₆₇	#124	1	4	yes	—
107	C₂×C₄×C₆₇	#125	1	8	yes	—
108	C₂×C₄×C₆₇	#131	1	4	yes	—
109	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
113	C₈×C₆₇	—	4	4	no	—
114	C₈×C₆₇	—	4	4	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	2	8	no	—
118	C₂²×C₆₇	#382	1	4	yes	—
119	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#117	1	2	yes	—
121	C₂²×C₆₇	C₂×#131	1	2	yes	—
122	C₂²×C₆₇	#568	1	2	yes	—
123	C₄×C₆₇	—	2	2	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	4	8	no	—
127	C₄×C₆₇	#382	1	4	yes	—
128	C₄×C₆₇	#424	1	4	yes	—
129	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
130	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
131	C₄×C₆₇	C₂×#125	1	2	yes	—
132	#20755	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₆₇	—	2	4	no	—
134	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
136	C₂×C₆₇	#2905	1	1	yes	characteristic
137	C₂×C₆₇	#2925	1	4	yes	—
138	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
139	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
140	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
141	#1832	C₂×C₆₇	1	4	yes	complemented
142	#1831	C₂×C₆₇	1	4	yes	complemented
143	#1729	C₂×C₆₇	1	4	yes	—
144	C₆₇	#20755	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
146	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
147	C₂×#117	C₂²×C₆₇	1	2	yes	—
148	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
149	#423	—	2	4	no	—
150	C₂×C₄⋊C₈	—	2	4	no	—
151	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
155	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
156	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
157	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
158	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
159	#376	C₂²×C₆₇	1	2	yes	—
160	C₄³	C₂²×C₆₇	1	2	yes	—
161	#336	C₂²×C₆₇	1	16	yes	—
162	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	8	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	2	8	no	—
169	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
170	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
171	#117	—	2	4	no	—
172	#117	—	2	4	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
175	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₆₇	1	4	yes	—
180	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
181	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
182	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	4	no	—
188	#95	C₂×C₄×C₆₇	1	4	yes	—
189	#95	C₂×C₄×C₆₇	1	4	yes	—
190	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
194	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
203	C₂²×C₄	#41×C₆₇	1	4	yes	—
204	C₂²×C₄	#41×C₆₇	1	4	yes	—
205	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
206	M₁₆	—	4	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	4	8	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	2	8	no	—
214	C₄²	—	2	8	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	4	4	no	—
218	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
219	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
220	C₄²	C₂×D₄×C₆₇	1	2	yes	—
221	C₄²	C₂×D₄×C₆₇	1	2	yes	—
222	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
223	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
227	C₂³	#117×C₆₇	1	4	yes	—
228	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
242	C₂×C₄	#117×C₆₇	1	4	yes	—
243	C₂×C₄	#124×C₆₇	1	4	yes	—
244	C₂×C₄	#124×C₆₇	1	4	yes	—
245	C₂×C₄	#125×C₆₇	1	8	yes	—
246	C₂×C₄	#131×C₆₇	1	4	yes	—
247	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
249	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
250	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₈	—	4	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	#382×C₆₇	1	4	yes	—
257	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
258	C₂²	C₂×#117×C₆₇	1	2	yes	—
259	C₂²	C₂×#131×C₆₇	1	2	yes	—
260	C₂²	#568×C₆₇	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	8	no	—
265	C₄	#382×C₆₇	1	4	yes	—
266	C₄	#424×C₆₇	1	4	yes	—
267	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
268	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
269	C₄	C₂×#125×C₆₇	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	C₂×#382×C₆₇	1	1	yes	characteristic
272	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
273	C₂	C₆₇×#2905	1	1	yes	characteristic
274	C₂	C₆₇×#2925	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1832	C₂	semidirect
5. C₆₇×#1831	C₂	semidirect
132. #20755	C₆₇	direct (N × H)
141. #1832	C₂×C₆₇	semidirect
142. #1831	C₂×C₆₇	semidirect
144. C₆₇	#20755	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×#423×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₄×#117×C₆₇	2
C₆₇×#1832	2
C₆₇×#1831	2
C₆₇×#1729	2
#20755	67

Subgroups — C67×#20755 #30700006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20755 #30700006