Subgroups — C₆₇×#21846 #30701097

Order 17152.

938 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3458	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2921	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2298	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1995	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1994	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1986	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1826	C₂	1	4	yes	maximal
8	#544×C₆₇	—	2	2	no	—
9	#544×C₆₇	C₄	1	8	yes	complemented
10	#543×C₆₇	—	2	2	no	—
11	#533×C₆₇	—	2	4	no	—
12	#533×C₆₇	C₂²	1	2	yes	—
13	#533×C₆₇	C₂²	1	2	yes	—
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	#433×C₆₇	—	2	4	no	—
17	#423×C₆₇	C₂²	1	2	yes	—
18	#407×C₆₇	C₂²	1	2	yes	—
19	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
20	#388×C₆₇	—	2	2	no	—
21	#387×C₆₇	—	2	2	no	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	C₄³×C₆₇	C₂²	1	1	yes	characteristic
26	#339×C₆₇	C₂²	1	8	yes	—
27	#339×C₆₇	C₂²	1	8	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
30	C₄⋊Q₈×C₆₇	—	2	8	no	—
31	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	#125×C₆₇	—	2	8	no	—
34	#125×C₆₇	C₂×C₄	1	4	yes	—
35	#125×C₆₇	C₂×C₄	1	4	yes	—
36	C₂²⋊Q₈×C₆₇	—	2	4	no	—
37	C₂²⋊Q₈×C₆₇	—	2	4	no	—
38	C₂²⋊Q₈×C₆₇	—	2	8	no	—
39	C₂²⋊Q₈×C₆₇	—	2	8	no	—
40	C₄×Q₈×C₆₇	—	2	4	no	—
41	C₄×Q₈×C₆₇	—	2	4	no	—
42	#117×C₆₇	—	2	2	no	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	C₂×C₄	1	4	yes	—
45	#117×C₆₇	C₂×C₄	1	4	yes	—
46	#117×C₆₇	D₄	1	4	yes	—
47	#117×C₆₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	—	2	2	no	—
55	C₂×C₄²×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	4	no	—
61	C₂²⋊C₈×C₆₇	—	2	4	no	—
62	#97×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	2	no	—
65	#95×C₆₇	—	2	2	no	—
66	#95×C₆₇	—	2	2	no	—
67	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Q₈×C₆₇	—	2	4	no	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₆₇	—	2	8	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	2	8	no	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	4	4	no	—
88	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
89	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₆₇	—	2	4	no	—
91	C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	4	4	no	—
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₄²×C₆₇	C₂×D₄	1	2	yes	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	—	4	2	no	—
105	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₆₇	—	8	4	no	—
108	Q₈×C₆₇	—	8	4	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	8	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	8	4	no	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
128	C₈×C₆₇	—	4	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	—	8	2	no	—
133	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₇	#536	1	2	yes	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	4	no	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	4	4	no	—
139	C₄×C₆₇	—	4	8	no	—
140	C₄×C₆₇	—	8	2	no	—
141	C₄×C₆₇	#519	1	2	yes	—
142	C₄×C₆₇	#536	1	2	yes	—
143	#21846	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₆₇	—	2	2	no	—
145	C₂×C₆₇	—	4	2	no	—
146	C₂×C₆₇	#2310	1	1	yes	characteristic
147	C₂×C₆₇	#2901	1	2	yes	—
148	#3458	C₂×C₆₇	1	1	yes	characteristic
149	#2921	C₂×C₆₇	1	1	yes	characteristic
150	#2298	C₂×C₆₇	1	1	yes	characteristic
151	#1995	C₂×C₆₇	1	2	yes	complemented
152	#1994	C₂×C₆₇	1	2	yes	complemented
153	#1986	C₂×C₆₇	1	4	yes	complemented
154	#1826	C₂×C₆₇	1	4	yes	—
155	C₆₇	#21846	1	1	yes	characteristic, direct product factor, Sylow
156	#544	—	2	2	no	—
157	#544	C₄×C₆₇	1	8	yes	complemented
158	#543	—	2	2	no	—
159	#533	—	2	4	no	—
160	#533	C₂²×C₆₇	1	2	yes	—
161	#533	C₂²×C₆₇	1	2	yes	—
162	C₂×#117	C₂²×C₆₇	1	2	yes	—
163	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
164	#433	—	2	4	no	—
165	#423	C₂²×C₆₇	1	2	yes	—
166	#407	C₂²×C₆₇	1	2	yes	—
167	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
168	#388	—	2	2	no	—
169	#387	—	2	2	no	—
170	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
171	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
172	#376	C₂²×C₆₇	1	1	yes	characteristic
173	C₄³	C₂²×C₆₇	1	1	yes	characteristic
174	#339	C₂²×C₆₇	1	8	yes	—
175	#339	C₂²×C₆₇	1	8	yes	—
176	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
177	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
178	C₄⋊Q₈	—	2	8	no	—
179	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
180	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
181	#125	—	2	8	no	—
182	#125	C₂×C₄×C₆₇	1	4	yes	—
183	#125	C₂×C₄×C₆₇	1	4	yes	—
184	C₂²⋊Q₈	—	2	4	no	—
185	C₂²⋊Q₈	—	2	4	no	—
186	C₂²⋊Q₈	—	2	8	no	—
187	C₂²⋊Q₈	—	2	8	no	—
188	C₄×Q₈	—	2	4	no	—
189	C₄×Q₈	—	2	4	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	C₂×C₄×C₆₇	1	4	yes	—
193	#117	C₂×C₄×C₆₇	1	4	yes	—
194	#117	D₄×C₆₇	1	4	yes	—
195	#117	C₂³×C₆₇	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
198	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
208	C₄⋊C₈	—	2	4	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
228	M₁₆	—	2	8	no	—
229	C₂×C₈	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
237	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
248	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
249	C₄²	C₂×D₄×C₆₇	1	2	yes	—
250	C₄²	C₂×D₄×C₆₇	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
254	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
255	Q₈	—	8	4	no	—
256	Q₈	—	8	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	8	2	no	—
281	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
282	C₂²	#536×C₆₇	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	8	no	—
288	C₄	—	8	2	no	—
289	C₄	#519×C₆₇	1	2	yes	—
290	C₄	#536×C₆₇	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	4	2	no	—
293	C₂	C₆₇×#2310	1	1	yes	characteristic
294	C₂	C₆₇×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1995	C₂	semidirect
5. C₆₇×#1994	C₂	semidirect
6. C₆₇×#1986	C₂	semidirect
9. #544×C₆₇	C₄	semidirect
143. #21846	C₆₇	direct (N × H)
151. #1995	C₂×C₆₇	semidirect
152. #1994	C₂×C₆₇	semidirect
153. #1986	C₂×C₆₇	semidirect
155. C₆₇	#21846	direct (N × H)
157. #544	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3458	2
C₆₇×#2921	2
C₆₇×#2298	2
C₆₇×#1995	2
C₆₇×#1994	2
C₆₇×#1986	2
C₆₇×#1826	2
#21846	67

Subgroups — C67×#21846 #30701097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21846 #30701097