Subgroups — C₆₇×#18072 #30697323

Order 17152.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2962	C₂	1	2	yes	maximal
3	C₆₇×#1593	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1587	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1579	C₂	1	2	yes	maximal, complemented
6	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1481	C₂	1	2	yes	maximal
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic
9	#533×C₆₇	C₂²	1	2	yes	—
10	#500×C₆₇	C₄	1	1	yes	characteristic
11	#494×C₆₇	C₄	1	1	yes	characteristic
12	#487×C₆₇	C₄	1	2	yes	—
13	#467×C₆₇	C₄	1	2	yes	—
14	#467×C₆₇	C₄	1	2	yes	—
15	#446×C₆₇	—	2	4	no	—
16	#446×C₆₇	C₂²	1	4	yes	—
17	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
18	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	#431×C₆₇	C₂²	1	1	yes	characteristic
21	#424×C₆₇	—	2	2	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	#405×C₆₇	—	2	2	no	—
24	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#334×C₆₇	C₂²	1	1	yes	characteristic
27	#334×C₆₇	C₂²	1	2	yes	—
28	#334×C₆₇	C₂²	1	1	yes	characteristic
29	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
30	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
31	#322×C₆₇	C₂²	1	2	yes	—
32	#322×C₆₇	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
41	#125×C₆₇	—	2	2	no	—
42	C₂²⋊Q₈×C₆₇	—	2	2	no	—
43	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₄×Q₈×C₆₇	—	2	2	no	—
45	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
46	#117×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	#107×C₆₇	C₂×C₄	1	2	yes	—
49	#107×C₆₇	C₂×C₄	1	2	yes	—
50	C₄⋊C₈×C₆₇	—	2	4	no	—
51	C₄⋊C₈×C₆₇	—	2	4	no	—
52	C₄⋊C₈×C₆₇	—	2	2	no	—
53	C₄⋊C₈×C₆₇	—	2	1	no	—
54	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
55	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
56	#103×C₆₇	C₂×C₄	1	4	yes	—
57	#103×C₆₇	C₂×C₄	1	4	yes	—
58	C₂²⋊C₈×C₆₇	—	2	1	no	—
59	#97×C₆₇	—	2	4	no	—
60	#97×C₆₇	D₄	1	2	yes	—
61	C₄×C₈×C₆₇	—	2	4	no	—
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
73	Q₁₆×C₆₇	—	2	4	no	—
74	Q₁₆×C₆₇	—	4	2	no	—
75	M₁₆×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	4	1	no	—
81	C₂×C₈×C₆₇	—	4	4	no	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
93	C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	#41	1	1	yes	characteristic
98	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₆₇	—	2	4	no	—
101	Q₈×C₆₇	—	2	4	no	—
102	Q₈×C₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#137	1	4	yes	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	2	4	no	—
119	C₈×C₆₇	—	4	2	no	—
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	—	4	2	no	—
122	C₈×C₆₇	#131	1	2	yes	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	#441	1	2	yes	—
126	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	4	4	no	—
131	C₄×C₆₇	—	4	4	no	—
132	C₄×C₆₇	#441	1	2	yes	—
133	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
134	C₄×C₆₇	#568	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#137	1	2	yes	—
136	#18072	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	#2935	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
140	C₂×C₆₇	#2957	1	1	yes	characteristic
141	#3170	C₂×C₆₇	1	1	yes	characteristic
142	#2962	C₂×C₆₇	1	2	yes	—
143	#1593	C₂×C₆₇	1	2	yes	complemented
144	#1587	C₂×C₆₇	1	4	yes	complemented
145	#1579	C₂×C₆₇	1	2	yes	complemented
146	C₂×#334	C₂×C₆₇	1	1	yes	characteristic
147	#1481	C₂×C₆₇	1	2	yes	—
148	#1451	C₂×C₆₇	1	1	yes	characteristic
149	C₆₇	#18072	1	1	yes	characteristic, direct product factor, Sylow
150	#533	C₂²×C₆₇	1	2	yes	—
151	#500	C₄×C₆₇	1	1	yes	characteristic
152	#494	C₄×C₆₇	1	1	yes	characteristic
153	#487	C₄×C₆₇	1	2	yes	—
154	#467	C₄×C₆₇	1	2	yes	—
155	#467	C₄×C₆₇	1	2	yes	—
156	#446	—	2	4	no	—
157	#446	C₂²×C₆₇	1	4	yes	—
158	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
159	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
160	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
161	#431	C₂²×C₆₇	1	1	yes	characteristic
162	#424	—	2	2	no	—
163	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
164	#405	—	2	2	no	—
165	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
167	#334	C₂²×C₆₇	1	1	yes	characteristic
168	#334	C₂²×C₆₇	1	2	yes	—
169	#334	C₂²×C₆₇	1	1	yes	characteristic
170	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
171	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
172	#322	C₂²×C₆₇	1	2	yes	—
173	#322	C₂²×C₆₇	1	2	yes	—
174	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
175	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
176	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
180	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
181	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
182	#125	—	2	2	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
187	#117	—	2	2	no	—
188	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
189	#107	C₂×C₄×C₆₇	1	2	yes	—
190	#107	C₂×C₄×C₆₇	1	2	yes	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
196	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
197	#103	C₂×C₄×C₆₇	1	4	yes	—
198	#103	C₂×C₄×C₆₇	1	4	yes	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	2	4	no	—
201	#97	D₄×C₆₇	1	2	yes	—
202	C₄×C₈	—	2	4	no	—
203	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₆₇	1	2	yes	—
206	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₆₇	1	2	yes	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	2	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
238	C₄²	#41×C₆₇	1	1	yes	characteristic
239	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
240	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	8	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄	#137×C₆₇	1	4	yes	—
253	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	2	no	—
263	C₈	#131×C₆₇	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#441×C₆₇	1	2	yes	—
267	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	—	4	4	no	—
273	C₄	#441×C₆₇	1	2	yes	—
274	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
275	C₄	#568×C₆₇	1	1	yes	characteristic
276	C₄	C₂×#137×C₆₇	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₆₇×#2935	1	1	yes	characteristic
279	C₂	C₂×#441×C₆₇	1	1	yes	characteristic
280	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1593	C₂	semidirect
4. C₆₇×#1587	C₂	semidirect
5. C₆₇×#1579	C₂	semidirect
136. #18072	C₆₇	direct (N × H)
143. #1593	C₂×C₆₇	semidirect
144. #1587	C₂×C₆₇	semidirect
145. #1579	C₂×C₆₇	semidirect
149. C₆₇	#18072	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3170	2
C₆₇×#2962	2
C₆₇×#1593	2
C₆₇×#1587	2
C₆₇×#1579	2
C₂×#334×C₆₇	2
C₆₇×#1481	2
C₆₇×#1451	2
#18072	67

Subgroups — C67×#18072 #30697323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18072 #30697323