Subgroups — C₆₇×#22619 #30701870

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2346	C₂	1	2	yes	maximal
3	C₆₇×#1943	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1919	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1825	C₂	1	2	yes	maximal
6	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1776	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	#427×C₆₇	C₄	1	8	yes	complemented
12	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#106×C₆₇	C₂²	1	2	yes	—
15	C₂×#106×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	#389×C₆₇	C₂²	1	4	yes	—
19	#388×C₆₇	—	2	4	no	—
20	#382×C₆₇	—	2	2	no	—
21	#380×C₆₇	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	#376×C₆₇	—	2	2	no	—
25	#340×C₆₇	C₂²	1	4	yes	—
26	#340×C₆₇	C₂²	1	2	yes	—
27	#340×C₆₇	C₂²	1	4	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	4	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	#117×C₆₇	—	2	4	no	—
36	#117×C₆₇	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
47	#107×C₆₇	C₂×C₄	1	4	yes	—
48	#107×C₆₇	C₂×C₄	1	4	yes	—
49	#106×C₆₇	—	2	4	no	—
50	#106×C₆₇	C₂×C₄	1	8	yes	—
51	C₂²⋊C₈×C₆₇	—	2	2	no	—
52	#97×C₆₇	—	2	4	no	—
53	#95×C₆₇	—	2	2	no	—
54	#95×C₆₇	—	2	4	no	—
55	#95×C₆₇	—	2	4	no	—
56	#95×C₆₇	D₄	1	2	yes	—
57	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	8	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₇	#41	1	4	yes	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	8	4	no	—
81	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
82	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₆₇	—	2	8	no	—
84	C₂²⋊C₄×C₆₇	—	4	2	no	—
85	C₂²⋊C₄×C₆₇	—	8	2	no	—
86	C₄²×C₆₇	—	2	8	no	—
87	C₄²×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	—	4	8	no	—
90	C₂³×C₆₇	—	2	2	no	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	C₄×D₄	1	2	yes	—
94	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₆₇	—	2	2	no	—
116	C₈×C₆₇	—	4	2	no	—
117	C₈×C₆₇	#117	1	4	yes	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	4	4	no	—
122	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₆₇	#547	1	2	yes	—
124	C₂²×C₆₇	#575	1	2	yes	—
125	C₂²×C₆₇	#577	1	2	yes	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	4	4	no	—
128	C₄×C₆₇	—	4	4	no	—
129	C₄×C₆₇	—	4	4	no	—
130	C₄×C₆₇	—	8	4	no	—
131	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
132	C₄×C₆₇	#519	1	1	yes	characteristic
133	C₄×C₆₇	#552	1	2	yes	—
134	#22619	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₆₇	—	4	2	no	—
136	C₂×C₆₇	#2344	1	1	yes	characteristic
137	C₂×C₆₇	#2943	1	1	yes	characteristic
138	C₂×C₆₇	#2952	1	1	yes	characteristic
139	C₂×C₆₇	#3348	1	4	yes	—
140	C₂×#427	C₂×C₆₇	1	1	yes	characteristic
141	#2346	C₂×C₆₇	1	2	yes	—
142	#1943	C₂×C₆₇	1	2	yes	complemented
143	#1919	C₂×C₆₇	1	4	yes	complemented
144	#1825	C₂×C₆₇	1	2	yes	—
145	#1807	C₂×C₆₇	1	1	yes	characteristic
146	#1776	C₂×C₆₇	1	2	yes	complemented
147	#1752	C₂×C₆₇	1	1	yes	characteristic
148	C₆₇	#22619	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
150	C₂×#117	C₂²×C₆₇	1	2	yes	—
151	#427	C₄×C₆₇	1	8	yes	complemented
152	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
153	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
154	C₂×#106	C₂²×C₆₇	1	2	yes	—
155	C₂×#106	C₂²×C₆₇	1	2	yes	—
156	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
157	C₂×#97	C₂²×C₆₇	1	2	yes	—
158	#389	C₂²×C₆₇	1	4	yes	—
159	#388	—	2	4	no	—
160	#382	—	2	2	no	—
161	#380	C₂²×C₆₇	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
163	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
164	#376	—	2	2	no	—
165	#340	C₂²×C₆₇	1	2	yes	—
166	#340	C₂²×C₆₇	1	4	yes	—
167	#340	C₂²×C₆₇	1	4	yes	—
168	#336	C₂²×C₆₇	1	2	yes	—
169	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
172	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
175	#117	—	2	4	no	—
176	#117	C₂×C₄×C₆₇	1	8	yes	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
179	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
187	#107	C₂×C₄×C₆₇	1	4	yes	—
188	#107	C₂×C₄×C₆₇	1	4	yes	—
189	#106	—	2	4	no	—
190	#106	C₂×C₄×C₆₇	1	8	yes	—
191	C₂²⋊C₈	—	2	2	no	—
192	#97	—	2	4	no	—
193	#95	—	2	2	no	—
194	#95	—	2	4	no	—
195	#95	—	2	4	no	—
196	#95	D₄×C₆₇	1	2	yes	—
197	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
205	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
206	C₂²×C₄	#41×C₆₇	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
215	C₂×C₈	#41×C₆₇	1	4	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	8	4	no	—
221	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
222	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
223	C₂²⋊C₄	—	2	8	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₄×D₄×C₆₇	1	2	yes	—
234	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
251	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
253	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
254	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	4	2	no	—
257	C₈	#117×C₆₇	1	4	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
263	C₂²	#547×C₆₇	1	2	yes	—
264	C₂²	#575×C₆₇	1	2	yes	—
265	C₂²	#577×C₆₇	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
272	C₄	#519×C₆₇	1	1	yes	characteristic
273	C₄	#552×C₆₇	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	C₆₇×#2344	1	1	yes	characteristic
276	C₂	C₆₇×#2943	1	1	yes	characteristic
277	C₂	C₆₇×#2952	1	1	yes	characteristic
278	C₂	C₆₇×#3348	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1943	C₂	semidirect
4. C₆₇×#1919	C₂	semidirect
7. C₆₇×#1776	C₂	semidirect
11. #427×C₆₇	C₄	semidirect
134. #22619	C₆₇	direct (N × H)
142. #1943	C₂×C₆₇	semidirect
143. #1919	C₂×C₆₇	semidirect
146. #1776	C₂×C₆₇	semidirect
148. C₆₇	#22619	direct (N × H)
151. #427	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427×C₆₇	2
C₆₇×#2346	2
C₆₇×#1943	2
C₆₇×#1919	2
C₆₇×#1825	2
C₆₇×#1807	2
C₆₇×#1776	2
C₆₇×#1752	2
#22619	67

Subgroups — C67×#22619 #30701870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22619 #30701870