Subgroups — C₆₇×#19971 #30699222

Order 17152.

706 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3334	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3283	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1673	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1669	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1647	C₂	1	4	yes	maximal, complemented
7	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	—	2	2	no	—
10	#547×C₆₇	C₂²	1	4	yes	—
11	#500×C₆₇	C₂²	1	2	yes	—
12	#500×C₆₇	C₂²	1	2	yes	—
13	#494×C₆₇	C₂²	1	2	yes	—
14	#480×C₆₇	—	2	4	no	—
15	#474×C₆₇	C₂²	1	4	yes	—
16	#467×C₆₇	—	2	2	no	—
17	#466×C₆₇	—	2	2	no	—
18	#461×C₆₇	C₂²	1	4	yes	—
19	#452×C₆₇	—	2	4	no	—
20	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₈×D₄×C₆₇	C₂²	1	2	yes	—
23	C₈×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂×#107×C₆₇	—	2	2	no	—
25	#335×C₆₇	C₂²	1	1	yes	characteristic
26	#335×C₆₇	C₂²	1	1	yes	characteristic
27	#332×C₆₇	C₂²	1	4	yes	—
28	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
29	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₈²×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#41×C₆₇	—	2	4	no	—
32	#135×C₆₇	—	4	8	no	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	2	no	—
35	C₂×SD₁₆×C₆₇	—	2	4	no	—
36	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
41	#123×C₆₇	—	4	4	no	—
42	C₂²⋊Q₈×C₆₇	—	2	4	no	—
43	#121×C₆₇	—	2	4	no	—
44	C₄×D₄×C₆₇	C₂³	1	2	yes	—
45	C₄×D₄×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	#107×C₆₇	—	2	2	no	—
48	#107×C₆₇	—	2	2	no	—
49	#107×C₆₇	—	2	4	no	—
50	#107×C₆₇	—	4	2	no	—
51	#107×C₆₇	D₄	1	1	yes	characteristic
52	#107×C₆₇	D₄	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
55	#103×C₆₇	—	2	4	no	—
56	#102×C₆₇	—	2	4	no	—
57	#102×C₆₇	D₄	1	2	yes	—
58	#102×C₆₇	D₄	1	2	yes	—
59	C₂²⋊C₈×C₆₇	—	2	2	no	—
60	C₂²⋊C₈×C₆₇	—	2	2	no	—
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	#41×C₆₇	—	4	8	no	—
65	#41×C₆₇	—	8	4	no	—
66	C₂×Q₈×C₆₇	—	2	4	no	—
67	C₂×D₄×C₆₇	—	4	4	no	—
68	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	4	4	no	—
73	Q₁₆×C₆₇	—	4	4	no	—
74	Q₁₆×C₆₇	—	8	2	no	—
75	SD₁₆×C₆₇	—	4	8	no	—
76	D₈×C₆₇	—	2	2	no	—
77	D₈×C₆₇	D₈	1	4	yes	complemented
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	4	4	no	—
82	C₂×C₈×C₆₇	—	4	4	no	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	4	4	no	—
90	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₂³×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	—	2	2	no	—
98	Q₈×C₆₇	—	4	4	no	—
99	Q₈×C₆₇	—	4	4	no	—
100	D₄×C₆₇	—	2	2	no	—
101	D₄×C₆₇	—	2	2	no	—
102	D₄×C₆₇	—	2	2	no	—
103	D₄×C₆₇	—	8	8	no	—
104	D₄×C₆₇	C₂×D₈	1	4	yes	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	8	8	no	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	2	1	no	—
119	C₈×C₆₇	—	4	2	no	—
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	C₂×D₈	1	2	yes	—
122	C₈×C₆₇	#135	1	2	yes	complemented
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	4	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	2	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	D₄²	1	1	yes	characteristic
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	4	2	no	—
132	C₄×C₆₇	—	4	2	no	—
133	C₄×C₆₇	—	8	4	no	—
134	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
136	C₄×C₆₇	#576	1	1	yes	characteristic
137	C₄×C₆₇	#578	1	1	yes	characteristic
138	#19971	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₆₇	—	4	4	no	—
140	C₂×C₆₇	—	8	2	no	—
141	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
142	C₂×C₆₇	#3283	1	1	yes	characteristic
143	C₂×C₆₇	#3293	1	1	yes	characteristic
144	#3334	C₂×C₆₇	1	2	yes	complemented
145	#3283	C₂×C₆₇	1	2	yes	complemented
146	#1713	C₂×C₆₇	1	1	yes	characteristic, complemented
147	#1673	C₂×C₆₇	1	2	yes	complemented
148	#1669	C₂×C₆₇	1	2	yes	complemented
149	#1647	C₂×C₆₇	1	4	yes	complemented
150	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
151	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
152	C₆₇	#19971	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#135	—	2	2	no	—
154	#547	C₂²×C₆₇	1	4	yes	—
155	#500	C₂²×C₆₇	1	2	yes	—
156	#500	C₂²×C₆₇	1	2	yes	—
157	#494	C₂²×C₆₇	1	2	yes	—
158	#480	—	2	4	no	—
159	#474	C₂²×C₆₇	1	4	yes	—
160	#467	—	2	2	no	—
161	#466	—	2	2	no	—
162	#461	C₂²×C₆₇	1	4	yes	—
163	#452	—	2	4	no	—
164	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
165	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
166	C₈×D₄	C₂²×C₆₇	1	2	yes	—
167	C₈×D₄	C₂²×C₆₇	1	2	yes	—
168	C₂×#107	—	2	2	no	—
169	#335	C₂²×C₆₇	1	1	yes	characteristic
170	#335	C₂²×C₆₇	1	1	yes	characteristic
171	#332	C₂²×C₆₇	1	4	yes	—
172	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
173	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
174	C₈²	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#41	—	2	4	no	—
176	#135	—	4	8	no	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×SD₁₆	—	2	4	no	—
180	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
181	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
185	#123	—	4	4	no	—
186	C₂²⋊Q₈	—	2	4	no	—
187	#121	—	2	4	no	—
188	C₄×D₄	C₂³×C₆₇	1	2	yes	—
189	C₄×D₄	C₂³×C₆₇	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	#107	—	2	2	no	—
192	#107	—	2	2	no	—
193	#107	—	2	4	no	—
194	#107	—	4	2	no	—
195	#107	D₄×C₆₇	1	1	yes	characteristic
196	#107	D₄×C₆₇	1	1	yes	characteristic
197	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
198	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
199	#103	—	2	4	no	—
200	#102	—	2	4	no	—
201	#102	D₄×C₆₇	1	2	yes	—
202	#102	D₄×C₆₇	1	2	yes	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
208	#41	—	4	8	no	—
209	#41	—	8	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
213	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	4	no	—
217	Q₁₆	—	4	4	no	—
218	Q₁₆	—	8	2	no	—
219	SD₁₆	—	4	8	no	—
220	D₈	—	2	2	no	—
221	D₈	D₈×C₆₇	1	4	yes	complemented
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
235	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	8	8	no	—
248	D₄	C₂×D₈×C₆₇	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×D₈×C₆₇	1	2	yes	—
266	C₈	#135×C₆₇	1	2	yes	complemented
267	C₂²	—	2	4	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	D₄²×C₆₇	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	8	4	no	—
278	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
279	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
280	C₄	#576×C₆₇	1	1	yes	characteristic
281	C₄	#578×C₆₇	1	1	yes	characteristic
282	C₂	—	4	4	no	—
283	C₂	—	8	2	no	—
284	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
285	C₂	C₆₇×#3283	1	1	yes	characteristic
286	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3334	C₂	semidirect
2. C₆₇×#3283	C₂	semidirect
3. C₆₇×#1713	C₂	semidirect
4. C₆₇×#1673	C₂	semidirect
5. C₆₇×#1669	C₂	semidirect
6. C₆₇×#1647	C₂	semidirect
7. C₈×D₈×C₆₇	C₂	semidirect
8. C₈×D₈×C₆₇	C₂	semidirect
77. D₈×C₆₇	D₈	semidirect
122. C₈×C₆₇	#135	semidirect
138. #19971	C₆₇	direct (N × H)
144. #3334	C₂×C₆₇	semidirect
145. #3283	C₂×C₆₇	semidirect
146. #1713	C₂×C₆₇	semidirect
147. #1673	C₂×C₆₇	semidirect
148. #1669	C₂×C₆₇	semidirect
149. #1647	C₂×C₆₇	semidirect
150. C₈×D₈	C₂×C₆₇	semidirect
151. C₈×D₈	C₂×C₆₇	semidirect
152. C₆₇	#19971	direct (N × H)
221. D₈	D₈×C₆₇	semidirect
266. C₈	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3334	2
C₆₇×#3283	2
C₆₇×#1713	2
C₆₇×#1673	2
C₆₇×#1669	2
C₆₇×#1647	2
C₈×D₈×C₆₇	2
#19971	67

Subgroups — C67×#19971 #30699222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19971 #30699222