Subgroups — C₆₇×#17897 #30697148

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2983	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1593	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1591	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1479	C₂	1	2	yes	maximal
7	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
10	#446×C₆₇	—	2	4	no	—
11	#446×C₆₇	C₂²	1	8	yes	—
12	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
13	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
14	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
16	#432×C₆₇	—	2	4	no	—
17	#431×C₆₇	C₂²	1	1	yes	characteristic
18	#431×C₆₇	C₂²	1	2	yes	—
19	#431×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#103×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#335×C₆₇	C₂²	1	2	yes	—
24	#334×C₆₇	C₂²	1	2	yes	—
25	Q₈⋊C₈×C₆₇	C₂²	1	8	yes	—
26	#322×C₆₇	C₂²	1	2	yes	—
27	C₈²×C₆₇	C₂²	1	2	yes	—
28	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×Q₁₆×C₆₇	—	2	4	no	—
30	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#107×C₆₇	C₂×C₄	1	2	yes	—
43	#107×C₆₇	C₂×C₄	1	2	yes	—
44	C₄⋊C₈×C₆₇	—	2	8	no	—
45	C₄⋊C₈×C₆₇	—	2	1	no	—
46	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
47	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
48	#103×C₆₇	C₂×C₄	1	4	yes	—
49	#103×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₆₇	—	2	2	no	—
51	C₂²⋊C₈×C₆₇	—	2	1	no	—
52	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
53	#97×C₆₇	—	2	8	no	—
54	C₄×C₈×C₆₇	—	2	2	no	—
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	2	4	no	—
63	C₂×Q₈×C₆₇	—	2	4	no	—
64	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	Q₁₆×C₆₇	—	2	4	no	—
71	Q₁₆×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₆₇	—	2	4	no	—
93	Q₈×C₆₇	—	2	4	no	—
94	Q₈×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	4	1	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	8	2	no	—
101	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₆₇	#131	1	2	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	2	4	no	—
112	C₈×C₆₇	—	4	2	no	—
113	C₈×C₆₇	—	4	4	no	—
114	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
115	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	#436	1	2	yes	—
119	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	4	4	no	—
124	C₄×C₆₇	—	4	4	no	—
125	C₄×C₆₇	#436	1	2	yes	—
126	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
128	C₄×C₆₇	#578	1	2	yes	—
129	#17897	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₆₇	—	2	2	no	—
131	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
132	C₂×C₆₇	#2950	1	1	yes	characteristic
133	C₂×C₆₇	#2957	1	1	yes	characteristic
134	#2983	C₂×C₆₇	1	2	yes	—
135	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
136	#1593	C₂×C₆₇	1	4	yes	complemented
137	#1591	C₂×C₆₇	1	4	yes	complemented
138	#1564	C₂×C₆₇	1	1	yes	characteristic
139	#1479	C₂×C₆₇	1	2	yes	—
140	#1450	C₂×C₆₇	1	1	yes	characteristic
141	C₆₇	#17897	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
143	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
144	#446	—	2	4	no	—
145	#446	C₂²×C₆₇	1	8	yes	—
146	C₄×Q₁₆	C₄×C₆₇	1	2	yes	—
147	C₄×Q₁₆	C₄×C₆₇	1	2	yes	—
148	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
149	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
150	#432	—	2	4	no	—
151	#431	C₂²×C₆₇	1	2	yes	—
152	#431	C₂²×C₆₇	1	1	yes	characteristic
153	#431	C₂²×C₆₇	1	1	yes	characteristic
154	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
155	C₂×#103	C₄×C₆₇	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
157	#335	C₂²×C₆₇	1	2	yes	—
158	#334	C₂²×C₆₇	1	2	yes	—
159	Q₈⋊C₈	C₂²×C₆₇	1	8	yes	—
160	#322	C₂²×C₆₇	1	2	yes	—
161	C₈²	C₂²×C₆₇	1	2	yes	—
162	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
165	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
171	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
176	#107	C₂×C₄×C₆₇	1	2	yes	—
177	#107	C₂×C₄×C₆₇	1	2	yes	—
178	C₄⋊C₈	—	2	8	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
181	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
182	#103	C₂×C₄×C₆₇	1	4	yes	—
183	#103	C₂×C₄×C₆₇	1	4	yes	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
187	#97	—	2	8	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
190	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₆₇	1	2	yes	—
192	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₆₇	1	2	yes	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	M₁₆×C₆₇	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
223	C₄²	#41×C₆₇	1	1	yes	characteristic
224	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
225	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	2	4	no	—
228	Q₈	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
239	C₂×C₄	#131×C₆₇	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	4	no	—
248	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
249	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#436×C₆₇	1	2	yes	—
253	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	#436×C₆₇	1	2	yes	—
260	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
261	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
262	C₄	#578×C₆₇	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
265	C₂	C₆₇×#2950	1	1	yes	characteristic
266	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 ◁ #431×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×#1591	C₂	semidirect
129. #17897	C₆₇	direct (N × H)
136. #1593	C₂×C₆₇	semidirect
137. #1591	C₂×C₆₇	semidirect
141. C₆₇	#17897	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×#2983	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#1593	2
C₆₇×#1591	2
C₆₇×#1564	2
C₆₇×#1479	2
C₆₇×#1450	2
#17897	67

Subgroups — C67×#17897 #30697148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17897 #30697148