Subgroups — C₆₇×#20520 #30699771

Order 17152.

938 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1771	C₂	1	4	yes	maximal
4	C₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#1767	C₂	1	2	yes	maximal
6	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
7	C₄×#97×C₆₇	C₂	1	2	yes	maximal
8	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	#446×C₆₇	C₄	1	16	yes	complemented
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×#97×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₂²	1	4	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₄³×C₆₇	C₂²	1	2	yes	—
26	C₄³×C₆₇	C₂²	1	1	yes	characteristic
27	#336×C₆₇	C₂²	1	4	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₄×Q₈×C₆₇	—	2	8	no	—
34	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	—
50	#97×C₆₇	—	2	4	no	—
51	#97×C₆₇	—	2	4	no	—
52	#97×C₆₇	—	2	2	no	—
53	#97×C₆₇	C₂×C₄	1	16	yes	—
54	#97×C₆₇	Q₈	1	4	yes	complemented
55	C₄×C₈×C₆₇	—	2	8	no	—
56	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
57	#95×C₆₇	C₂×C₄	1	4	yes	—
58	#95×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×Q₈×C₆₇	—	2	4	no	—
60	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₆₇	#41	1	4	yes	—
78	C₄⋊C₄×C₆₇	—	2	8	no	—
79	C₄⋊C₄×C₆₇	C₄²	1	8	yes	—
80	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
81	C₄⋊C₄×C₆₇	M₁₆	1	8	yes	—
82	C₄²×C₆₇	—	2	8	no	—
83	C₄²×C₆₇	—	2	4	no	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	4	4	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
90	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
91	C₄²×C₆₇	#41	1	2	yes	—
92	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₆₇	—	2	8	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₈×C₆₇	—	2	8	no	—
116	C₈×C₆₇	—	2	4	no	—
117	C₈×C₆₇	C₄×Q₈	1	8	yes	complemented
118	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
121	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
122	C₂²×C₆₇	#568	1	2	yes	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	4	no	—
126	C₄×C₆₇	—	2	4	no	—
127	C₄×C₆₇	—	2	8	no	—
128	C₄×C₆₇	—	4	2	no	—
129	C₄×C₆₇	C₄×M₁₆	1	4	yes	—
130	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
131	C₄×C₆₇	C₄×#41	1	2	yes	—
132	#20520	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₆₇	C₄²×Q₈	1	1	yes	characteristic
134	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
135	C₂×C₆₇	#2875	1	1	yes	characteristic
136	C₂×C₆₇	Q₈×M₁₆	1	2	yes	—
137	C₂×C₆₇	#2970	1	2	yes	—
138	C₂×#446	C₂×C₆₇	1	2	yes	—
139	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
140	#1771	C₂×C₆₇	1	4	yes	—
141	C₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
142	#1767	C₂×C₆₇	1	2	yes	—
143	#1767	C₂×C₆₇	1	1	yes	characteristic
144	C₄×#97	C₂×C₆₇	1	2	yes	—
145	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
146	C₆₇	#20520	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	—
148	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
149	#446	C₄×C₆₇	1	16	yes	complemented
150	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
152	C₂×#97	C₂²×C₆₇	1	2	yes	—
153	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
154	C₂×#97	C₂²×C₆₇	1	2	yes	—
155	C₂×#97	C₂²×C₆₇	1	4	yes	—
156	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
158	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
159	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	—
160	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
162	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
163	C₄³	C₂²×C₆₇	1	2	yes	—
164	C₄³	C₂²×C₆₇	1	1	yes	characteristic
165	#336	C₂²×C₆₇	1	4	yes	—
166	#336	C₂²×C₆₇	1	2	yes	—
167	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
169	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
170	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
171	C₄×Q₈	—	2	8	no	—
172	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
179	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
181	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
182	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
185	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
187	C₄⋊C₈	C₂×C₄×C₆₇	1	16	yes	—
188	#97	—	2	4	no	—
189	#97	—	2	4	no	—
190	#97	—	2	2	no	—
191	#97	C₂×C₄×C₆₇	1	16	yes	—
192	#97	Q₈×C₆₇	1	4	yes	complemented
193	C₄×C₈	—	2	8	no	—
194	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
195	#95	C₂×C₄×C₆₇	1	4	yes	—
196	#95	C₂×C₄×C₆₇	1	2	yes	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₄²×C₆₇	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
212	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
213	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
214	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
215	C₂×C₈	#41×C₆₇	1	4	yes	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	C₄²×C₆₇	1	8	yes	—
218	C₄⋊C₄	C₄²×C₆₇	1	4	yes	—
219	C₄⋊C₄	M₁₆×C₆₇	1	8	yes	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	4	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
227	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
228	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
229	C₄²	#41×C₆₇	1	2	yes	—
230	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
244	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
245	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
246	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
247	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
248	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
249	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
250	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
252	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
253	C₈	—	2	8	no	—
254	C₈	—	2	4	no	—
255	C₈	C₄×Q₈×C₆₇	1	8	yes	complemented
256	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
257	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
258	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
259	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
260	C₂²	#568×C₆₇	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	8	no	—
266	C₄	—	4	2	no	—
267	C₄	C₄×M₁₆×C₆₇	1	4	yes	—
268	C₄	C₂×C₄×Q₈×C₆₇	1	2	yes	—
269	C₄	C₄×#41×C₆₇	1	2	yes	—
270	C₂	C₄²×Q₈×C₆₇	1	1	yes	characteristic
271	C₂	C₂×C₄×M₁₆×C₆₇	1	1	yes	characteristic
272	C₂	C₆₇×#2875	1	1	yes	characteristic
273	C₂	Q₈×M₁₆×C₆₇	1	2	yes	—
274	C₂	C₆₇×#2970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. #446×C₆₇	C₄	semidirect
54. #97×C₆₇	Q₈	semidirect
117. C₈×C₆₇	C₄×Q₈	semidirect
132. #20520	C₆₇	direct (N × H)
146. C₆₇	#20520	direct (N × H)
149. #446	C₄×C₆₇	semidirect
192. #97	Q₈×C₆₇	semidirect
255. C₈	C₄×Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₆₇	2
C₄²×Q₈×C₆₇	2
C₆₇×#1771	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1767	2
C₄×#97×C₆₇	2
#20520	67

Subgroups — C67×#20520 #30699771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20520 #30699771