Subgroups — C₆₇×#20020 #30699271

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3393	C₂	1	2	yes	maximal
2	C₆₇×#3342	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×#1670	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1650	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#556×C₆₇	C₂²	1	4	yes	—
10	#500×C₆₇	C₂²	1	2	yes	—
11	#500×C₆₇	C₂²	1	2	yes	—
12	#495×C₆₇	—	2	2	no	—
13	#494×C₆₇	C₂²	1	2	yes	—
14	#485×C₆₇	—	2	4	no	—
15	#475×C₆₇	C₂²	1	4	yes	—
16	#467×C₆₇	—	2	2	no	—
17	#464×C₆₇	—	2	4	no	—
18	#461×C₆₇	C₂²	1	4	yes	—
19	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₆₇	—	2	2	no	—
21	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₆₇	C₂²	1	2	yes	—
24	#427×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	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
29	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₈²×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₆₇	—	2	2	no	—
32	C₂×Q₁₆×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₆₇	—	2	4	no	—
35	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
38	#126×C₆₇	—	4	4	no	—
39	#124×C₆₇	—	2	4	no	—
40	C₂²⋊Q₈×C₆₇	—	2	4	no	—
41	C₄×Q₈×C₆₇	—	2	4	no	—
42	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₄×D₄×C₆₇	C₂³	1	2	yes	—
44	#117×C₆₇	—	2	4	no	—
45	#107×C₆₇	—	2	2	no	—
46	#107×C₆₇	—	2	2	no	—
47	#107×C₆₇	—	4	2	no	—
48	#107×C₆₇	D₄	1	1	yes	characteristic
49	#107×C₆₇	D₄	1	1	yes	characteristic
50	#106×C₆₇	—	2	4	no	—
51	C₄⋊C₈×C₆₇	—	2	2	no	—
52	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
53	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
54	#103×C₆₇	—	2	4	no	—
55	#103×C₆₇	—	2	4	no	—
56	#103×C₆₇	D₄	1	2	yes	—
57	#102×C₆₇	D₄	1	2	yes	—
58	C₂²⋊C₈×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	—	2	2	no	—
60	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₆₇	—	2	4	no	—
64	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	Q₁₆×C₆₇	—	2	2	no	—
68	Q₁₆×C₆₇	—	4	4	no	—
69	Q₁₆×C₆₇	—	8	2	no	—
70	SD₁₆×C₆₇	—	8	4	no	—
71	D₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	4	4	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂²⋊C₄×C₆₇	—	4	4	no	—
88	C₂²⋊C₄×C₆₇	—	4	4	no	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	4	4	no	—
91	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₂³×C₆₇	—	2	2	no	—
93	Q₈×C₆₇	—	2	2	no	—
94	Q₈×C₆₇	—	2	2	no	—
95	Q₈×C₆₇	—	4	4	no	—
96	Q₈×C₆₇	—	4	4	no	—
97	D₄×C₆₇	—	2	2	no	—
98	D₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	4	2	no	—
112	C₈×C₆₇	—	4	2	no	—
113	C₈×C₆₇	#135	1	2	yes	—
114	C₈×C₆₇	#135	1	2	yes	—
115	C₂²×C₆₇	—	4	2	no	—
116	C₂²×C₆₇	—	4	2	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	D₄²	1	1	yes	characteristic
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	4	2	no	—
121	C₄×C₆₇	—	4	2	no	—
122	C₄×C₆₇	—	4	2	no	—
123	C₄×C₆₇	—	8	4	no	—
124	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
125	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
126	C₄×C₆₇	#576	1	1	yes	characteristic
127	C₄×C₆₇	#578	1	1	yes	characteristic
128	#20020	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₆₇	—	8	2	no	—
130	C₂×C₆₇	#3283	1	1	yes	characteristic
131	C₂×C₆₇	#3287	1	1	yes	characteristic
132	C₂×C₆₇	#3293	1	1	yes	characteristic
133	#3393	C₂×C₆₇	1	2	yes	—
134	#3342	C₂×C₆₇	1	2	yes	complemented
135	#1713	C₂×C₆₇	1	1	yes	characteristic, complemented
136	#1673	C₂×C₆₇	1	2	yes	complemented
137	#1670	C₂×C₆₇	1	2	yes	complemented
138	#1650	C₂×C₆₇	1	4	yes	complemented
139	C₈×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
140	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic
141	C₆₇	#20020	1	1	yes	characteristic, direct product factor, Sylow
142	#556	C₂²×C₆₇	1	4	yes	—
143	#500	C₂²×C₆₇	1	2	yes	—
144	#500	C₂²×C₆₇	1	2	yes	—
145	#495	—	2	2	no	—
146	#494	C₂²×C₆₇	1	2	yes	—
147	#485	—	2	4	no	—
148	#475	C₂²×C₆₇	1	4	yes	—
149	#467	—	2	2	no	—
150	#464	—	2	4	no	—
151	#461	C₂²×C₆₇	1	4	yes	—
152	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
153	C₄×Q₁₆	—	2	2	no	—
154	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
155	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
156	C₈×D₄	C₂²×C₆₇	1	2	yes	—
157	#427	—	2	2	no	—
158	#335	C₂²×C₆₇	1	1	yes	characteristic
159	#335	C₂²×C₆₇	1	1	yes	characteristic
160	#332	C₂²×C₆₇	1	4	yes	—
161	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
162	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
163	C₈²	C₂²×C₆₇	1	1	yes	characteristic
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
167	C₂×SD₁₆	—	2	4	no	—
168	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
171	#126	—	4	4	no	—
172	#124	—	2	4	no	—
173	C₂²⋊Q₈	—	2	4	no	—
174	C₄×Q₈	—	2	4	no	—
175	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
176	C₄×D₄	C₂³×C₆₇	1	2	yes	—
177	#117	—	2	4	no	—
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#107	—	4	2	no	—
181	#107	D₄×C₆₇	1	1	yes	characteristic
182	#107	D₄×C₆₇	1	1	yes	characteristic
183	#106	—	2	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
186	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
187	#103	—	2	4	no	—
188	#103	—	2	4	no	—
189	#103	D₄×C₆₇	1	2	yes	—
190	#102	D₄×C₆₇	1	2	yes	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
195	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
198	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
199	C₂²×C₄	—	2	2	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	4	4	no	—
202	Q₁₆	—	8	2	no	—
203	SD₁₆	—	8	4	no	—
204	D₈	—	2	2	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
218	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
225	C₂³	—	2	2	no	—
226	Q₈	—	2	2	no	—
227	Q₈	—	2	2	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	4	4	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
240	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	4	2	no	—
246	C₈	#135×C₆₇	1	2	yes	—
247	C₈	#135×C₆₇	1	2	yes	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	D₄²×C₆₇	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	4	no	—
257	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
258	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
259	C₄	#576×C₆₇	1	1	yes	characteristic
260	C₄	#578×C₆₇	1	1	yes	characteristic
261	C₂	—	8	2	no	—
262	C₂	C₆₇×#3283	1	1	yes	characteristic
263	C₂	C₆₇×#3287	1	1	yes	characteristic
264	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×D₈×C₆₇	D₄
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
2. C₆₇×#3342	C₂	semidirect
3. C₆₇×#1713	C₂	semidirect
4. C₆₇×#1673	C₂	semidirect
5. C₆₇×#1670	C₂	semidirect
6. C₆₇×#1650	C₂	semidirect
7. C₈×Q₁₆×C₆₇	C₂	semidirect
128. #20020	C₆₇	direct (N × H)
134. #3342	C₂×C₆₇	semidirect
135. #1713	C₂×C₆₇	semidirect
136. #1673	C₂×C₆₇	semidirect
137. #1670	C₂×C₆₇	semidirect
138. #1650	C₂×C₆₇	semidirect
139. C₈×Q₁₆	C₂×C₆₇	semidirect
141. C₆₇	#20020	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×#3393	2
C₆₇×#3342	2
C₆₇×#1713	2
C₆₇×#1673	2
C₆₇×#1670	2
C₆₇×#1650	2
C₈×Q₁₆×C₆₇	2
C₈×D₈×C₆₇	2
#20020	67

Subgroups — C67×#20020 #30699271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20020 #30699271