Subgroups — C₆₇×#21344 #30700595

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1985	C₂	1	2	yes	maximal
3	C₆₇×#1940	C₂	1	2	yes	maximal
4	C₆₇×#1916	C₂	1	4	yes	maximal
5	C₆₇×#1768	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1746	C₂	1	2	yes	maximal
8	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×#97×C₆₇	C₂²	1	2	yes	—
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	#391×C₆₇	C₄	1	2	yes	—
18	#389×C₆₇	C₄	1	2	yes	—
19	#387×C₆₇	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	C₂²	1	4	yes	—
25	#336×C₆₇	C₂²	1	4	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	#336×C₆₇	C₂²	1	2	yes	—
28	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₄×Q₈×C₆₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	—	2	8	no	—
45	C₄⋊C₈×C₆₇	—	2	4	no	—
46	#97×C₆₇	—	2	4	no	—
47	#97×C₆₇	—	2	2	no	—
48	#97×C₆₇	Q₈	1	4	yes	complemented
49	C₄×C₈×C₆₇	—	2	4	no	—
50	C₄×C₈×C₆₇	—	2	4	no	—
51	#95×C₆₇	C₂×C₄	1	4	yes	—
52	#95×C₆₇	C₂×C₄	1	4	yes	—
53	#95×C₆₇	C₂×C₄	1	2	yes	—
54	#95×C₆₇	C₂×C₄	1	4	yes	—
55	C₂×Q₈×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₆₇	#41	1	4	yes	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	—	2	8	no	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	—	2	4	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	4	2	no	—
83	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₆₇	#41	1	4	yes	—
85	C₄²×C₆₇	#41	1	2	yes	—
86	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	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	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	#117	1	2	yes	—
102	C₂×C₄×C₆₇	#117	1	2	yes	—
103	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
104	C₂×C₄×C₆₇	#131	1	4	yes	—
105	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
109	C₈×C₆₇	—	2	4	no	—
110	C₈×C₆₇	—	2	4	no	—
111	C₈×C₆₇	—	4	4	no	—
112	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#131	1	2	yes	—
116	C₂²×C₆₇	#568	1	2	yes	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	4	no	—
120	C₄×C₆₇	—	2	4	no	—
121	C₄×C₆₇	—	4	2	no	—
122	C₄×C₆₇	—	4	2	no	—
123	C₄×C₆₇	—	4	4	no	—
124	C₄×C₆₇	#529	1	2	yes	—
125	C₄×C₆₇	#533	1	2	yes	—
126	#21344	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₆₇	#2295	1	1	yes	characteristic
128	C₂×C₆₇	#2921	1	1	yes	characteristic
129	C₂×C₆₇	#2922	1	1	yes	characteristic
130	C₂×C₆₇	#2962	1	2	yes	—
131	C₂×C₆₇	#2968	1	2	yes	—
132	#2295	C₂×C₆₇	1	1	yes	characteristic
133	#1985	C₂×C₆₇	1	2	yes	—
134	#1940	C₂×C₆₇	1	2	yes	—
135	#1916	C₂×C₆₇	1	4	yes	—
136	#1768	C₂×C₆₇	1	2	yes	—
137	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
138	#1746	C₂×C₆₇	1	2	yes	—
139	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
140	C₆₇	#21344	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
142	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
143	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
144	C₂×#97	C₂²×C₆₇	1	2	yes	—
145	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
146	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
149	#391	C₄×C₆₇	1	2	yes	—
150	#389	C₄×C₆₇	1	2	yes	—
151	#387	C₄×C₆₇	1	4	yes	—
152	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
153	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
154	#376	C₂²×C₆₇	1	2	yes	—
155	C₄³	C₂²×C₆₇	1	1	yes	characteristic
156	#336	C₂²×C₆₇	1	2	yes	—
157	#336	C₂²×C₆₇	1	4	yes	—
158	#336	C₂²×C₆₇	1	4	yes	—
159	#336	C₂²×C₆₇	1	4	yes	—
160	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
162	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
163	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
164	C₄×Q₈	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
171	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
174	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	8	no	—
177	C₄⋊C₈	—	2	4	no	—
178	#97	—	2	4	no	—
179	#97	—	2	2	no	—
180	#97	Q₈×C₆₇	1	4	yes	complemented
181	C₄×C₈	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	#95	C₂×C₄×C₆₇	1	4	yes	—
184	#95	C₂×C₄×C₆₇	1	4	yes	—
185	#95	C₂×C₄×C₆₇	1	2	yes	—
186	#95	C₂×C₄×C₆₇	1	4	yes	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
196	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
205	C₂×C₈	#41×C₆₇	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
216	C₄²	#41×C₆₇	1	4	yes	—
217	C₄²	#41×C₆₇	1	2	yes	—
218	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	#117×C₆₇	1	2	yes	—
234	C₂×C₄	#117×C₆₇	1	2	yes	—
235	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
236	C₂×C₄	#131×C₆₇	1	4	yes	—
237	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
240	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
241	C₈	—	2	4	no	—
242	C₈	—	2	4	no	—
243	C₈	—	4	4	no	—
244	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
245	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
246	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
247	C₂²	C₂×#131×C₆₇	1	2	yes	—
248	C₂²	#568×C₆₇	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	4	no	—
256	C₄	#529×C₆₇	1	2	yes	—
257	C₄	#533×C₆₇	1	2	yes	—
258	C₂	C₆₇×#2295	1	1	yes	characteristic
259	C₂	C₆₇×#2921	1	1	yes	characteristic
260	C₂	C₆₇×#2922	1	1	yes	characteristic
261	C₂	C₆₇×#2962	1	2	yes	—
262	C₂	C₆₇×#2968	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
48. #97×C₆₇	Q₈	semidirect
126. #21344	C₆₇	direct (N × H)
140. C₆₇	#21344	direct (N × H)
180. #97	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2295	2
C₆₇×#1985	2
C₆₇×#1940	2
C₆₇×#1916	2
C₆₇×#1768	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1746	2
C₄×#97×C₆₇	2
#21344	67

Subgroups — C67×#21344 #30700595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21344 #30700595