Subgroups — C₆₇×#17958 #30697209

Order 17152.

614 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	2	yes	maximal
3	C₆₇×#1591	C₂	1	4	yes	maximal, complemented
4	C₈×Q₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1493	C₂	1	2	yes	maximal
7	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#533×C₆₇	C₂²	1	2	yes	—
9	#500×C₆₇	C₄	1	1	yes	characteristic
10	#494×C₆₇	C₄	1	1	yes	characteristic
11	#487×C₆₇	C₄	1	2	yes	—
12	#467×C₆₇	C₄	1	4	yes	—
13	#446×C₆₇	—	2	2	no	—
14	#446×C₆₇	C₂²	1	4	yes	—
15	C₈×Q₈×C₆₇	—	2	2	no	—
16	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
18	#424×C₆₇	—	2	2	no	—
19	#423×C₆₇	C₂²	1	2	yes	—
20	#405×C₆₇	—	2	2	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#335×C₆₇	C₂²	1	4	yes	—
25	Q₈⋊C₈×C₆₇	C₂²	1	8	yes	—
26	C₈²×C₆₇	C₂²	1	2	yes	—
27	C₈²×C₆₇	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	#125×C₆₇	—	2	2	no	—
37	C₂²⋊Q₈×C₆₇	—	2	2	no	—
38	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₆₇	—	2	2	no	—
40	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
41	#117×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	C₂×C₄	1	4	yes	—
44	C₄⋊C₈×C₆₇	—	2	4	no	—
45	C₄⋊C₈×C₆₇	—	2	4	no	—
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
48	#103×C₆₇	C₂×C₄	1	8	yes	—
49	#97×C₆₇	—	2	4	no	—
50	C₄×C₈×C₆₇	—	2	4	no	—
51	C₄×C₈×C₆₇	—	2	1	no	—
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂³	1	2	yes	—
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	C₂×Q₈×C₆₇	—	2	2	no	—
61	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
65	Q₁₆×C₆₇	—	2	8	no	—
66	M₁₆×C₆₇	—	4	4	no	—
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
73	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×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	8	no	—
93	Q₈×C₆₇	—	4	4	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	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₆₇	#135	1	2	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
106	C₈×C₆₇	—	2	4	no	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	4	4	no	—
111	C₈×C₆₇	#131	1	4	yes	—
112	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
113	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
117	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	4	4	no	—
121	C₄×C₆₇	—	4	4	no	—
122	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
123	C₄×C₆₇	#443	1	2	yes	—
124	C₄×C₆₇	C₂×#131	1	2	yes	—
125	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
126	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
127	#17958	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₆₇	—	2	2	no	—
129	C₂×C₆₇	#2931	1	1	yes	characteristic
130	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
131	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
132	#3170	C₂×C₆₇	1	1	yes	characteristic
133	#2960	C₂×C₆₇	1	2	yes	—
134	#1591	C₂×C₆₇	1	4	yes	complemented
135	C₈×Q₁₆	C₂×C₆₇	1	4	yes	complemented
136	#1567	C₂×C₆₇	1	1	yes	characteristic
137	#1493	C₂×C₆₇	1	2	yes	—
138	C₂×C₈²	C₂×C₆₇	1	1	yes	characteristic
139	C₆₇	#17958	1	1	yes	characteristic, direct product factor, Sylow
140	#533	C₂²×C₆₇	1	2	yes	—
141	#500	C₄×C₆₇	1	1	yes	characteristic
142	#494	C₄×C₆₇	1	1	yes	characteristic
143	#487	C₄×C₆₇	1	2	yes	—
144	#467	C₄×C₆₇	1	4	yes	—
145	#446	—	2	2	no	—
146	#446	C₂²×C₆₇	1	4	yes	—
147	C₈×Q₈	—	2	2	no	—
148	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
149	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
150	#424	—	2	2	no	—
151	#423	C₂²×C₆₇	1	2	yes	—
152	#405	—	2	2	no	—
153	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
156	#335	C₂²×C₆₇	1	4	yes	—
157	Q₈⋊C₈	C₂²×C₆₇	1	8	yes	—
158	C₈²	C₂²×C₆₇	1	2	yes	—
159	C₈²	C₂²×C₆₇	1	2	yes	—
160	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
164	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
166	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
167	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
168	#125	—	2	2	no	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
173	#117	—	2	2	no	—
174	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
175	#107	C₂×C₄×C₆₇	1	4	yes	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
180	#103	C₂×C₄×C₆₇	1	8	yes	—
181	#97	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
186	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
187	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₆₇	1	2	yes	—
189	C₄×C₈	C₂³×C₆₇	1	2	yes	—
190	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
196	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
197	Q₁₆	—	2	8	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	Q₁₆×C₆₇	1	4	yes	complemented
205	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
221	C₄²	#41×C₆₇	1	1	yes	characteristic
222	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
223	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	4	4	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	C₄×D₄×C₆₇	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
234	C₂×C₄	#135×C₆₇	1	2	yes	—
235	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	4	no	—
243	C₈	#131×C₆₇	1	4	yes	—
244	C₈	C₂×Q₁₆×C₆₇	1	2	yes	—
245	C₈	C₂×Q₁₆×C₆₇	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Q₁₆×C₆₇	1	2	yes	—
249	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	C₄×Q₁₆×C₆₇	1	2	yes	—
255	C₄	#443×C₆₇	1	2	yes	—
256	C₄	C₂×#131×C₆₇	1	2	yes	—
257	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
258	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	C₆₇×#2931	1	1	yes	characteristic
261	C₂	C₂×C₄×Q₁₆×C₆₇	1	1	yes	characteristic
262	C₂	C₂×#443×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×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×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1591	C₂	semidirect
4. C₈×Q₁₆×C₆₇	C₂	semidirect
72. C₂×C₈×C₆₇	Q₁₆	semidirect
127. #17958	C₆₇	direct (N × H)
134. #1591	C₂×C₆₇	semidirect
135. C₈×Q₁₆	C₂×C₆₇	semidirect
139. C₆₇	#17958	direct (N × H)
204. C₂×C₈	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3170	2
C₆₇×#2960	2
C₆₇×#1591	2
C₈×Q₁₆×C₆₇	2
C₆₇×#1567	2
C₆₇×#1493	2
C₂×C₈²×C₆₇	2
#17958	67

Subgroups — C67×#17958 #30697209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17958 #30697209