Subgroups — C₆₇×#18090 #30697341

Order 17152.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#2986	C₂	1	2	yes	maximal
3	C₆₇×#1593	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1584	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1579	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1485	C₂	1	2	yes	maximal
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic
9	#533×C₆₇	C₂²	1	2	yes	—
10	#500×C₆₇	C₄	1	1	yes	characteristic
11	#494×C₆₇	C₄	1	1	yes	characteristic
12	#487×C₆₇	C₄	1	2	yes	—
13	#467×C₆₇	C₄	1	2	yes	—
14	#467×C₆₇	C₄	1	2	yes	—
15	#446×C₆₇	—	2	2	no	—
16	#446×C₆₇	C₂²	1	4	yes	—
17	C₈×Q₈×C₆₇	—	2	2	no	—
18	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
21	#433×C₆₇	—	2	4	no	—
22	#431×C₆₇	C₂²	1	2	yes	—
23	#431×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#335×C₆₇	C₂²	1	2	yes	—
27	#334×C₆₇	C₂²	1	1	yes	characteristic
28	#334×C₆₇	C₂²	1	1	yes	characteristic
29	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
30	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
31	#322×C₆₇	C₂²	1	2	yes	—
32	#322×C₆₇	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
40	#125×C₆₇	—	2	2	no	—
41	C₂²⋊Q₈×C₆₇	—	2	2	no	—
42	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₆₇	—	2	2	no	—
44	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
45	#117×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#107×C₆₇	C₂×C₄	1	2	yes	—
48	#107×C₆₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₈×C₆₇	—	2	4	no	—
50	C₄⋊C₈×C₆₇	—	2	4	no	—
51	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
52	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
53	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
54	#103×C₆₇	C₂×C₄	1	4	yes	—
55	#103×C₆₇	C₂×C₄	1	4	yes	—
56	C₂²⋊C₈×C₆₇	—	2	2	no	—
57	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
58	#97×C₆₇	—	2	4	no	—
59	#97×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	—	2	4	no	—
61	C₄×C₈×C₆₇	—	2	2	no	—
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
73	Q₁₆×C₆₇	—	2	4	no	—
74	Q₁₆×C₆₇	—	4	2	no	—
75	M₁₆×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	4	1	no	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₇	#41	1	1	yes	characteristic
97	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₆₇	—	2	4	no	—
100	Q₈×C₆₇	—	2	4	no	—
101	Q₈×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₈×C₆₇	—	2	1	no	—
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	4	no	—
118	C₈×C₆₇	—	4	2	no	—
119	C₈×C₆₇	—	4	2	no	—
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	#131	1	2	yes	—
122	C₂²×C₆₇	—	2	1	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	4	4	no	—
130	C₄×C₆₇	—	4	4	no	—
131	C₄×C₆₇	#444	1	2	yes	—
132	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
133	C₄×C₆₇	#568	1	1	yes	characteristic
134	C₄×C₆₇	#578	1	2	yes	—
135	#18090	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₆₇	—	4	1	no	—
137	C₂×C₆₇	#2935	1	1	yes	characteristic
138	C₂×C₆₇	#2950	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
140	#3170	C₂×C₆₇	1	1	yes	characteristic
141	#2986	C₂×C₆₇	1	2	yes	—
142	#1593	C₂×C₆₇	1	2	yes	complemented
143	#1584	C₂×C₆₇	1	4	yes	complemented
144	#1579	C₂×C₆₇	1	2	yes	complemented
145	#1564	C₂×C₆₇	1	1	yes	characteristic
146	#1485	C₂×C₆₇	1	2	yes	—
147	#1451	C₂×C₆₇	1	1	yes	characteristic
148	C₆₇	#18090	1	1	yes	characteristic, direct product factor, Sylow
149	#533	C₂²×C₆₇	1	2	yes	—
150	#500	C₄×C₆₇	1	1	yes	characteristic
151	#494	C₄×C₆₇	1	1	yes	characteristic
152	#487	C₄×C₆₇	1	2	yes	—
153	#467	C₄×C₆₇	1	2	yes	—
154	#467	C₄×C₆₇	1	2	yes	—
155	#446	—	2	2	no	—
156	#446	C₂²×C₆₇	1	4	yes	—
157	C₈×Q₈	—	2	2	no	—
158	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
159	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
160	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
161	#433	—	2	4	no	—
162	#431	C₂²×C₆₇	1	2	yes	—
163	#431	C₂²×C₆₇	1	1	yes	characteristic
164	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
166	#335	C₂²×C₆₇	1	2	yes	—
167	#334	C₂²×C₆₇	1	1	yes	characteristic
168	#334	C₂²×C₆₇	1	1	yes	characteristic
169	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
170	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
171	#322	C₂²×C₆₇	1	2	yes	—
172	#322	C₂²×C₆₇	1	2	yes	—
173	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
174	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
179	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
180	#125	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
185	#117	—	2	2	no	—
186	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
187	#107	C₂×C₄×C₆₇	1	2	yes	—
188	#107	C₂×C₄×C₆₇	1	2	yes	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
192	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
193	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
194	#103	C₂×C₄×C₆₇	1	4	yes	—
195	#103	C₂×C₄×C₆₇	1	4	yes	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
198	#97	—	2	4	no	—
199	#97	—	2	1	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
203	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₆₇	1	2	yes	—
205	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₆₇	1	2	yes	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
212	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
235	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
236	C₄²	#41×C₆₇	1	1	yes	characteristic
237	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
238	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
239	Q₈	—	2	4	no	—
240	Q₈	—	2	4	no	—
241	Q₈	—	4	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	#131×C₆₇	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	#444×C₆₇	1	2	yes	—
272	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
273	C₄	#568×C₆₇	1	1	yes	characteristic
274	C₄	#578×C₆₇	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	C₆₇×#2935	1	1	yes	characteristic
277	C₂	C₆₇×#2950	1	1	yes	characteristic
278	C₂	C₂×#444×C₆₇	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×#1584	C₂	semidirect
5. C₆₇×#1579	C₂	semidirect
135. #18090	C₆₇	direct (N × H)
142. #1593	C₂×C₆₇	semidirect
143. #1584	C₂×C₆₇	semidirect
144. #1579	C₂×C₆₇	semidirect
148. C₆₇	#18090	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×#3170	2
C₆₇×#2986	2
C₆₇×#1593	2
C₆₇×#1584	2
C₆₇×#1579	2
C₆₇×#1564	2
C₆₇×#1485	2
C₆₇×#1451	2
#18090	67

Subgroups — C67×#18090 #30697341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18090 #30697341