Subgroups — C₆₇×#17980 #30697231

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#1587	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1493	C₂	1	2	yes	maximal
6	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#533×C₆₇	C₂²	1	2	yes	—
8	#500×C₆₇	C₄	1	1	yes	characteristic
9	#494×C₆₇	C₄	1	1	yes	characteristic
10	#487×C₆₇	C₄	1	2	yes	—
11	#467×C₆₇	C₄	1	4	yes	—
12	#446×C₆₇	—	2	2	no	—
13	#446×C₆₇	C₂²	1	4	yes	—
14	C₈×Q₈×C₆₇	—	2	2	no	—
15	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
17	#424×C₆₇	—	2	2	no	—
18	#423×C₆₇	C₂²	1	2	yes	—
19	#405×C₆₇	—	2	2	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#334×C₆₇	C₂²	1	4	yes	—
24	Q₈⋊C₈×C₆₇	C₂²	1	8	yes	—
25	#322×C₆₇	C₂²	1	4	yes	—
26	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	1	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
33	#125×C₆₇	—	2	2	no	—
34	C₂²⋊Q₈×C₆₇	—	2	2	no	—
35	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
38	#117×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#107×C₆₇	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	C₄⋊C₈×C₆₇	—	2	4	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
45	#103×C₆₇	C₂×C₄	1	8	yes	—
46	#97×C₆₇	—	2	4	no	—
47	C₄×C₈×C₆₇	—	2	4	no	—
48	C₄×C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
61	Q₁₆×C₆₇	—	4	4	no	—
62	M₁₆×C₆₇	—	4	4	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	—	2	4	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	4	2	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
78	C₂²⋊C₄×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	#41	1	1	yes	characteristic
83	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
85	Q₈×C₆₇	—	2	8	no	—
86	Q₈×C₆₇	—	4	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#137	1	4	yes	—
95	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
98	C₈×C₆₇	—	2	4	no	—
99	C₈×C₆₇	—	2	2	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	2	2	no	—
103	C₈×C₆₇	—	4	4	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	#441	1	2	yes	—
107	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	4	4	no	—
111	C₄×C₆₇	—	4	4	no	—
112	C₄×C₆₇	#441	1	2	yes	—
113	C₄×C₆₇	#443	1	2	yes	—
114	C₄×C₆₇	#568	1	2	yes	—
115	C₄×C₆₇	C₂×#137	1	2	yes	—
116	#17980	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	#2932	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
121	#3170	C₂×C₆₇	1	1	yes	characteristic
122	#2960	C₂×C₆₇	1	2	yes	—
123	#1587	C₂×C₆₇	1	8	yes	complemented
124	#1566	C₂×C₆₇	1	1	yes	characteristic
125	#1493	C₂×C₆₇	1	2	yes	—
126	C₂×#322	C₂×C₆₇	1	1	yes	characteristic
127	C₆₇	#17980	1	1	yes	characteristic, direct product factor, Sylow
128	#533	C₂²×C₆₇	1	2	yes	—
129	#500	C₄×C₆₇	1	1	yes	characteristic
130	#494	C₄×C₆₇	1	1	yes	characteristic
131	#487	C₄×C₆₇	1	2	yes	—
132	#467	C₄×C₆₇	1	4	yes	—
133	#446	—	2	2	no	—
134	#446	C₂²×C₆₇	1	4	yes	—
135	C₈×Q₈	—	2	2	no	—
136	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
137	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
138	#424	—	2	2	no	—
139	#423	C₂²×C₆₇	1	2	yes	—
140	#405	—	2	2	no	—
141	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
144	#334	C₂²×C₆₇	1	4	yes	—
145	Q₈⋊C₈	C₂²×C₆₇	1	8	yes	—
146	#322	C₂²×C₆₇	1	4	yes	—
147	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
151	C₂²×C₈	D₄×C₆₇	1	2	yes	—
152	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
153	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
154	#125	—	2	2	no	—
155	C₂²⋊Q₈	—	2	2	no	—
156	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
159	#117	—	2	2	no	—
160	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
161	#107	C₂×C₄×C₆₇	1	4	yes	—
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
166	#103	C₂×C₄×C₆₇	1	8	yes	—
167	#97	—	2	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
171	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
172	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
173	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
174	C₄×C₈	C₂³×C₆₇	1	2	yes	—
175	C₄×C₈	C₂³×C₆₇	1	2	yes	—
176	C₄×C₈	C₂³×C₆₇	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
179	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
181	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
182	Q₁₆	—	4	4	no	—
183	M₁₆	—	4	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
190	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
203	C₄²	#41×C₆₇	1	1	yes	characteristic
204	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
205	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
214	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
215	C₂×C₄	#137×C₆₇	1	4	yes	—
216	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#441×C₆₇	1	2	yes	—
228	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	4	4	no	—
233	C₄	#441×C₆₇	1	2	yes	—
234	C₄	#443×C₆₇	1	2	yes	—
235	C₄	#568×C₆₇	1	2	yes	—
236	C₄	C₂×#137×C₆₇	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	C₆₇×#2932	1	1	yes	characteristic
239	C₂	C₂×#441×C₆₇	1	1	yes	characteristic
240	C₂	C₂×#443×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 ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1587	C₂	semidirect
116. #17980	C₆₇	direct (N × H)
123. #1587	C₂×C₆₇	semidirect
127. C₆₇	#17980	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×#2960	2
C₆₇×#1587	2
C₆₇×#1566	2
C₆₇×#1493	2
C₂×#322×C₆₇	2
#17980	67

Subgroups — C67×#17980 #30697231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17980 #30697231