Subgroups — C₆₇×#17961 #30697212

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 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×#1593	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1579	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1493	C₂	1	2	yes	maximal
7	C₂×#322×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	#334×C₆₇	C₂²	1	4	yes	—
25	Q₈⋊C₈×C₆₇	C₂²	1	8	yes	—
26	#322×C₆₇	C₂²	1	2	yes	—
27	#322×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	2	yes	—
33	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	#125×C₆₇	—	2	2	no	—
36	C₂²⋊Q₈×C₆₇	—	2	2	no	—
37	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₆₇	—	2	2	no	—
39	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
40	#117×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#107×C₆₇	C₂×C₄	1	4	yes	—
43	C₄⋊C₈×C₆₇	—	2	4	no	—
44	C₄⋊C₈×C₆₇	—	2	4	no	—
45	C₄⋊C₈×C₆₇	—	2	2	no	—
46	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
47	#103×C₆₇	C₂×C₄	1	8	yes	—
48	#97×C₆₇	—	2	4	no	—
49	C₄×C₈×C₆₇	—	2	4	no	—
50	C₄×C₈×C₆₇	—	2	1	no	—
51	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
63	Q₁₆×C₆₇	—	2	8	no	—
64	M₁₆×C₆₇	—	4	4	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	4	2	no	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
80	C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₄²×C₆₇	—	2	4	no	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
87	Q₈×C₆₇	—	2	8	no	—
88	Q₈×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#137	1	4	yes	—
97	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	2	2	no	—
103	C₈×C₆₇	—	4	2	no	—
104	C₈×C₆₇	—	4	4	no	—
105	C₈×C₆₇	#131	1	4	yes	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	#441	1	2	yes	—
109	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	4	4	no	—
113	C₄×C₆₇	—	4	4	no	—
114	C₄×C₆₇	#441	1	2	yes	—
115	C₄×C₆₇	#444	1	2	yes	—
116	C₄×C₆₇	C₂×#131	1	2	yes	—
117	C₄×C₆₇	C₂×#137	1	2	yes	—
118	#17961	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	#2931	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
123	#3170	C₂×C₆₇	1	1	yes	characteristic
124	#2960	C₂×C₆₇	1	2	yes	—
125	#1593	C₂×C₆₇	1	4	yes	complemented
126	#1579	C₂×C₆₇	1	4	yes	complemented
127	#1566	C₂×C₆₇	1	1	yes	characteristic
128	#1493	C₂×C₆₇	1	2	yes	—
129	C₂×#322	C₂×C₆₇	1	1	yes	characteristic
130	C₆₇	#17961	1	1	yes	characteristic, direct product factor, Sylow
131	#533	C₂²×C₆₇	1	2	yes	—
132	#500	C₄×C₆₇	1	1	yes	characteristic
133	#494	C₄×C₆₇	1	1	yes	characteristic
134	#487	C₄×C₆₇	1	2	yes	—
135	#467	C₄×C₆₇	1	4	yes	—
136	#446	—	2	2	no	—
137	#446	C₂²×C₆₇	1	4	yes	—
138	C₈×Q₈	—	2	2	no	—
139	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
140	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
141	#424	—	2	2	no	—
142	#423	C₂²×C₆₇	1	2	yes	—
143	#405	—	2	2	no	—
144	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
147	#334	C₂²×C₆₇	1	4	yes	—
148	Q₈⋊C₈	C₂²×C₆₇	1	8	yes	—
149	#322	C₂²×C₆₇	1	2	yes	—
150	#322	C₂²×C₆₇	1	2	yes	—
151	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
155	C₂²×C₈	D₄×C₆₇	1	2	yes	—
156	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
157	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
158	#125	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
163	#117	—	2	2	no	—
164	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
165	#107	C₂×C₄×C₆₇	1	4	yes	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
170	#103	C₂×C₄×C₆₇	1	8	yes	—
171	#97	—	2	4	no	—
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
175	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
176	C₄×C₈	D₄×C₆₇	1	2	yes	—
177	C₄×C₈	C₂³×C₆₇	1	2	yes	—
178	C₄×C₈	C₂³×C₆₇	1	2	yes	—
179	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
185	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
186	Q₁₆	—	2	8	no	—
187	M₁₆	—	4	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
194	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
207	C₄²	#41×C₆₇	1	1	yes	characteristic
208	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
209	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
219	C₂×C₄	#137×C₆₇	1	4	yes	—
220	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	4	4	no	—
228	C₈	#131×C₆₇	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#441×C₆₇	1	2	yes	—
232	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	4	4	no	—
237	C₄	#441×C₆₇	1	2	yes	—
238	C₄	#444×C₆₇	1	2	yes	—
239	C₄	C₂×#131×C₆₇	1	2	yes	—
240	C₄	C₂×#137×C₆₇	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	C₆₇×#2931	1	1	yes	characteristic
243	C₂	C₂×#441×C₆₇	1	1	yes	characteristic
244	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 ◁ C₂×C₄×C₈×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×#1579	C₂	semidirect
118. #17961	C₆₇	direct (N × H)
125. #1593	C₂×C₆₇	semidirect
126. #1579	C₂×C₆₇	semidirect
130. C₆₇	#17961	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×#1593	2
C₆₇×#1579	2
C₆₇×#1566	2
C₆₇×#1493	2
C₂×#322×C₆₇	2
#17961	67

Subgroups — C67×#17961 #30697212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17961 #30697212