Subgroups — C₂²×#501×C₆₇ #30739907

Order 17152.

2142 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#501×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#107×C₆₇	C₂	1	2	yes	maximal
5	C₂²×#106×C₆₇	C₂	1	2	yes	maximal
6	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₆₇	—	2	1	no	—
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	6	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	—
11	C₂×#125×C₆₇	—	2	6	no	—
12	C₂×#125×C₆₇	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
14	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#501×C₆₇	C₂²	1	64	yes	direct product factor
21	C₂×#107×C₆₇	C₂²	1	24	yes	—
22	C₂×#106×C₆₇	C₂²	1	24	yes	—
23	C₂×#97×C₆₇	C₂²	1	12	yes	—
24	C₂²×Q₈×C₆₇	—	2	12	no	—
25	C₂²×Q₈×C₆₇	—	4	8	no	—
26	C₂³×C₄×C₆₇	—	2	1	no	—
27	C₂³×C₄×C₆₇	—	2	1	no	—
28	C₂³×C₄×C₆₇	—	2	2	no	—
29	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	Q₈	1	16	yes	complemented
33	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
34	C₄⋊Q₈×C₆₇	—	2	24	no	—
35	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	—
36	#125×C₆₇	—	2	24	no	—
37	#125×C₆₇	C₂³	1	16	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
47	C₂×C₄²×C₆₇	D₄	1	6	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
49	#107×C₆₇	C₂³	1	32	yes	—
50	#106×C₆₇	C₂³	1	32	yes	—
51	#97×C₆₇	C₂³	1	16	yes	—
52	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Q₈×C₆₇	—	2	16	no	—
54	C₂×Q₈×C₆₇	—	4	48	no	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	—	2	6	no	—
58	C₂²×C₄×C₆₇	—	2	6	no	—
59	C₂²×C₄×C₆₇	—	2	12	no	—
60	C₂²×C₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	4	8	no	—
62	C₂²×C₄×C₆₇	—	4	4	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
68	C₂×C₈×C₆₇	C₂×Q₈	1	48	yes	complemented
69	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
70	C₄⋊C₄×C₆₇	—	2	16	no	—
71	C₄⋊C₄×C₆₇	—	2	16	no	—
72	C₄⋊C₄×C₆₇	—	2	16	no	—
73	C₄⋊C₄×C₆₇	—	2	6	no	—
74	C₄⋊C₄×C₆₇	—	2	6	no	—
75	C₄⋊C₄×C₆₇	—	2	12	no	—
76	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
77	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
78	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
79	C₄²×C₆₇	C₂×D₄	1	12	yes	—
80	C₄²×C₆₇	C₂⁴	1	4	yes	—
81	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
82	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
83	C₂³×C₆₇	#137	1	4	yes	—
84	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
85	Q₈×C₆₇	—	4	32	no	—
86	C₂×C₄×C₆₇	—	2	12	no	—
87	C₂×C₄×C₆₇	—	2	12	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	8	no	—
91	C₂×C₄×C₆₇	—	4	12	no	—
92	C₂×C₄×C₆₇	—	4	24	no	—
93	C₂×C₄×C₆₇	—	4	12	no	—
94	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
97	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
98	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
99	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
100	C₈×C₆₇	C₂²×Q₈	1	16	yes	complemented
101	C₂²×C₆₇	#501	1	16	yes	direct product factor
102	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
103	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
104	C₂²×C₆₇	C₂×#137	1	6	yes	—
105	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	4	4	no	—
109	C₄×C₆₇	—	4	8	no	—
110	C₄×C₆₇	—	4	4	no	—
111	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
112	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
113	C₂²×#501	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₆₇	C₂×#501	1	12	yes	direct product factor
115	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
118	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
119	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic
120	C₂×#501	C₂×C₆₇	1	24	yes	direct product factor
121	C₂²×#107	C₂×C₆₇	1	2	yes	—
122	C₂²×#106	C₂×C₆₇	1	2	yes	—
123	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
124	C₆₇	C₂²×#501	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×Q₈	—	2	1	no	—
126	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
127	C₂×C₄⋊Q₈	—	2	6	no	—
128	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	—
129	C₂×#125	—	2	6	no	—
130	C₂×#125	C₂²×C₆₇	1	12	yes	—
131	C₂²×C₄⋊C₄	—	2	1	no	—
132	C₂²×C₄⋊C₄	—	2	1	no	—
133	C₂²×C₄⋊C₄	—	2	1	no	—
134	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
135	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
136	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
137	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
138	#501	C₂²×C₆₇	1	64	yes	direct product factor
139	C₂×#107	C₂²×C₆₇	1	24	yes	—
140	C₂×#106	C₂²×C₆₇	1	24	yes	—
141	C₂×#97	C₂²×C₆₇	1	12	yes	—
142	C₂²×Q₈	—	2	12	no	—
143	C₂²×Q₈	—	4	8	no	—
144	C₂³×C₄	—	2	1	no	—
145	C₂³×C₄	—	2	1	no	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
148	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
149	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
150	C₂²×C₈	Q₈×C₆₇	1	16	yes	complemented
151	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
152	C₄⋊Q₈	—	2	24	no	—
153	C₄⋊Q₈	C₂³×C₆₇	1	16	yes	—
154	#125	—	2	24	no	—
155	#125	C₂³×C₆₇	1	16	yes	—
156	C₂×C₄⋊C₄	—	2	12	no	—
157	C₂×C₄⋊C₄	—	2	12	no	—
158	C₂×C₄⋊C₄	—	2	12	no	—
159	C₂×C₄⋊C₄	—	2	3	no	—
160	C₂×C₄⋊C₄	—	2	3	no	—
161	C₂×C₄⋊C₄	—	2	6	no	—
162	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
163	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
164	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
165	C₂×C₄²	D₄×C₆₇	1	6	yes	—
166	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
167	#107	C₂³×C₆₇	1	32	yes	—
168	#106	C₂³×C₆₇	1	32	yes	—
169	#97	C₂³×C₆₇	1	16	yes	—
170	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
171	C₂×Q₈	—	2	16	no	—
172	C₂×Q₈	—	4	48	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	6	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	4	8	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
182	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
183	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
184	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
185	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
186	C₂×C₈	C₂×Q₈×C₆₇	1	48	yes	complemented
187	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
188	C₄⋊C₄	—	2	16	no	—
189	C₄⋊C₄	—	2	16	no	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	2	6	no	—
192	C₄⋊C₄	—	2	6	no	—
193	C₄⋊C₄	—	2	12	no	—
194	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
195	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
196	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
197	C₄²	C₂×D₄×C₆₇	1	12	yes	—
198	C₄²	C₂⁴×C₆₇	1	4	yes	—
199	C₂³	C₄⋊Q₈×C₆₇	1	4	yes	—
200	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
201	C₂³	#137×C₆₇	1	4	yes	—
202	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
203	Q₈	—	4	32	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	4	24	no	—
210	C₂×C₄	—	4	12	no	—
211	C₂×C₄	—	4	12	no	—
212	C₂×C₄	C₄⋊Q₈×C₆₇	1	12	yes	—
213	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
214	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
215	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
216	C₂×C₄	C₂²×Q₈×C₆₇	1	12	yes	—
217	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
218	C₈	C₂²×Q₈×C₆₇	1	16	yes	complemented
219	C₂²	#501×C₆₇	1	16	yes	direct product factor
220	C₂²	C₂×C₄⋊Q₈×C₆₇	1	6	yes	—
221	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	6	yes	—
222	C₂²	C₂×#137×C₆₇	1	6	yes	—
223	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	4	8	no	—
227	C₄	—	4	4	no	—
228	C₄	—	4	4	no	—
229	C₄	C₂×C₄⋊Q₈×C₆₇	1	6	yes	—
230	C₄	C₂³×Q₈×C₆₇	1	2	yes	—
231	C₂	C₂×#501×C₆₇	1	12	yes	direct product factor
232	C₂	C₂²×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
233	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
234	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#501×C₆₇	C₂	direct (N × H)
20. #501×C₆₇	C₂²	direct (N × H)
32. C₂²×C₈×C₆₇	Q₈	semidirect
68. C₂×C₈×C₆₇	C₂×Q₈	semidirect
100. C₈×C₆₇	C₂²×Q₈	semidirect
101. C₂²×C₆₇	#501	direct (N × H)
113. C₂²×#501	C₆₇	direct (N × H)
114. C₂×C₆₇	C₂×#501	direct (N × H)
120. C₂×#501	C₂×C₆₇	direct (N × H)
124. C₆₇	C₂²×#501	direct (N × H)
138. #501	C₂²×C₆₇	direct (N × H)
150. C₂²×C₈	Q₈×C₆₇	semidirect
186. C₂×C₈	C₂×Q₈×C₆₇	semidirect
218. C₈	C₂²×Q₈×C₆₇	semidirect
219. C₂²	#501×C₆₇	direct (N × H)
231. C₂	C₂×#501×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×#125×C₆₇	2
C₂×#501×C₆₇	2
C₂²×#107×C₆₇	2
C₂²×#106×C₆₇	2
C₂²×#97×C₆₇	2
C₂²×#501	67

Subgroups — C22×#501×C67 #30739907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#501×C₆₇ #30739907