Subgroups — C₆₇×#33242 #30712493

Order 17152.

926 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#487×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2862	C₂	1	2	yes	maximal
5	C₆₇×#1969	C₂	1	8	yes	maximal
6	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	#500×C₆₇	—	2	4	no	—
11	#494×C₆₇	—	2	4	no	—
12	#487×C₆₇	—	2	8	no	—
13	C₂×#107×C₆₇	C₂²	1	4	yes	—
14	C₂×#103×C₆₇	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#391×C₆₇	—	2	4	no	—
19	#391×C₆₇	C₂²	1	8	yes	—
20	#382×C₆₇	—	2	4	no	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
24	C₂×Q₁₆×C₆₇	—	2	8	no	—
25	C₂×Q₁₆×C₆₇	—	2	16	no	—
26	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₆₇	—	2	4	no	—
29	C₄⋊Q₈×C₆₇	—	2	4	no	—
30	C₄⋊Q₈×C₆₇	—	2	4	no	—
31	C₄⋊Q₈×C₆₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	D₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#107×C₆₇	—	2	8	no	—
41	#103×C₆₇	—	2	16	no	—
42	C₄×C₈×C₆₇	—	2	8	no	—
43	C₄×C₈×C₆₇	D₄	1	4	yes	—
44	C₄×C₈×C₆₇	D₄	1	4	yes	—
45	#95×C₆₇	—	2	16	no	—
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂×Q₈×C₆₇	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	2	16	no	—
49	C₂×Q₈×C₆₇	—	4	4	no	—
50	C₂×Q₈×C₆₇	—	4	8	no	—
51	C₂²×C₄×C₆₇	—	2	4	no	—
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
57	Q₁₆×C₆₇	—	4	16	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₆₇	#41	1	8	yes	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄⋊C₄×C₆₇	—	4	4	no	—
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	Q₁₆	1	8	yes	complemented
70	C₄²×C₆₇	C₂×D₄	1	2	yes	—
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
73	Q₈×C₆₇	—	4	8	no	—
74	Q₈×C₆₇	—	4	8	no	—
75	Q₈×C₆₇	—	8	8	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	8	no	—
81	C₂×C₄×C₆₇	—	4	8	no	—
82	C₂×C₄×C₆₇	#127	1	2	yes	—
83	C₂×C₄×C₆₇	#127	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×Q₁₆	1	8	yes	—
85	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
86	C₂×C₄×C₆₇	#135	1	4	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
90	C₈×C₆₇	—	2	4	no	—
91	C₈×C₆₇	#124	1	8	yes	—
92	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
93	C₂²×C₆₇	#532	1	2	yes	—
94	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
95	C₂²×C₆₇	C₂×#135	1	2	yes	—
96	C₄×C₆₇	—	2	8	no	—
97	C₄×C₆₇	—	8	4	no	—
98	C₄×C₆₇	—	8	4	no	—
99	C₄×C₆₇	#494	1	4	yes	—
100	C₄×C₆₇	#495	1	4	yes	—
101	C₄×C₆₇	C₂×#124	1	2	yes	—
102	C₄×C₆₇	#532	1	2	yes	—
103	#33242	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	#2858	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
106	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
107	C₂×C₆₇	#3170	1	4	yes	—
108	C₂×#500	C₂×C₆₇	1	1	yes	characteristic
109	C₂×#494	C₂×C₆₇	1	1	yes	characteristic
110	C₂×#487	C₂×C₆₇	1	2	yes	—
111	#2862	C₂×C₆₇	1	2	yes	—
112	#1969	C₂×C₆₇	1	8	yes	—
113	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
114	C₆₇	#33242	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
116	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
117	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
118	#500	—	2	4	no	—
119	#494	—	2	4	no	—
120	#487	—	2	8	no	—
121	C₂×#107	C₂²×C₆₇	1	4	yes	—
122	C₂×#103	C₂²×C₆₇	1	8	yes	—
123	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
124	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	#391	—	2	4	no	—
127	#391	C₂²×C₆₇	1	8	yes	—
128	#382	—	2	4	no	—
129	C₄³	C₂²×C₆₇	1	1	yes	characteristic
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
132	C₂×Q₁₆	—	2	8	no	—
133	C₂×Q₁₆	—	2	16	no	—
134	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
135	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	2	4	no	—
139	C₄⋊Q₈	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
144	C₂×C₄²	D₄×C₆₇	1	2	yes	—
145	C₂×C₄²	D₄×C₆₇	1	2	yes	—
146	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
147	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
148	#107	—	2	8	no	—
149	#103	—	2	16	no	—
150	C₄×C₈	—	2	8	no	—
151	C₄×C₈	D₄×C₆₇	1	4	yes	—
152	C₄×C₈	D₄×C₆₇	1	4	yes	—
153	#95	—	2	16	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	16	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	8	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
162	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
164	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
165	Q₁₆	—	4	16	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
169	C₂×C₈	#41×C₆₇	1	8	yes	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	Q₁₆×C₆₇	1	8	yes	complemented
178	C₄²	C₂×D₄×C₆₇	1	2	yes	—
179	C₄²	C₂×D₄×C₆₇	1	2	yes	—
180	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
181	Q₈	—	4	8	no	—
182	Q₈	—	4	8	no	—
183	Q₈	—	8	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	#127×C₆₇	1	2	yes	—
191	C₂×C₄	#127×C₆₇	1	2	yes	—
192	C₂×C₄	C₂×Q₁₆×C₆₇	1	8	yes	—
193	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
194	C₂×C₄	#135×C₆₇	1	4	yes	—
195	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
197	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
198	C₈	—	2	4	no	—
199	C₈	#124×C₆₇	1	8	yes	—
200	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
201	C₂²	#532×C₆₇	1	2	yes	—
202	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
203	C₂²	C₂×#135×C₆₇	1	2	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	8	4	no	—
206	C₄	—	8	4	no	—
207	C₄	#494×C₆₇	1	4	yes	—
208	C₄	#495×C₆₇	1	4	yes	—
209	C₄	C₂×#124×C₆₇	1	2	yes	—
210	C₄	#532×C₆₇	1	2	yes	—
211	C₂	C₆₇×#2858	1	1	yes	characteristic
212	C₂	C₂×#494×C₆₇	1	1	yes	characteristic
213	C₂	C₂×#495×C₆₇	1	1	yes	characteristic
214	C₂	C₆₇×#3170	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
69. C₄²×C₆₇	Q₁₆	semidirect
103. #33242	C₆₇	direct (N × H)
114. C₆₇	#33242	direct (N × H)
177. C₄²	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#494×C₆₇	2
C₂×#487×C₆₇	2
C₆₇×#2862	2
C₆₇×#1969	2
C₄²×C₈×C₆₇	2
#33242	67

Subgroups — C67×#33242 #30712493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33242 #30712493