Subgroups — C₆₇×#21787 #30701038

Order 17152.

1402 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1954	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1780	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₆₇	C₂²	1	2	yes	—
6	#532×C₆₇	—	2	8	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×#127×C₆₇	C₄	1	1	yes	characteristic
9	C₂×#124×C₆₇	C₄	1	2	yes	—
10	C₂×#121×C₆₇	C₄	1	4	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#431×C₆₇	—	2	8	no	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	16	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
18	C₂×#41×C₆₇	—	2	8	no	—
19	C₂×#41×C₆₇	—	4	8	no	—
20	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
22	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
28	C₄⋊Q₈×C₆₇	—	2	4	no	—
29	#127×C₆₇	—	2	4	no	—
30	#124×C₆₇	—	2	8	no	—
31	#121×C₆₇	—	2	16	no	—
32	C₄×D₄×C₆₇	—	2	8	no	—
33	C₄×D₄×C₆₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	8	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₄⋊C₈×C₆₇	—	2	8	no	—
41	C₂²⋊C₈×C₆₇	—	2	8	no	—
42	C₂²⋊C₈×C₆₇	—	4	16	no	—
43	C₄×C₈×C₆₇	—	2	8	no	—
44	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	#41×C₆₇	—	4	32	no	—
47	C₂×Q₈×C₆₇	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	4	4	no	—
49	C₂×D₄×C₆₇	—	2	8	no	—
50	C₂×D₄×C₆₇	—	2	8	no	—
51	C₂×D₄×C₆₇	—	2	4	no	—
52	C₂×D₄×C₆₇	—	2	16	no	—
53	C₂×D₄×C₆₇	—	4	4	no	—
54	C₂×D₄×C₆₇	—	4	8	no	—
55	C₂²×C₄×C₆₇	—	2	8	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	—	2	8	no	—
58	C₂²×C₄×C₆₇	—	2	16	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	—	2	16	no	—
68	C₄⋊C₄×C₆₇	—	2	8	no	—
69	C₂²⋊C₄×C₆₇	—	2	16	no	—
70	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
71	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
72	C₄²×C₆₇	C₂×D₄	1	4	yes	—
73	C₄²×C₆₇	C₂×D₄	1	4	yes	—
74	C₂³×C₆₇	—	2	2	no	—
75	C₂³×C₆₇	—	2	4	no	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	—	2	8	no	—
78	C₂³×C₆₇	—	2	16	no	—
79	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
80	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	Q₈×C₆₇	—	4	8	no	—
82	D₄×C₆₇	—	4	8	no	—
83	D₄×C₆₇	—	4	16	no	—
84	D₄×C₆₇	—	4	16	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	—	2	16	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	8	no	—
91	C₂×C₄×C₆₇	—	4	16	no	—
92	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
95	C₂×C₄×C₆₇	#131	1	8	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
98	C₈×C₆₇	—	4	8	no	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	—	2	4	no	—
101	C₂²×C₆₇	—	4	4	no	—
102	C₂²×C₆₇	—	4	4	no	—
103	C₂²×C₆₇	—	4	8	no	—
104	C₂²×C₆₇	—	4	16	no	—
105	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₆₇	#518	1	2	yes	—
107	C₂²×C₆₇	C₂×#131	1	4	yes	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	2	4	no	—
110	C₄×C₆₇	—	4	4	no	—
111	C₄×C₆₇	#408	1	8	yes	—
112	C₄×C₆₇	#535	1	4	yes	—
113	#21787	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₆₇	—	4	2	no	—
115	C₂×C₆₇	—	8	4	no	—
116	C₂×C₆₇	#2309	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#408	1	2	yes	—
118	C₂×C₆₇	#2979	1	4	yes	—
119	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
120	C₂×#431	C₂×C₆₇	1	2	yes	complemented
121	#1954	C₂×C₆₇	1	8	yes	complemented
122	#1780	C₂×C₆₇	1	4	yes	complemented
123	C₆₇	#21787	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×#41	C₂²×C₆₇	1	2	yes	—
125	#532	—	2	8	no	—
126	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
127	C₂×#127	C₄×C₆₇	1	1	yes	characteristic
128	C₂×#124	C₄×C₆₇	1	2	yes	—
129	C₂×#121	C₄×C₆₇	1	4	yes	—
130	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
131	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
132	#431	—	2	8	no	—
133	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
134	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
135	C₂×C₂²⋊C₈	C₂²×C₆₇	1	16	yes	—
136	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
137	C₂×#41	—	2	8	no	—
138	C₂×#41	—	4	8	no	—
139	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
140	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
141	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
142	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
143	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
144	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
145	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
147	C₄⋊Q₈	—	2	4	no	—
148	#127	—	2	4	no	—
149	#124	—	2	8	no	—
150	#121	—	2	16	no	—
151	C₄×D₄	—	2	8	no	—
152	C₄×D₄	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
155	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
157	C₂×C₄²	D₄×C₆₇	1	8	yes	—
158	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
159	C₄⋊C₈	—	2	8	no	—
160	C₂²⋊C₈	—	2	8	no	—
161	C₂²⋊C₈	—	4	16	no	—
162	C₄×C₈	—	2	8	no	—
163	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
164	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
165	#41	—	4	32	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	4	no	—
168	C₂×D₄	—	2	8	no	—
169	C₂×D₄	—	2	8	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	16	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	16	no	—
178	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
179	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
180	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
184	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	16	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₂²⋊C₄	—	2	16	no	—
189	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
190	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
191	C₄²	C₂×D₄×C₆₇	1	4	yes	—
192	C₄²	C₂×D₄×C₆₇	1	4	yes	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	8	no	—
197	C₂³	—	2	16	no	—
198	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
199	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
200	Q₈	—	4	8	no	—
201	D₄	—	4	8	no	—
202	D₄	—	4	16	no	—
203	D₄	—	4	16	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	4	16	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
214	C₂×C₄	#131×C₆₇	1	8	yes	—
215	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
216	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
217	C₈	—	4	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	4	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	16	no	—
224	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
225	C₂²	#518×C₆₇	1	2	yes	—
226	C₂²	C₂×#131×C₆₇	1	4	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	4	no	—
230	C₄	#408×C₆₇	1	8	yes	—
231	C₄	#535×C₆₇	1	4	yes	—
232	C₂	—	4	2	no	—
233	C₂	—	8	4	no	—
234	C₂	C₆₇×#2309	1	1	yes	characteristic
235	C₂	C₂×#408×C₆₇	1	2	yes	—
236	C₂	C₆₇×#2979	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#431×C₆₇	C₂	semidirect
3. C₆₇×#1954	C₂	semidirect
4. C₆₇×#1780	C₂	semidirect
113. #21787	C₆₇	direct (N × H)
120. C₂×#431	C₂×C₆₇	semidirect
121. #1954	C₂×C₆₇	semidirect
122. #1780	C₂×C₆₇	semidirect
123. C₆₇	#21787	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₂×#431×C₆₇	2
C₆₇×#1954	2
C₆₇×#1780	2
#21787	67

Subgroups — C67×#21787 #30701038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21787 #30701038