Subgroups — C₂×C₆₇×#1848 #30700055

Order 17152.

1046 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₆₇	C₂	1	4	yes	maximal
2	C₂×#423×C₆₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1848	C₂	1	8	yes	maximal, direct product factor
5	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
6	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₆₇	—	2	16	no	—
9	C₂×#108×C₆₇	C₂²	1	16	yes	—
10	#423×C₆₇	—	2	4	no	—
11	#423×C₆₇	C₂²	1	8	yes	—
12	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
15	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
17	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
18	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	—	2	4	no	—
20	C₂×M₁₆×C₆₇	—	2	16	no	—
21	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
22	C₂²×C₈×C₆₇	—	2	4	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₇	D₄	1	8	yes	—
26	C₂²×C₈×C₆₇	Q₈	1	8	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
35	#108×C₆₇	—	2	32	no	—
36	C₄⋊C₈×C₆₇	—	2	16	no	—
37	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
39	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	—	2	8	no	—
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	16	no	—
54	M₁₆×C₆₇	—	4	8	no	—
55	C₂×C₈×C₆₇	—	2	8	no	—
56	C₂×C₈×C₆₇	—	4	8	no	—
57	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	—
58	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
61	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
62	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
64	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₆₇	C₂×D₄	1	4	yes	—
68	C₂³×C₆₇	—	2	4	no	—
69	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	2	8	no	—
75	C₂×C₄×C₆₇	—	2	8	no	—
76	C₂×C₄×C₆₇	#108	1	8	yes	complemented
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₆₇	#127	1	2	yes	—
84	C₂×C₄×C₆₇	#127	1	2	yes	—
85	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
92	C₈×C₆₇	—	4	8	no	—
93	C₈×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
94	C₂²×C₆₇	—	2	8	no	—
95	C₂²×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	#384	1	2	yes	—
98	C₂²×C₆₇	#384	1	2	yes	—
99	C₂²×C₆₇	C₂×#108	1	4	yes	—
100	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
102	C₄×C₆₇	—	2	4	no	—
103	C₄×C₆₇	#384	1	2	yes	—
104	C₄×C₆₇	#384	1	2	yes	—
105	C₄×C₆₇	C₂×#108	1	8	yes	complemented
106	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
107	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
108	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×#1848	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₆₇	—	2	2	no	—
111	C₂×C₆₇	—	2	2	no	—
112	C₂×C₆₇	#1848	1	4	yes	direct product factor
113	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
114	C₂×C₆₇	C₂²×#108	1	2	yes	—
115	C₂²×#108	C₂×C₆₇	1	4	yes	—
116	C₂×#423	C₂×C₆₇	1	2	yes	—
117	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
118	#1848	C₂×C₆₇	1	8	yes	direct product factor
119	C₆₇	C₂×#1848	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
121	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
122	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
123	C₂×#108	—	2	16	no	—
124	C₂×#108	C₂²×C₆₇	1	16	yes	—
125	#423	—	2	4	no	—
126	#423	C₂²×C₆₇	1	8	yes	—
127	C₂×C₄⋊C₈	—	2	4	no	—
128	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
129	C₂×C₄×C₈	C₄×C₆₇	1	4	yes	—
130	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
132	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
133	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	2	16	no	—
136	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
139	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
140	C₂²×C₈	D₄×C₆₇	1	8	yes	—
141	C₂²×C₈	Q₈×C₆₇	1	8	yes	—
142	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
146	C₂×C₄²	D₄×C₆₇	1	2	yes	—
147	C₂×C₄²	D₄×C₆₇	1	2	yes	—
148	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
149	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
150	#108	—	2	32	no	—
151	C₄⋊C₈	—	2	16	no	—
152	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
153	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
154	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
157	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
158	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
159	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
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	C₂×Q₈×C₆₇	1	4	yes	—
165	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
167	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
168	M₁₆	—	2	16	no	—
169	M₁₆	—	4	8	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	4	8	no	—
172	C₂×C₈	C₄⋊C₄×C₆₇	1	16	yes	—
173	C₂×C₈	C₄⋊C₄×C₆₇	1	16	yes	—
174	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
175	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
176	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
177	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
178	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
179	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
180	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
181	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
182	C₄²	C₂×D₄×C₆₇	1	4	yes	—
183	C₂³	—	2	4	no	—
184	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
185	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
186	C₂³	C₄⋊Q₈×C₆₇	1	4	yes	—
187	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	#108×C₆₇	1	8	yes	complemented
192	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
193	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
194	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
195	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
198	C₂×C₄	#127×C₆₇	1	2	yes	—
199	C₂×C₄	#127×C₆₇	1	2	yes	—
200	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
201	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
202	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
203	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
205	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
207	C₈	—	4	8	no	—
208	C₈	C₂×C₄⋊C₄×C₆₇	1	16	yes	—
209	C₂²	—	2	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#384×C₆₇	1	2	yes	—
213	C₂²	#384×C₆₇	1	2	yes	—
214	C₂²	C₂×#108×C₆₇	1	4	yes	—
215	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
216	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
217	C₄	—	2	4	no	—
218	C₄	#384×C₆₇	1	2	yes	—
219	C₄	#384×C₆₇	1	2	yes	—
220	C₄	C₂×#108×C₆₇	1	8	yes	complemented
221	C₄	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
222	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
223	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	2	2	no	—
226	C₂	C₆₇×#1848	1	4	yes	direct product factor
227	C₂	C₂×#384×C₆₇	1	1	yes	characteristic
228	C₂	C₂²×#108×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1848	C₂	direct (N × H)
76. C₂×C₄×C₆₇	#108	semidirect
105. C₄×C₆₇	C₂×#108	semidirect
109. C₂×#1848	C₆₇	direct (N × H)
112. C₂×C₆₇	#1848	direct (N × H)
118. #1848	C₂×C₆₇	direct (N × H)
119. C₆₇	C₂×#1848	direct (N × H)
191. C₂×C₄	#108×C₆₇	semidirect
220. C₄	C₂×#108×C₆₇	semidirect
226. C₂	C₆₇×#1848	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×#108×C₆₇	2
C₂×#423×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#1848	2
C₂×#1848	67

Subgroups — C2×C67×#1848 #30700055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1848 #30700055