Subgroups — C₆₇×#21927 #30701178

Order 17152.

628 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	C₆₇×#2298	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1924	C₂	1	4	yes	maximal
3	C₆₇×#1917	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1798	C₂	1	2	yes	maximal
7	C₆₇×#1767	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×#97×C₆₇	C₂²	1	4	yes	—
14	#388×C₆₇	C₄	1	4	yes	—
15	#387×C₆₇	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
18	#376×C₆₇	C₂²	1	1	yes	characteristic
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	#336×C₆₇	C₂²	1	4	yes	—
21	#336×C₆₇	C₂²	1	4	yes	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
26	#117×C₆₇	—	2	2	no	—
27	#117×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	4	no	—
38	C₄⋊C₈×C₆₇	—	2	4	no	—
39	C₄⋊C₈×C₆₇	—	4	4	no	—
40	C₂²⋊C₈×C₆₇	—	4	4	no	—
41	#97×C₆₇	—	4	4	no	—
42	#95×C₆₇	C₂×C₄	1	4	yes	—
43	#95×C₆₇	C₂×C₄	1	4	yes	—
44	#95×C₆₇	C₂×C₄	1	4	yes	—
45	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₂²⋊C₄×C₆₇	—	2	4	no	—
65	C₂²⋊C₄×C₆₇	—	4	4	no	—
66	C₄²×C₆₇	—	2	1	no	—
67	C₄²×C₆₇	—	2	1	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	4	4	no	—
73	C₄²×C₆₇	#41	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	—	2	1	no	—
76	C₂³×C₆₇	—	2	2	no	—
77	C₂³×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	4	1	no	—
91	C₂×C₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	#117	1	4	yes	—
94	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
97	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
98	C₈×C₆₇	—	4	4	no	—
99	C₈×C₆₇	—	4	4	no	—
100	C₂²×C₆₇	—	4	1	no	—
101	C₂²×C₆₇	—	4	2	no	—
102	C₂²×C₆₇	—	4	4	no	—
103	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
104	C₂²×C₆₇	#520	1	2	yes	—
105	C₂²×C₆₇	#568	1	4	yes	—
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	4	1	no	—
108	C₄×C₆₇	—	4	2	no	—
109	C₄×C₆₇	—	4	4	no	—
110	C₄×C₆₇	—	4	4	no	—
111	C₄×C₆₇	#540	1	1	yes	characteristic
112	C₄×C₆₇	#541	1	1	yes	characteristic
113	C₄×C₆₇	#542	1	2	yes	—
114	#21927	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₇	—	8	1	no	—
116	C₂×C₆₇	#2314	1	1	yes	characteristic
117	C₂×C₆₇	#2923	1	2	yes	—
118	C₂×C₆₇	#2995	1	4	yes	—
119	#2298	C₂×C₆₇	1	1	yes	characteristic
120	#1924	C₂×C₆₇	1	4	yes	—
121	#1917	C₂×C₆₇	1	4	yes	complemented
122	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
123	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
124	#1798	C₂×C₆₇	1	2	yes	—
125	#1767	C₂×C₆₇	1	2	yes	complemented
126	C₆₇	#21927	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
131	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
132	C₂×#97	C₂²×C₆₇	1	4	yes	—
133	#388	C₄×C₆₇	1	4	yes	—
134	#387	C₄×C₆₇	1	4	yes	—
135	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
136	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
137	#376	C₂²×C₆₇	1	1	yes	characteristic
138	C₄³	C₂²×C₆₇	1	1	yes	characteristic
139	#336	C₂²×C₆₇	1	4	yes	—
140	#336	C₂²×C₆₇	1	4	yes	—
141	#336	C₂²×C₆₇	1	4	yes	—
142	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
144	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
149	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
151	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
152	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
154	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
155	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	—	4	4	no	—
159	C₂²⋊C₈	—	4	4	no	—
160	#97	—	4	4	no	—
161	#95	C₂×C₄×C₆₇	1	4	yes	—
162	#95	C₂×C₄×C₆₇	1	4	yes	—
163	#95	C₂×C₄×C₆₇	1	4	yes	—
164	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
170	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
174	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	4	4	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	4	4	no	—
192	C₄²	#41×C₆₇	1	2	yes	—
193	C₄²	#41×C₆₇	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	#117×C₆₇	1	4	yes	—
213	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
215	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
216	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	4	4	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	4	no	—
222	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
223	C₂²	#520×C₆₇	1	2	yes	—
224	C₂²	#568×C₆₇	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	4	no	—
230	C₄	#540×C₆₇	1	1	yes	characteristic
231	C₄	#541×C₆₇	1	1	yes	characteristic
232	C₄	#542×C₆₇	1	2	yes	—
233	C₂	—	8	1	no	—
234	C₂	C₆₇×#2314	1	1	yes	characteristic
235	C₂	C₆₇×#2923	1	2	yes	—
236	C₂	C₆₇×#2995	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	C₂⁴×C₆₇	C₂²×C₄
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
3. C₆₇×#1917	C₂	semidirect
4. C₆₇×#1843	C₂	semidirect
5. C₆₇×#1832	C₂	semidirect
7. C₆₇×#1767	C₂	semidirect
114. #21927	C₆₇	direct (N × H)
121. #1917	C₂×C₆₇	semidirect
122. #1843	C₂×C₆₇	semidirect
123. #1832	C₂×C₆₇	semidirect
125. #1767	C₂×C₆₇	semidirect
126. C₆₇	#21927	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×#2298	2
C₆₇×#1924	2
C₆₇×#1917	2
C₆₇×#1843	2
C₆₇×#1832	2
C₆₇×#1798	2
C₆₇×#1767	2
#21927	67

Subgroups — C67×#21927 #30701178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21927 #30701178