Subgroups — C₆₇×#22139 #30701390

Order 17152.

942 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1955	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1870	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1851	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#436×C₆₇	—	2	4	no	—
18	#431×C₆₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#390×C₆₇	C₄	1	1	yes	characteristic
34	#389×C₆₇	C₄	1	1	yes	characteristic
35	#388×C₆₇	C₄	1	2	yes	—
36	#386×C₆₇	C₄	1	2	yes	—
37	#380×C₆₇	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×M₁₆×C₆₇	—	2	4	no	—
50	C₂×M₁₆×C₆₇	—	4	2	no	—
51	C₂×M₁₆×C₆₇	—	4	2	no	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	#121×C₆₇	—	2	4	no	—
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	4	2	no	—
62	C₄×D₄×C₆₇	—	4	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	C₄⋊C₈×C₆₇	—	2	2	no	—
83	C₄⋊C₈×C₆₇	—	4	2	no	—
84	C₂²⋊C₈×C₆₇	—	2	2	no	—
85	C₂²⋊C₈×C₆₇	—	2	2	no	—
86	C₂²⋊C₈×C₆₇	—	2	4	no	—
87	C₂²⋊C₈×C₆₇	—	4	2	no	—
88	C₂²⋊C₈×C₆₇	—	4	2	no	—
89	C₂²⋊C₈×C₆₇	—	4	2	no	—
90	#97×C₆₇	—	4	2	no	—
91	C₄×C₈×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
93	#95×C₆₇	C₂×C₄	1	2	yes	—
94	#95×C₆₇	C₂×C₄	1	2	yes	—
95	#95×C₆₇	C₂×C₄	1	2	yes	—
96	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	4	1	no	—
120	C₂²×C₄×C₆₇	—	4	1	no	—
121	C₂²×C₄×C₆₇	M₁₆	1	4	yes	complemented
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₆₇	—	4	8	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	1	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	—	2	1	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	4	2	no	—
154	C₂²⋊C₄×C₆₇	—	2	4	no	—
155	C₂²⋊C₄×C₆₇	—	2	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	2	no	—
157	C₂²⋊C₄×C₆₇	—	4	2	no	—
158	C₂²⋊C₄×C₆₇	—	4	2	no	—
159	C₄²×C₆₇	—	2	2	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	—	2	1	no	—
162	C₄²×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	2	no	—
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	C₂×D₄	1	2	yes	—
166	C₄²×C₆₇	#41	1	2	yes	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	1	no	—
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
181	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
182	D₄×C₆₇	—	4	8	no	—
183	D₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	4	4	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	1	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
217	C₂×C₄×C₆₇	#131	1	2	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
223	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	2	2	no	—
226	C₈×C₆₇	—	4	1	no	—
227	C₈×C₆₇	—	4	2	no	—
228	C₈×C₆₇	—	4	1	no	—
229	C₈×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	4	1	no	—
234	C₂²×C₆₇	—	4	1	no	—
235	C₂²×C₆₇	—	4	1	no	—
236	C₂²×C₆₇	—	4	1	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₆₇	#518	1	1	yes	characteristic
246	C₂²×C₆₇	#520	1	1	yes	characteristic
247	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
248	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
249	C₂²×C₆₇	#568	1	1	yes	characteristic
250	C₂²×C₆₇	#568	1	1	yes	characteristic
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	2	2	no	—
253	C₄×C₆₇	—	4	1	no	—
254	C₄×C₆₇	—	4	2	no	—
255	C₄×C₆₇	—	4	1	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	#435	1	4	yes	—
259	C₄×C₆₇	#535	1	1	yes	characteristic
260	C₄×C₆₇	#537	1	1	yes	characteristic
261	C₄×C₆₇	#540	1	1	yes	characteristic
262	C₄×C₆₇	#542	1	1	yes	characteristic
263	#22139	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂×C₆₇	—	4	2	no	—
265	C₂×C₆₇	—	8	2	no	—
266	C₂×C₆₇	#2324	1	1	yes	characteristic
267	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
268	C₂×C₆₇	#2930	1	1	yes	characteristic
269	C₂×C₆₇	#2975	1	1	yes	characteristic
270	C₂×C₆₇	#2981	1	1	yes	characteristic
271	C₂×C₆₇	#2989	1	1	yes	characteristic
272	C₂×C₆₇	#2993	1	1	yes	characteristic
273	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
274	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₄×#121	C₂×C₆₇	1	1	yes	characteristic
276	#1955	C₂×C₆₇	1	2	yes	complemented
277	#1954	C₂×C₆₇	1	1	yes	characteristic, complemented
278	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#1924	C₂×C₆₇	1	2	yes	complemented
280	#1870	C₂×C₆₇	1	2	yes	complemented
281	#1851	C₂×C₆₇	1	2	yes	complemented
282	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₆₇	#22139	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
286	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
289	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
290	#436	—	2	4	no	—
291	#431	—	2	4	no	—
292	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
293	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
297	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
300	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
301	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
302	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
303	C₂×#97	C₂²×C₆₇	1	2	yes	—
304	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
306	#390	C₄×C₆₇	1	1	yes	characteristic
307	#389	C₄×C₆₇	1	1	yes	characteristic
308	#388	C₄×C₆₇	1	2	yes	—
309	#386	C₄×C₆₇	1	2	yes	—
310	#380	C₄×C₆₇	1	1	yes	characteristic
311	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
312	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
313	#336	C₂²×C₆₇	1	2	yes	—
314	#336	C₂²×C₆₇	1	2	yes	—
315	#336	C₂²×C₆₇	1	2	yes	—
316	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
318	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
319	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
321	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	—	4	2	no	—
324	C₂×M₁₆	—	4	2	no	—
325	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
326	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
327	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
328	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	#121	—	2	4	no	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	—	4	2	no	—
335	C₄×D₄	—	4	2	no	—
336	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
340	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
341	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
342	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
343	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
344	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	—	2	2	no	—
347	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
350	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
352	C₄⋊C₈	—	2	2	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	—	2	2	no	—
356	C₄⋊C₈	—	4	2	no	—
357	C₂²⋊C₈	—	2	2	no	—
358	C₂²⋊C₈	—	2	2	no	—
359	C₂²⋊C₈	—	2	4	no	—
360	C₂²⋊C₈	—	4	2	no	—
361	C₂²⋊C₈	—	4	2	no	—
362	C₂²⋊C₈	—	4	2	no	—
363	#97	—	4	2	no	—
364	C₄×C₈	—	2	2	no	—
365	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
366	#95	C₂×C₄×C₆₇	1	2	yes	—
367	#95	C₂×C₄×C₆₇	1	2	yes	—
368	#95	C₂×C₄×C₆₇	1	2	yes	—
369	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	4	2	no	—
376	C₂×D₄	—	4	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	4	1	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	M₁₆×C₆₇	1	4	yes	complemented
395	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
397	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
399	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
404	M₁₆	—	4	8	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
422	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	4	2	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₂²⋊C₄	—	4	2	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	C₂×D₄×C₆₇	1	2	yes	—
439	C₄²	#41×C₆₇	1	2	yes	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
454	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
455	D₄	—	4	8	no	—
456	D₄	—	4	4	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
487	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
488	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
489	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
490	C₂×C₄	#131×C₆₇	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
493	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
494	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
495	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
496	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	2	2	no	—
499	C₈	—	4	2	no	—
500	C₈	—	4	1	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
518	C₂²	#518×C₆₇	1	1	yes	characteristic
519	C₂²	#520×C₆₇	1	1	yes	characteristic
520	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
521	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
522	C₂²	#568×C₆₇	1	1	yes	characteristic
523	C₂²	#568×C₆₇	1	1	yes	characteristic
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	#435×C₆₇	1	4	yes	—
532	C₄	#535×C₆₇	1	1	yes	characteristic
533	C₄	#537×C₆₇	1	1	yes	characteristic
534	C₄	#540×C₆₇	1	1	yes	characteristic
535	C₄	#542×C₆₇	1	1	yes	characteristic
536	C₂	—	4	2	no	—
537	C₂	—	8	2	no	—
538	C₂	C₆₇×#2324	1	1	yes	characteristic
539	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
540	C₂	C₆₇×#2930	1	1	yes	characteristic
541	C₂	C₆₇×#2975	1	1	yes	characteristic
542	C₂	C₆₇×#2981	1	1	yes	characteristic
543	C₂	C₆₇×#2989	1	1	yes	characteristic
544	C₂	C₆₇×#2993	1	1	yes	characteristic

	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×#121×C₆₇	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
1. C₂×#436×C₆₇	C₂	semidirect
2. C₂×#431×C₆₇	C₂	semidirect
4. C₆₇×#1955	C₂	semidirect
5. C₆₇×#1954	C₂	semidirect
6. C₆₇×#1939	C₂	semidirect
7. C₆₇×#1924	C₂	semidirect
8. C₆₇×#1870	C₂	semidirect
9. C₆₇×#1851	C₂	semidirect
10. C₆₇×#1797	C₂	semidirect
11. C₆₇×#1770	C₂	semidirect
92. C₄×C₈×C₆₇	D₄	semidirect
121. C₂²×C₄×C₆₇	M₁₆	semidirect
263. #22139	C₆₇	direct (N × H)
273. C₂×#436	C₂×C₆₇	semidirect
274. C₂×#431	C₂×C₆₇	semidirect
276. #1955	C₂×C₆₇	semidirect
277. #1954	C₂×C₆₇	semidirect
278. #1939	C₂×C₆₇	semidirect
279. #1924	C₂×C₆₇	semidirect
280. #1870	C₂×C₆₇	semidirect
281. #1851	C₂×C₆₇	semidirect
282. #1797	C₂×C₆₇	semidirect
283. #1770	C₂×C₆₇	semidirect
284. C₆₇	#22139	direct (N × H)
365. C₄×C₈	D₄×C₆₇	semidirect
394. C₂²×C₄	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₆₇	2
C₂×#431×C₆₇	2
C₄×#121×C₆₇	2
C₆₇×#1955	2
C₆₇×#1954	2
C₆₇×#1939	2
C₆₇×#1924	2
C₆₇×#1870	2
C₆₇×#1851	2
C₆₇×#1797	2
C₆₇×#1770	2
#22139	67

Subgroups — C67×#22139 #30701390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22139 #30701390