Subgroups — C₆₇×#22104 #30701355

Order 17152.

794 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1940	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1924	C₂	1	2	yes	maximal
8	C₆₇×#1924	C₂	1	2	yes	maximal
9	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	—	2	4	no	—
16	#431×C₆₇	—	2	4	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
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	2	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	C₂×#97×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#391×C₆₇	C₄	1	1	yes	characteristic
31	#389×C₆₇	C₄	1	1	yes	characteristic
32	#387×C₆₇	C₄	1	2	yes	—
33	#385×C₆₇	C₄	1	2	yes	—
34	#380×C₆₇	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	#336×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×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²⋊Q₈×C₆₇	—	2	4	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	—	2	2	no	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	—	2	4	no	—
75	C₄⋊C₈×C₆₇	—	2	2	no	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	4	2	no	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	C₂²⋊C₈×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	—	4	2	no	—
81	#97×C₆₇	—	2	4	no	—
82	#97×C₆₇	—	4	2	no	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	C₄×C₈×C₆₇	D₄	1	4	yes	—
85	#95×C₆₇	C₂×C₄	1	2	yes	—
86	#95×C₆₇	C₂×C₄	1	2	yes	—
87	#95×C₆₇	C₂×C₄	1	2	yes	—
88	#95×C₆₇	C₂×C₄	1	2	yes	—
89	#95×C₆₇	C₂×C₄	1	2	yes	—
90	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	4	1	no	—
105	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	4	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	4	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	1	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	C₂×D₄	1	2	yes	—
151	C₄²×C₆₇	#41	1	2	yes	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	1	no	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	—	2	1	no	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
159	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	Q₈×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	4	4	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₆₇	#131	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
200	C₈×C₆₇	—	2	2	no	—
201	C₈×C₆₇	—	2	2	no	—
202	C₈×C₆₇	—	4	1	no	—
203	C₈×C₆₇	—	4	2	no	—
204	C₈×C₆₇	—	4	1	no	—
205	C₈×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	—	4	1	no	—
213	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂²×C₆₇	#519	1	1	yes	characteristic
215	C₂²×C₆₇	#520	1	1	yes	characteristic
216	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
218	C₂²×C₆₇	#568	1	1	yes	characteristic
219	C₂²×C₆₇	#568	1	1	yes	characteristic
220	C₄×C₆₇	—	2	2	no	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	4	1	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	4	2	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	—	4	2	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	#435	1	4	yes	—
229	C₄×C₆₇	#536	1	1	yes	characteristic
230	C₄×C₆₇	#537	1	1	yes	characteristic
231	C₄×C₆₇	#541	1	1	yes	characteristic
232	C₄×C₆₇	#542	1	1	yes	characteristic
233	#22104	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂×C₆₇	—	4	2	no	—
235	C₂×C₆₇	#2323	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
237	C₂×C₆₇	#2930	1	1	yes	characteristic
238	C₂×C₆₇	#2983	1	1	yes	characteristic
239	C₂×C₆₇	#2985	1	1	yes	characteristic
240	C₂×C₆₇	#2989	1	1	yes	characteristic
241	C₂×C₆₇	#2993	1	1	yes	characteristic
242	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
243	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
244	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
245	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#1940	C₂×C₆₇	1	2	yes	complemented
247	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#1924	C₂×C₆₇	1	2	yes	—
249	#1924	C₂×C₆₇	1	2	yes	—
250	#1917	C₂×C₆₇	1	2	yes	complemented
251	#1797	C₂×C₆₇	1	1	yes	characteristic
252	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	#22104	1	1	yes	characteristic, direct product factor, Sylow
254	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
255	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
256	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
257	#446	—	2	4	no	—
258	#431	—	2	4	no	—
259	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
265	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
267	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
268	C₂×#97	C₂²×C₆₇	1	2	yes	—
269	C₂×#97	C₂²×C₆₇	1	2	yes	—
270	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
272	#391	C₄×C₆₇	1	1	yes	characteristic
273	#389	C₄×C₆₇	1	1	yes	characteristic
274	#387	C₄×C₆₇	1	2	yes	—
275	#385	C₄×C₆₇	1	2	yes	—
276	#380	C₄×C₆₇	1	1	yes	characteristic
277	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
279	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
280	#336	C₂²×C₆₇	1	2	yes	—
281	#336	C₂²×C₆₇	1	2	yes	—
282	#336	C₂²×C₆₇	1	2	yes	—
283	#336	C₂²×C₆₇	1	2	yes	—
284	#336	C₂²×C₆₇	1	2	yes	—
285	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
291	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
293	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₂²⋊Q₈	—	2	4	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
300	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
309	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
311	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	C₄⋊C₈	—	2	2	no	—
315	C₄⋊C₈	—	2	2	no	—
316	C₄⋊C₈	—	2	4	no	—
317	C₄⋊C₈	—	2	2	no	—
318	C₄⋊C₈	—	2	2	no	—
319	C₄⋊C₈	—	4	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₂²⋊C₈	—	4	2	no	—
323	#97	—	2	4	no	—
324	#97	—	4	2	no	—
325	C₄×C₈	—	2	2	no	—
326	C₄×C₈	D₄×C₆₇	1	4	yes	—
327	#95	C₂×C₄×C₆₇	1	2	yes	—
328	#95	C₂×C₄×C₆₇	1	2	yes	—
329	#95	C₂×C₄×C₆₇	1	2	yes	—
330	#95	C₂×C₄×C₆₇	1	2	yes	—
331	#95	C₂×C₄×C₆₇	1	2	yes	—
332	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	—	4	1	no	—
347	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
348	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	4	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	2	4	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	C₂×D₄×C₆₇	1	2	yes	—
393	C₄²	#41×C₆₇	1	2	yes	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	1	no	—
400	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
401	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
402	Q₈	—	4	4	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	8	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
432	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
433	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
434	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
435	C₂×C₄	#131×C₆₇	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
438	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
439	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
440	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
441	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	2	2	no	—
444	C₈	—	4	1	no	—
445	C₈	—	4	1	no	—
446	C₈	—	4	2	no	—
447	C₈	—	4	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	4	1	no	—
455	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
456	C₂²	#519×C₆₇	1	1	yes	characteristic
457	C₂²	#520×C₆₇	1	1	yes	characteristic
458	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
459	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
460	C₂²	#568×C₆₇	1	1	yes	characteristic
461	C₂²	#568×C₆₇	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	#435×C₆₇	1	4	yes	—
471	C₄	#536×C₆₇	1	1	yes	characteristic
472	C₄	#537×C₆₇	1	1	yes	characteristic
473	C₄	#541×C₆₇	1	1	yes	characteristic
474	C₄	#542×C₆₇	1	1	yes	characteristic
475	C₂	—	4	2	no	—
476	C₂	C₆₇×#2323	1	1	yes	characteristic
477	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
478	C₂	C₆₇×#2930	1	1	yes	characteristic
479	C₂	C₆₇×#2983	1	1	yes	characteristic
480	C₂	C₆₇×#2985	1	1	yes	characteristic
481	C₂	C₆₇×#2989	1	1	yes	characteristic
482	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	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
1. C₂×#446×C₆₇	C₂	semidirect
4. C₆₇×#1984	C₂	semidirect
5. C₆₇×#1940	C₂	semidirect
6. C₆₇×#1939	C₂	semidirect
9. C₆₇×#1917	C₂	semidirect
11. C₆₇×#1770	C₂	semidirect
233. #22104	C₆₇	direct (N × H)
242. C₂×#446	C₂×C₆₇	semidirect
245. #1984	C₂×C₆₇	semidirect
246. #1940	C₂×C₆₇	semidirect
247. #1939	C₂×C₆₇	semidirect
250. #1917	C₂×C₆₇	semidirect
252. #1770	C₂×C₆₇	semidirect
253. C₆₇	#22104	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×#446×C₆₇	2
C₂×#431×C₆₇	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#1984	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1924	2
C₆₇×#1917	2
C₆₇×#1797	2
C₆₇×#1770	2
#22104	67

Subgroups — C67×#22104 #30701355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22104 #30701355