Subgroups — C₆₇×#21994 #30701245

Order 17152.

704 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1950	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1855	C₂	1	4	yes	maximal
6	C₆₇×#1849	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1838	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#432×C₆₇	—	2	1	no	—
11	#432×C₆₇	—	2	1	no	—
12	#432×C₆₇	C₄	1	2	yes	—
13	#432×C₆₇	C₄	1	2	yes	—
14	#432×C₆₇	C₄	1	2	yes	—
15	#432×C₆₇	C₄	1	2	yes	—
16	#431×C₆₇	—	2	1	no	—
17	#431×C₆₇	—	2	1	no	—
18	C₂×#108×C₆₇	C₂²	1	8	yes	—
19	#423×C₆₇	—	2	1	no	—
20	#423×C₆₇	—	2	1	no	—
21	#423×C₆₇	C₂²	1	1	yes	characteristic
22	#423×C₆₇	C₂²	1	2	yes	—
23	#423×C₆₇	C₂²	1	1	yes	characteristic
24	#423×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
26	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
28	#407×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
31	C₂×#97×C₆₇	C₂²	1	2	yes	—
32	#341×C₆₇	C₂²	1	8	yes	—
33	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	4	4	no	—
38	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
39	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
40	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	—	4	8	no	—
56	C₄⋊C₈×C₆₇	—	2	2	no	—
57	C₄⋊C₈×C₆₇	—	2	2	no	—
58	C₄⋊C₈×C₆₇	—	2	2	no	—
59	C₄⋊C₈×C₆₇	—	2	2	no	—
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
65	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
66	C₂²⋊C₈×C₆₇	—	2	2	no	—
67	C₂²⋊C₈×C₆₇	—	2	2	no	—
68	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
70	#97×C₆₇	—	2	2	no	—
71	#97×C₆₇	C₂×C₄	1	4	yes	—
72	#97×C₆₇	C₂×C₄	1	4	yes	—
73	C₄×C₈×C₆₇	—	2	2	no	—
74	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	4	2	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₆₇	—	2	4	no	—
98	M₁₆×C₆₇	—	2	8	no	—
99	M₁₆×C₆₇	—	2	4	no	—
100	M₁₆×C₆₇	—	4	4	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	4	4	no	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
108	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	4	1	no	—
127	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
162	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
163	C₈×C₆₇	—	2	8	no	—
164	C₈×C₆₇	—	4	4	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₈×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	2	1	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	4	4	no	—
175	C₂²×C₆₇	#430	1	2	yes	—
176	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₆₇	#544	1	2	yes	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	#430	1	2	yes	—
188	C₄×C₆₇	#520	1	2	yes	—
189	C₄×C₆₇	#535	1	1	yes	characteristic
190	C₄×C₆₇	#536	1	1	yes	characteristic
191	#21994	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂×C₆₇	—	2	2	no	—
193	C₂×C₆₇	—	2	2	no	—
194	C₂×C₆₇	—	4	1	no	—
195	C₂×C₆₇	#2317	1	1	yes	characteristic
196	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
198	#2920	C₂×C₆₇	1	1	yes	characteristic
199	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
201	#1950	C₂×C₆₇	1	4	yes	complemented
202	#1855	C₂×C₆₇	1	4	yes	—
203	#1849	C₂×C₆₇	1	2	yes	complemented
204	#1838	C₂×C₆₇	1	2	yes	complemented
205	C₆₇	#21994	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
207	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
208	#432	—	2	1	no	—
209	#432	—	2	1	no	—
210	#432	C₄×C₆₇	1	2	yes	—
211	#432	C₄×C₆₇	1	2	yes	—
212	#432	C₄×C₆₇	1	2	yes	—
213	#432	C₄×C₆₇	1	2	yes	—
214	#431	—	2	1	no	—
215	#431	—	2	1	no	—
216	C₂×#108	C₂²×C₆₇	1	8	yes	—
217	#423	—	2	1	no	—
218	#423	—	2	1	no	—
219	#423	C₂²×C₆₇	1	1	yes	characteristic
220	#423	C₂²×C₆₇	1	2	yes	—
221	#423	C₂²×C₆₇	1	1	yes	characteristic
222	#423	C₂²×C₆₇	1	2	yes	—
223	C₂×C₄⋊C₈	—	2	1	no	—
224	C₂×C₄⋊C₈	—	2	1	no	—
225	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
226	#407	C₂²×C₆₇	1	2	yes	—
227	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
228	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
229	C₂×#97	C₂²×C₆₇	1	2	yes	—
230	#341	C₂²×C₆₇	1	8	yes	—
231	C₂³×C₄	Q₈×C₆₇	1	1	yes	characteristic
232	C₂³×C₄	Q₈×C₆₇	1	1	yes	characteristic
233	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	4	4	no	—
236	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
237	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
238	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
241	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
246	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
247	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄²	Q₈×C₆₇	1	1	yes	characteristic
250	C₂×C₄²	Q₈×C₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
253	#108	—	4	8	no	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
263	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
264	C₂²⋊C₈	—	2	2	no	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
267	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
268	#97	—	2	2	no	—
269	#97	C₂×C₄×C₆₇	1	4	yes	—
270	#97	C₂×C₄×C₆₇	1	4	yes	—
271	C₄×C₈	—	2	2	no	—
272	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
289	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
295	M₁₆	—	2	4	no	—
296	M₁₆	—	2	8	no	—
297	M₁₆	—	2	4	no	—
298	M₁₆	—	4	4	no	—
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	4	4	no	—
305	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
306	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
307	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
308	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
309	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
310	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
311	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
312	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
313	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
314	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
315	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
316	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
317	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	—	2	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
326	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
327	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
360	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
361	C₈	—	2	8	no	—
362	C₈	—	4	4	no	—
363	C₈	—	4	2	no	—
364	C₈	—	4	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	4	4	no	—
373	C₂²	#430×C₆₇	1	2	yes	—
374	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
375	C₂²	#544×C₆₇	1	2	yes	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	#430×C₆₇	1	2	yes	—
386	C₄	#520×C₆₇	1	2	yes	—
387	C₄	#535×C₆₇	1	1	yes	characteristic
388	C₄	#536×C₆₇	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	2	2	no	—
391	C₂	—	4	1	no	—
392	C₂	C₆₇×#2317	1	1	yes	characteristic
393	C₂	C₂×#430×C₆₇	1	1	yes	characteristic
394	C₂	C₂×#430×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1950	C₂	semidirect
6. C₆₇×#1849	C₂	semidirect
7. C₆₇×#1838	C₂	semidirect
191. #21994	C₆₇	direct (N × H)
201. #1950	C₂×C₆₇	semidirect
203. #1849	C₂×C₆₇	semidirect
204. #1838	C₂×C₆₇	semidirect
205. C₆₇	#21994	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×#2920	2
C₂×#432×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#1950	2
C₆₇×#1855	2
C₆₇×#1849	2
C₆₇×#1838	2
#21994	67

Subgroups — C67×#21994 #30701245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21994 #30701245