Subgroups — C₆₇×#20446 #30699697

Order 17152.

1010 conjugacy classes of subgroups.

The subgroup lattice has 348 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	1	yes	maximal, characteristic
2	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2873	C₂	1	2	yes	maximal
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1773	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1772	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1733	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	2	yes	—
12	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₆₇	C₄	1	8	yes	complemented
14	C₂×#108×C₆₇	C₂²	1	2	yes	—
15	C₂×#108×C₆₇	C₂²	1	2	yes	—
16	#427×C₆₇	C₄	1	2	yes	complemented
17	#427×C₆₇	C₄	1	2	yes	complemented
18	#427×C₆₇	C₂²	1	2	yes	—
19	#427×C₆₇	C₂²	1	2	yes	—
20	C₂×#107×C₆₇	C₄	1	2	yes	complemented
21	C₂×#106×C₆₇	C₄	1	2	yes	complemented
22	#405×C₆₇	—	2	4	no	—
23	#405×C₆₇	C₄	1	8	yes	complemented
24	#405×C₆₇	C₂²	1	8	yes	—
25	C₄×M₁₆×C₆₇	—	2	4	no	—
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	#339×C₆₇	C₂²	1	8	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	2	4	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
33	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
35	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
36	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	#117×C₆₇	—	2	2	no	—
49	#117×C₆₇	—	2	2	no	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	C₂×C₄	1	4	yes	—
52	#117×C₆₇	C₂×C₄	1	4	yes	—
53	#117×C₆₇	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	#108×C₆₇	C₂×C₄	1	8	yes	complemented
61	#108×C₆₇	C₂×C₄	1	8	yes	complemented
62	#107×C₆₇	C₂×C₄	1	4	yes	—
63	#107×C₆₇	C₂×C₄	1	4	yes	—
64	#106×C₆₇	C₂×C₄	1	4	yes	—
65	#106×C₆₇	C₂×C₄	1	4	yes	—
66	#97×C₆₇	—	2	8	no	—
67	#97×C₆₇	C₂×C₄	1	8	yes	complemented
68	C₄×C₈×C₆₇	—	2	8	no	—
69	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₂⁴×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₆₇	—	2	8	no	—
84	M₁₆×C₆₇	—	2	4	no	—
85	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
86	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
87	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
88	C₂×C₈×C₆₇	—	2	8	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	C₄²	1	4	yes	—
91	C₂×C₈×C₆₇	C₄²	1	4	yes	—
92	C₂×C₈×C₆₇	C₄²	1	4	yes	—
93	C₂×C₈×C₆₇	C₄²	1	4	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
104	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
108	C₂²⋊C₄×C₆₇	—	2	4	no	—
109	C₂²⋊C₄×C₆₇	—	2	4	no	—
110	C₂²⋊C₄×C₆₇	—	2	4	no	—
111	C₄²×C₆₇	—	2	4	no	—
112	C₄²×C₆₇	—	2	4	no	—
113	C₄²×C₆₇	—	2	2	no	—
114	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	#117	1	2	yes	—
117	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
120	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	8	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	8	4	no	—
125	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₇	#117	1	2	yes	—
130	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₈×C₆₇	—	2	8	no	—
144	C₈×C₆₇	C₂×C₄²	1	2	yes	—
145	C₈×C₆₇	C₂×C₄²	1	2	yes	—
146	C₈×C₆₇	C₂×C₄²	1	2	yes	—
147	C₈×C₆₇	C₂×C₄²	1	2	yes	—
148	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×C₆₇	—	4	2	no	—
154	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
155	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
156	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₄×C₆₇	—	4	4	no	—
160	C₄×C₆₇	—	4	8	no	—
161	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
164	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	#20446	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	2	1	no	—
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	—	2	2	no	—
172	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₆₇	#2957	1	2	yes	—
174	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
175	C₂×#427	C₂×C₆₇	1	1	yes	characteristic
176	#2873	C₂×C₆₇	1	2	yes	—
177	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
178	#1773	C₂×C₆₇	1	4	yes	complemented
179	#1772	C₂×C₆₇	1	4	yes	complemented
180	#1733	C₂×C₆₇	1	2	yes	—
181	C₆₇	#20446	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
183	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
184	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×#117	C₂²×C₆₇	1	2	yes	—
186	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
187	C₂×#108	C₄×C₆₇	1	8	yes	complemented
188	C₂×#108	C₂²×C₆₇	1	2	yes	—
189	C₂×#108	C₂²×C₆₇	1	2	yes	—
190	#427	C₄×C₆₇	1	2	yes	complemented
191	#427	C₄×C₆₇	1	2	yes	complemented
192	#427	C₂²×C₆₇	1	2	yes	—
193	#427	C₂²×C₆₇	1	2	yes	—
194	C₂×#107	C₄×C₆₇	1	2	yes	complemented
195	C₂×#106	C₄×C₆₇	1	2	yes	complemented
196	#405	—	2	4	no	—
197	#405	C₄×C₆₇	1	8	yes	complemented
198	#405	C₂²×C₆₇	1	8	yes	—
199	C₄×M₁₆	—	2	4	no	—
200	C₂×#97	C₂²×C₆₇	1	2	yes	—
201	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
202	#339	C₂²×C₆₇	1	8	yes	—
203	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
207	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
208	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
209	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
210	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
211	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
214	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
219	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
222	#117	—	2	2	no	—
223	#117	—	2	2	no	—
224	#117	—	2	2	no	—
225	#117	C₂×C₄×C₆₇	1	4	yes	—
226	#117	C₂×C₄×C₆₇	1	4	yes	—
227	#117	C₂³×C₆₇	1	4	yes	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
230	C₂×C₂²⋊C₄	—	2	1	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
234	#108	C₂×C₄×C₆₇	1	8	yes	complemented
235	#108	C₂×C₄×C₆₇	1	8	yes	complemented
236	#107	C₂×C₄×C₆₇	1	4	yes	—
237	#107	C₂×C₄×C₆₇	1	4	yes	—
238	#106	C₂×C₄×C₆₇	1	4	yes	—
239	#106	C₂×C₄×C₆₇	1	4	yes	—
240	#97	—	2	8	no	—
241	#97	C₂×C₄×C₆₇	1	8	yes	complemented
242	C₄×C₈	—	2	8	no	—
243	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
244	C₂⁴	#41×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
254	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	#41×C₆₇	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
257	M₁₆	—	2	8	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	C₄⋊C₄×C₆₇	1	8	yes	—
260	M₁₆	C₄⋊C₄×C₆₇	1	8	yes	—
261	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
262	C₂×C₈	—	2	8	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₄²×C₆₇	1	4	yes	—
265	C₂×C₈	C₄²×C₆₇	1	4	yes	—
266	C₂×C₈	C₄²×C₆₇	1	4	yes	—
267	C₂×C₈	C₄²×C₆₇	1	4	yes	—
268	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
269	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
270	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
273	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
277	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
278	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
289	C₂³	—	4	2	no	—
290	C₂³	#117×C₆₇	1	2	yes	—
291	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
292	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
293	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
294	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
300	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
303	C₂×C₄	#117×C₆₇	1	2	yes	—
304	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
305	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
306	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
309	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
310	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
311	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
317	C₈	—	2	8	no	—
318	C₈	C₂×C₄²×C₆₇	1	2	yes	—
319	C₈	C₂×C₄²×C₆₇	1	2	yes	—
320	C₈	C₂×C₄²×C₆₇	1	2	yes	—
321	C₈	C₂×C₄²×C₆₇	1	2	yes	—
322	C₈	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
329	C₂²	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
330	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
331	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
332	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
333	C₄	—	4	4	no	—
334	C₄	—	4	8	no	—
335	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
336	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
337	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
338	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
339	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
340	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	C₂×C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
346	C₂	C₆₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂⁴×C₆₇	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1773	C₂	semidirect
6. C₆₇×#1772	C₂	semidirect
13. C₂×#108×C₆₇	C₄	semidirect
16. #427×C₆₇	C₄	semidirect
17. #427×C₆₇	C₄	semidirect
20. C₂×#107×C₆₇	C₄	semidirect
21. C₂×#106×C₆₇	C₄	semidirect
23. #405×C₆₇	C₄	semidirect
60. #108×C₆₇	C₂×C₄	semidirect
61. #108×C₆₇	C₂×C₄	semidirect
67. #97×C₆₇	C₂×C₄	semidirect
69. C₄×C₈×C₆₇	C₂×C₄	semidirect
167. #20446	C₆₇	direct (N × H)
178. #1773	C₂×C₆₇	semidirect
179. #1772	C₂×C₆₇	semidirect
181. C₆₇	#20446	direct (N × H)
187. C₂×#108	C₄×C₆₇	semidirect
190. #427	C₄×C₆₇	semidirect
191. #427	C₄×C₆₇	semidirect
194. C₂×#107	C₄×C₆₇	semidirect
195. C₂×#106	C₄×C₆₇	semidirect
197. #405	C₄×C₆₇	semidirect
234. #108	C₂×C₄×C₆₇	semidirect
235. #108	C₂×C₄×C₆₇	semidirect
241. #97	C₂×C₄×C₆₇	semidirect
243. C₄×C₈	C₂×C₄×C₆₇	semidirect

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×#427×C₆₇	2
C₆₇×#2873	2
C₂×#405×C₆₇	2
C₆₇×#1773	2
C₆₇×#1772	2
C₆₇×#1733	2
#20446	67

Subgroups — C67×#20446 #30699697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20446 #30699697