Subgroups — C₆₇×#21178 #30700429

Order 17152.

866 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic
4	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1950	C₂	1	8	yes	maximal, complemented
7	C₆₇×#1738	C₂	1	1	yes	maximal, characteristic
8	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	#430×C₆₇	—	2	4	no	—
17	C₂×#108×C₆₇	—	2	4	no	—
18	C₂×#108×C₆₇	C₂²	1	4	yes	—
19	C₂×#108×C₆₇	C₂²	1	4	yes	—
20	#424×C₆₇	—	2	2	no	—
21	#424×C₆₇	—	2	2	no	—
22	#424×C₆₇	C₄	1	4	yes	—
23	#424×C₆₇	C₄	1	4	yes	—
24	#423×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
26	#405×C₆₇	—	2	2	no	—
27	#405×C₆₇	—	2	2	no	—
28	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
29	#341×C₆₇	C₂²	1	4	yes	—
30	#341×C₆₇	C₂²	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₆₇	—	2	1	no	—
33	C₂×M₁₆×C₆₇	—	2	1	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	1	no	—
39	C₂×M₁₆×C₆₇	—	2	1	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
47	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
48	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
49	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	#117×C₆₇	C₂×C₄	1	2	yes	—
63	#117×C₆₇	C₂×C₄	1	2	yes	—
64	#117×C₆₇	D₄	1	2	yes	—
65	#117×C₆₇	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	#108×C₆₇	—	2	8	no	—
70	#108×C₆₇	—	2	8	no	—
71	C₄⋊C₈×C₆₇	—	2	8	no	—
72	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	—
73	#97×C₆₇	—	2	4	no	—
74	C₄×C₈×C₆₇	—	2	4	no	—
75	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	M₁₆×C₆₇	—	2	4	no	—
90	M₁₆×C₆₇	—	2	4	no	—
91	M₁₆×C₆₇	—	2	2	no	—
92	M₁₆×C₆₇	—	2	2	no	—
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	—	2	2	no	—
95	M₁₆×C₆₇	—	2	2	no	—
96	M₁₆×C₆₇	—	2	2	no	—
97	M₁₆×C₆₇	—	4	2	no	—
98	M₁₆×C₆₇	—	4	2	no	—
99	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
100	M₁₆×C₆₇	C₂×Q₈	1	4	yes	—
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	4	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	4	2	no	—
114	C₂×C₈×C₆₇	—	4	2	no	—
115	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
117	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
118	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
119	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
120	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
121	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
122	C₄²×C₆₇	C₂×D₄	1	4	yes	—
123	C₂³×C₆₇	—	4	2	no	—
124	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₆₇	#125	1	2	yes	—
127	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
128	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₄×D₄	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	#121	1	2	yes	—
138	C₂×C₄×C₆₇	#121	1	2	yes	—
139	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₆₇	#125	1	2	yes	—
142	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₆₇	—	2	4	no	—
150	C₈×C₆₇	—	2	4	no	—
151	C₈×C₆₇	—	2	4	no	—
152	C₈×C₆₇	—	2	4	no	—
153	C₈×C₆₇	—	4	2	no	—
154	C₈×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	—	2	2	no	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	4	2	no	—
159	C₂²×C₆₇	—	4	2	no	—
160	C₂²×C₆₇	#389	1	2	yes	—
161	C₂²×C₆₇	#389	1	2	yes	—
162	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₄×C₆₇	—	2	8	no	—
166	C₄×C₆₇	#389	1	2	yes	—
167	C₄×C₆₇	#389	1	2	yes	—
168	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
171	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	#21178	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	—	2	1	no	—
174	C₂×C₆₇	—	2	1	no	—
175	C₂×C₆₇	—	2	2	no	—
176	C₂×C₆₇	—	2	2	no	—
177	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
178	C₂×C₆₇	#2916	1	2	yes	—
179	C₂×#430	C₂×C₆₇	1	1	yes	characteristic
180	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
181	#2905	C₂×C₆₇	1	1	yes	characteristic
182	C₂×#424	C₂×C₆₇	1	1	yes	characteristic
183	#2873	C₂×C₆₇	1	1	yes	characteristic
184	#1950	C₂×C₆₇	1	8	yes	complemented
185	#1738	C₂×C₆₇	1	1	yes	characteristic
186	C₂×#341	C₂×C₆₇	1	1	yes	characteristic
187	C₆₇	#21178	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
189	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
190	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
191	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
192	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
193	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
195	#430	—	2	4	no	—
196	C₂×#108	—	2	4	no	—
197	C₂×#108	C₂²×C₆₇	1	4	yes	—
198	C₂×#108	C₂²×C₆₇	1	4	yes	—
199	#424	—	2	2	no	—
200	#424	—	2	2	no	—
201	#424	C₄×C₆₇	1	4	yes	—
202	#424	C₄×C₆₇	1	4	yes	—
203	#423	C₂²×C₆₇	1	4	yes	—
204	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
205	#405	—	2	2	no	—
206	#405	—	2	2	no	—
207	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
208	#341	C₂²×C₆₇	1	4	yes	—
209	#341	C₂²×C₆₇	1	4	yes	—
210	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂×M₁₆	—	2	2	no	—
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
224	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
225	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
226	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
227	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
228	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
229	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
230	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
238	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
239	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
240	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
241	#117	C₂×C₄×C₆₇	1	2	yes	—
242	#117	C₂×C₄×C₆₇	1	2	yes	—
243	#117	D₄×C₆₇	1	2	yes	—
244	#117	D₄×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	Q₈×C₆₇	1	2	yes	—
246	C₂×C₂²⋊C₄	Q₈×C₆₇	1	2	yes	—
247	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
248	#108	—	2	8	no	—
249	#108	—	2	8	no	—
250	C₄⋊C₈	—	2	8	no	—
251	C₄⋊C₈	C₂×C₄×C₆₇	1	16	yes	—
252	#97	—	2	4	no	—
253	C₄×C₈	—	2	4	no	—
254	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
260	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
261	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
263	C₂²×C₄	#41×C₆₇	1	2	yes	—
264	C₂²×C₄	#41×C₆₇	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
266	M₁₆	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	—	2	4	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	4	2	no	—
277	M₁₆	—	4	2	no	—
278	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
279	M₁₆	C₂×Q₈×C₆₇	1	4	yes	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
295	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
296	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
297	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
298	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
299	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
300	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
301	C₄²	C₂×D₄×C₆₇	1	4	yes	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
304	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
305	C₂³	#125×C₆₇	1	2	yes	—
306	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
307	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
310	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
313	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
314	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
315	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
316	C₂×C₄	#121×C₆₇	1	2	yes	—
317	C₂×C₄	#121×C₆₇	1	2	yes	—
318	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
319	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
320	C₂×C₄	#125×C₆₇	1	2	yes	—
321	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
322	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
325	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
326	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	—	2	4	no	—
331	C₈	—	2	4	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	#389×C₆₇	1	2	yes	—
340	C₂²	#389×C₆₇	1	2	yes	—
341	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
342	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
343	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
344	C₄	—	2	8	no	—
345	C₄	#389×C₆₇	1	2	yes	—
346	C₄	#389×C₆₇	1	2	yes	—
347	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
348	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
349	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
350	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	—	2	2	no	—
355	C₂	C₂×#389×C₆₇	1	1	yes	characteristic
356	C₂	C₆₇×#2916	1	2	yes	—

	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 ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₆₇×#1950	C₂	semidirect
172. #21178	C₆₇	direct (N × H)
184. #1950	C₂×C₆₇	semidirect
187. C₆₇	#21178	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×#430×C₆₇	2
C₂²×#108×C₆₇	2
C₆₇×#2905	2
C₂×#424×C₆₇	2
C₆₇×#2873	2
C₆₇×#1950	2
C₆₇×#1738	2
C₂×#341×C₆₇	2
#21178	67

Subgroups — C67×#21178 #30700429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21178 #30700429